Summary

The Standard Model

$$Z = \int \mathcal{D}(\text{Fields}) \exp \left( i \int d^4x \sqrt{-g} \left( R - F_{\mu\nu} F^{\mu\nu} - G_{\mu\nu} G^{\mu\nu} - W_{\mu\nu} W^{\mu\nu} + \sum_i \bar{\psi}_i \mathcal{D} \psi_i + D_\mu H^\dagger D^\mu H - V(H) - \lambda_{ij} \psi_i H \psi_j \right) \right)$$

Introduction and Overview

In this lecture, theoretical physicist David Tong addresses one of the most fundamental questions in science—a question dating back over 2,500 years to the ancient Greeks: What is the universe made of?

While conventional education teaches that the universe is built from indivisible subatomic particles acting like microscopic LEGO bricks, modern theoretical physics reveals a fundamentally different reality. The underlying building blocks of nature are not discrete particles, but continuous, fluid-like entities spread throughout the entirety of space, known as fields. Particles are merely localized ripples or discrete bundles of energy within these ubiquitous fields.

Historical Evolution of the Atom

• Ancient Greek Philosophy: Early thinkers such as Democritus and Lucretius hypothesized that matter consists of tiny, indivisible units called atoms.

• The Periodic Table (19th Century): Chemistry organized all known matter into roughly 120 chemical elements. While a major milestone, it represented a complex and fragmented classification of nature rather than a simple fundamental framework.

• Discovery of the Electron (1897): J.J. Thomson discovered the electron at the Royal Institution, demonstrating that atoms are divisible.

• The Atomic Nucleus (Early 20th Century): Ernest Rutherford demonstrated that an atom consists of a tiny nucleus surrounded by distant, orbiting electrons ("a fly in the cathedral").

• Protons, Neutrons, and Quarks (1970s): Scientists realized the nucleus comprises protons and neutrons, which in turn are composed of even smaller entities called quarks:

• Up Quark (+2/3 charge)

• Down Quark (-1/3 charge)

• A proton consists of two up quarks and one down quark ($uud$).

• A neutron consists of two down quarks and one up quark ($udd$).

For decades, the standard scientific narrative held that three fundamental particles—the electron, the up quark, and the down quark—form all stable matter in the universe.

The Field Paradigm and Quantum Field Theory (QFT)

The traditional particle model is a conceptual simplification. Modern physics demonstrates that nature's basic constituents are smooth, universe-spanning fluid-like entities called fields, which take values at every point in space and evolve over time.

• Michael Faraday's Breakthrough (1820s–1840s): Investigating magnetism and electricity at the Royal Institution, Faraday deduced the presence of invisible "lines of force" filling space—the electric and magnetic fields. He famously proposed that light itself consists of ripples propagating through these electromagnetic fields.

• Quantum Mechanics Integration: In the 1920s, quantum mechanics (developed by Heisenberg, Schrödinger, and others) established that energy at microscopic scales is quantized into discrete lumps (quanta).

• Quantum Field Theory (QFT): Combining classical field theory with quantum mechanics yields QFT. In QFT, continuous fields become quantized:

• Ripples in the electromagnetic field manifest as discrete light particles called photons.

• Ripples in the universal electron field manifest as localized energy packets called electrons.

• Ripples in the quark fields manifest as up and down quarks.

Consequently, all electrons in the universe are not isolated individual objects; they are interconnected oscillations of the single, underlying electron field that permeates space.

The Nature of the Quantum Vacuum and Mathematical Complexity

• The Quantum Vacuum: If all particles and energy are removed from a container, empty space is not barren. Due to the Heisenberg Uncertainty Principle, quantum fields cannot remain static. The pure vacuum is a dynamic environment constantly roiling with quantum vacuum fluctuations.

• Experimental Proof: These vacuum fluctuations produce physically measurable phenomena, such as the Casimir force (an attractive pressure between two uncharged parallel metal plates in a vacuum).

• Mathematical Challenges: Quantum field theory involves immense mathematical complexity:

• Formally understanding QFT equations and their vacuum structure from first principles remains one of the six unsolved Millennium Prize Problems in mathematics (Yang–Mills and Mass Gap problem).

• Successes: QFT yields predictions of astonishing precision when fluctuations are calm. The magnetic moment ($g$-factor) of the electron matches theoretical calculations to 12–13 significant figures.

• Limitations: When fluctuations are strong, exact calculations become intractable. Computing the mass of a proton from first principles using supercomputers currently achieves only ~3% accuracy.

The Standard Model of Particle Physics

The fundamental architecture of reality is summarized by the Standard Model, which accounts for 17 fundamental fields divided into matter, forces, and the Higgs mechanism:

1. Matter Fields (12 Fermionic Fields)

Nature replicates the basic set of four matter particles across three distinct "generations" or families (for reasons unknown):

• First Generation (Stable Matter): Up Quark, Down Quark, Electron, Electron Neutrino.

• Second Generation (Heavier Copies): Charm Quark, Strange Quark, Muon, Muon Neutrino.

• Third Generation (Heaviest Copies): Top Quark, Bottom Quark, Tau, Tau Neutrino.

2. Force Fields (4 Bosonic Interactions)

• Electromagnetism: Governed by the photon / electromagnetic field.

• Strong Nuclear Force: Holds quarks together inside hadrons, mediated by the gluon field.

• Weak Nuclear Force: Drives radioactive decay and nuclear fusion in stars, mediated by W and Z bosons.

• Gravity: Encoded in general relativity as the curvature and dynamics of space and time itself.

3. The Higgs Field

• Proposed by Peter Higgs and others in the 1960s, the Higgs field permeates space and interacts with matter fields. A particle's mass is a measure of how strongly its underlying field interacts with the background Higgs field.

• Confirmed experimentally in 2012 at CERN's Large Hadron Collider (LHC) via the ATLAS and CMS detectors through the discovery of the Higgs boson.

The Master Equation

The Standard Model can be compactly expressed in a single Lagrangian equation combining General Relativity, Maxwellian Electromagnetism, Quantum Chromodynamics, the Dirac Equation for matter, and the Higgs Mechanism. It correctly predicts the outcome of every terrestrial particle experiment conducted to date.

Cosmological Connections and the Early Universe

Despite its success, the Standard Model cannot account for several cosmic phenomena:

• Dark Matter: Invisible matter providing gravitational mass to galaxies.

• Dark Energy: An unknown vacuum energy driving the accelerated expansion of the universe.

• Cosmic Inflation: A fraction of a second ($10^{-30}$ s) after the Big Bang, microscopic quantum vacuum fluctuations were exponentially stretched across space, becoming frozen as density variations. These ancient ripples created the temperature fluctuations visible in the Cosmic Microwave Background (CMB) radiation—the universe's early fireball 380,000 years after the Big Bang—and seeded the large-scale cosmic web of galaxies.

Frontiers of Physics and the LHC Dilemma

In 2015, the LHC resumed operations at higher energies (13 TeV) to look for physics beyond the Standard Model, such as:

• Grand Unified Theories (GUTs): Merging the electromagnetic, strong, and weak forces into a single interaction.

• Supersymmetry (SUSY): A proposed symmetry linking matter fields (fermions) and force fields (bosons).

• String Theory: A framework replacing point-like field excitations with vibrating one-dimensional strings to unify quantum mechanics and general relativity.

The Current Empirical Crisis and Future Perspectives

The upgraded LHC has detected no evidence of new physics beyond the Standard Model, challenging decades of theoretical predictions. Physicists are divided into three main perspectives regarding how to proceed:

• Patience: Expecting new particles to emerge in future high-luminosity LHC runs.

• Larger Colliders: Constructing a 100 km collider (e.g., proposed projects in China or at CERN) costing ~$10 billion to probe higher energy scales.

• Paradigm Shift: Re-evaluating foundational theoretical assumptions, exploring new mathematical structures within existing equations, and integrating insights from condensed matter physics and quantum information science.

Transcript

Introduction: What Are We Made Of?

Tonight, I'd like to tell you about one of the big questions in science. It's a question that goes back at least two and a half thousand years, to the ancient Greeks. And it's a question that has been discussed in this room many, many times over the past 200 years, but it's an important question. And I think it's important that we revisit it. And the question is simply this. It's, what are we made of? What are the fundamental building blocks of nature that you and me and everything else in the universe are constructed from?

That's the story I'd like to tell you. So what I'd like to do is try and give you an overview of our current understanding. I'd also like to try and give you an overview of where we hope to go in the future, of what progress we can hope to make in the next few years and few decades. And we're going to cover quite a lot of ground in this talk. I should warn you now, not least because I'm going to discuss every single thing in the universe, quite literally.

We're going to talk, amongst other things, about what's happening at the world's most powerful particle collider. This is a machine that's called the Large Hadron Collider, or the LHC for short. It'll come up a lot in this talk. And it's a machine which is based underground in a place called CERN, which is just outside Geneva. We'll also talk about experiments in the last few years that look backwards in time towards the Big Bang, that give us some understanding about what was happening in the first few fractions of a second after time itself started to exist.

And on top of all this, I also want to give you some idea about the theoretical abstract ideas, and even a little bit of an idea about the mathematics that underlies our current understanding of the universe. Because I'm a theoretical physicist. What I do is study the equations, try to understand the equations, that govern the world we live in. And so, I'd just like to give you a flavour of what that's about.

At some point—I should warn you now. At some point, I'm even going to show you an equation. You know, you can get sent on training courses for this kind of thing. There's a number one rule. The number one rule is never show them any equations. If you show them equations, you'll just terrify them. At some point in this lecture, you're all going to be terrified, so just prepare yourselves. OK? OK.

From Democritus to the Periodic Table and Subatomic Particles

You know, there's a traditional way to start talks like this. The traditional way is to be very cultured and talk about what Democritus and Lucretius said two and a half thousand years ago and the ideas that the ancient Greeks had about atoms. But you know, I don't want to start like this. We've made a lot of progress in two and a half thousand years, and you know, there's just better places to kick off a science talk.

So the first modern picture that we had of what the universe is made of, everything we're made of, is this.

So I hope this is familiar to most people here. This is the periodic table of elements. It's one of the most iconic images in all of science. What we have here are 120-ish different elements. I should point out, no less than 10 of which were discovered in this very building, and which constitute, or at least in the 1800s were thought to constitute, everything that existed in nature.

So it's certainly true that any material you get, you can distill it down into its component parts, and you'll find that all of those component parts are made of one of these 120 elements. So it's a great moment in science. It's really one of the triumphs of science. It's also, I should add, the reason that I stopped doing chemistry in school. Because if you're a chemist, this is basically as good as it gets. You know, if we're honest, it's kind of a mess. Everything in the universe is classified into things on the left that go bang if you put them in water through things on the right which, really if we're honest, don't do very much at all. You kind of organise everything into these stupid shapes. And it looks a little bit like Australia. There's a big dip in the top, and then there's these two strips of elements that you have to put along the bottom, because there's no room for them in the middle where they belong.

You know, I don't know about you, if I was asked to come up with a fundamental classification of everything in the universe, this isn't what I would have gone for. Are there any chemists in the audience? I'm sorry for you. OK. But you know, I'm not alone in this. It's not just me that thinks this is a silly way to organise nature. Nature itself thinks this is a silly way to organise nature. Of course, we know this isn't the fundamental—this isn't the end of the story. This isn't the fundamental building blocks.

And the first person to realise that there's something deeper than this was a Cambridge physicist called J.J. Thomson. So at the end of the 1800s, J.J. Thomson discovered a particle that was smaller than an atom that we now call the electron. And in 1897, he announced this in this room—in fact, in this very lecture series—to a stunned audience, an audience that was so stunned at least half of them didn't believe what he was saying. There was one very distinguished scientist who afterwards told J.J. Thomson he thought the whole thing was a hoax, that J.J. Thomson had just been pulling their leg.

But of course, it's not a hoax. This isn't the fundamental elements of nature. And within 15 years of J.J. Thomson's discovery, his successor in Cambridge, a man called Ernest Rutherford, had figured out exactly what these atoms are made of. And this is the picture that Rutherford came up with. So we now know that each of these elements consists of a nucleus, which is tiny. The metaphor that Rutherford himself used was it's like a fly in the centre of the cathedral. And then orbiting this nucleus in, I should add, fairly blurry orbits, are the electrons, which sort of fill out very sparsely the rest of the space.

So that's a picture of these atoms. Subsequently, we learned that the nucleus is not itself fundamental. The nucleus contains smaller particles. They're particles that we call protons and neutrons. And in the 1970s, we learned that the protons and neutrons aren't fundamental either. So in the 1970s, we learned that inside each proton and neutron are three smaller particles that we call quarks. There are two different kinds of quarks. By the 1970s, I'm guessing physicists didn't have a classical Greek education, and had kind of run out of classy names. So we call these quarks the up quark and the down quark. OK? For no good reason. It's not like the up quark is higher than the down quark. It's not like it points up. Just no good reason at all. The up quark and the down quark.

So the proton consists of two up quarks and a down quark. And the neutron consists of two down quarks and an up quark. This, as far as we know, are the fundamental building blocks of nature. We've never discovered anything smaller than the electron, and we've never discovered anything smaller than the quarks. So we have three particles of which everything we know is made.

And it's worth stressing, that's kind of astonishing. You know? We sort of take it for granted. We learn this in school. We don't really think about it deeply. Everything we see in the world, all the diversity in the natural world, you, me, everything around us, just the same three particles with slightly different rearrangements repeated over and over and over again. It's an amazing lesson to draw about how the world is put together.

The Lie of Particles and the Discovery of Fields

So that's what we have. We have an electron and two quarks. And you know, these aren't the fundamental building blocks that the Greeks had thought about, and they're certainly not the fundamental building blocks that the Victorians had thought about. But you know, the spirit of the issue really hasn't changed. The spirit is exactly what Democritus said 2,500 years ago, that they're like LEGO bricks from which everything in the world is constructed. These LEGO bricks are particles, and the particles are the electron and two quarks.

It's a very nice picture. It's a very comforting picture. It's the picture we teach kids at school. It's the picture we even teach students in undergraduate university. And there's a problem with it. The problem is it's a lie. It's a white lie. It's a white lie that we tell our children because we don't want to expose them to the difficult and horrible truth too early on. It makes it easier to learn if you believe that these particles are the fundamental building blocks of the universe.

But it's simply not true. The best theories that we have of physics do not have underlying them the electron particle and the two quark particles. In fact, the very best theories we have of physics don't rely on particles at all. The best theories we have tell us that the fundamental building blocks of nature are not particles, but something much more nebulous and abstract. The fundamental building blocks of nature are fluid-like substances which are spread throughout the entire universe and ripple in strange and interesting ways. That's the fundamental reality in which we live.

These fluid-like substances we have a name for. We call them fields.

The physicist's definition of a field is the following. It's something that, as I said, is spread everywhere throughout the universe. It's something that takes a particular value at every point in space. And what's more, that value can change in time. So a good picture to have in your mind is fluid, which ripples and sways throughout the universe.

Now, it's not a new idea. It's not an idea that we've just come up with. It's an idea which dates back almost 200 years. And like so many other things in science, it's an idea which originated in this very room. Because as I'm sure many of you are aware, this is the home of Michael Faraday. And Michael Faraday initiated this lecture series in 1825. He gave over a hundred of these Friday evening discourses, and the vast majority of these were on his own discoveries, on the experiments he did on electricity and magnetism.

So he did many, many things in electricity and magnetism over many decades. And in doing so, he built up an intuition for how electric and magnetic phenomena work. And the intuition is what we now call the electric and magnetic field. So what he envisaged was that threaded everywhere throughout space were these invisible objects called the electric and magnetic fields. Now, we learned this in school. Again, it's something that we sort of take for granted because we learned it at an early age, and we don't sort of appreciate just how big of a radical step this idea of Faraday's is. I want to stress, it's one of the most revolutionary abstract ideas in the history of science, that these electric and magnetic fields exist.

So let me just—there's supposed to be demonstrations in this. I'm not just a theoretical physicist. I'm a very theoretical physicist. It's very hard for me to do any kind of experiment that's going to work. But I'm just going to show you something that you've all seen. They're magnets. OK? And we all played these games when we were kids or when we were in school. You take these magnets, and you move them together. And as they get closer and closer, there's this force that you can sort of just feel building up that pushes, the pressure that pushes against these two magnets.

And it doesn't matter how often you do it, and it doesn't matter how many degrees you have in physics. It's just a little bit magical. You know? And you all know this. There's something just special about this weird feeling that you get between magnets. And this was Faraday's genius. It was to appreciate that even though you can't see anything in between, even though no matter how closely you look, the space between these magnets will seem to be empty, he said nonetheless, there's something real there. There's something real and physical, which is invisible, but is building up, and that's what's responsible for the force. So he called them lines of force. We now call it the magnetic field.

Here is a drawing from one of Michael Faraday's papers. When you leave, there's a carpet just here. The carpet has this pattern, this picture just repeated on it over and over and over again. And on the bottom here is one of Michael Faraday's most famous demonstrations that he did here. So I'll just walk you through what Faraday did.

The thing on the right, there's a small coil with a hand on it. This is a battery, and the battery passes a current around this coil. And in doing so, there's a magnetic field that's induced in this. It's what's called a solenoid. And then Faraday did the following thing. He simply moved this small coil A through this big coil B like this. And something miraculous happened. When you do that, there's a moving magnetic field. Faraday's great discovery was induction. It gives rise to a current in B, which then over on this end of the table, makes a needle flicker like this.

So extremely simple. You move a magnetic field, and it gives rise to a current, which makes a needle flicker on the other side of the table. This astounded audiences in the 1800s. Because you were doing something and affecting the needle on the other end of the table, yet you never touched the needle. It was amazing. You could make something move without ever going near it, without ever touching it.

We're kind of jaded these days. You can do the same experiment. You can pick up your cell phone. You can press a few buttons. You can call somebody on the other end of the earth within seconds. But it's the same principle. But this was the first time it was demonstrated that the field is real. You can communicate using the field. You can affect things far away using the field without ever touching it.

So this is Michael Faraday's legacy. There's not just particles in the world. There's other objects that are slightly more subtle that are called fields that are spread throughout all of space.

By the way, if you ever want to really appreciate the genius of Michael Faraday, he gave this lecture in 1846. He gave many lectures in 1846. But there was one in particular where he finished 20 minutes early. He ran out of things to say, so he engaged in some idle speculation for 20 minutes. And Faraday suggested that these invisible, electric, and magnetic fields that he'd postulated were quite literally the only thing we've ever seen. He suggested that it's ripples of the electric and magnetic field, which is what we call light. So it took of course 50 years for people like Maxwell and Hertz to confirm that this is indeed what light is made of, but it was Faraday's genius that appreciated this, that there were waves in the electric magnetic field, and those waves are the light that we see around us.

Combining Quantum Mechanics and Fields: Quantum Field Theory

So this is Faraday's legacy. But it turns out this idea of fields was much more important than Faraday had realised. And it took over 150 years for us to appreciate the importance of these fields.

So what happened in these 150 years was that there was a small revolution in science. In the 1920s, we realised that the world is very, very different from the common sense ideas that Newton and Galileo had handed down to us centuries before. So in the 1920s, people like Heisenberg and Schrödinger realised that on the smallest scales, on the microscopic scales, the world is much more mysterious and counter-intuitive than we ever really imagined it could be. This, of course, is the theory that we now know as quantum mechanics.

So there's a lot I could say about quantum mechanics. Let me tell you one of the punch lines of quantum mechanics. One of the punch lines is that energy isn't continuous. Energy in the world is always parcelled up into some little discrete lump. That's actually what the word quantum means. Quantum means discrete or a lump.

So the real fun starts when you try and take the ideas of quantum mechanics, which say that things should be discrete, and you try to combine them with Faraday's ideas of fields, which are very much continuous, smooth objects, which are waving and oscillating in space. So the idea of trying to combine these two theories together is what we call quantum field theory.

And here's the implication of quantum field theory. The first implication is what happens for the electric and magnetic field. So Faraday taught us, and Maxwell later, that waves of the electromagnetic field are what we call light. But when you apply quantum mechanics to this, you find that these light waves aren't quite as smooth and continuous as they appeared. So if you look closely at light waves, you'll find that they're made of particles. They're little particles of light, and these are particles that we call the photon.

The magic of this idea is that that same principle applies to every single other particle in the universe. So there is spread everywhere throughout this room something that we call the electron field. It's like a fluid that fills this room and, in fact, fills the entire universe. And the ripples of this electron fluid, the ripples of the waves of this fluid, get tied into little bundles of energy by the rules of quantum mechanics, and those bundles of energy are what we call the particle, the electron.

All the electrons that are in your body are not fundamental. All the electrons that exist in your body are waves of the same underlying field. And we're all connected to each other. Just like the waves on the ocean all belong to the same underlying ocean, the electrons in your body are ripples of the same field as the electrons in my body.

There's more than this. There's also in this room two quark fields. And the ripples of these two quark fields give rise to what we call the up quark and the down quark. And the same is true for every other kind of particle in the universe. There are fields that underlie everything. And what we think of as particles aren't really particles at all, they're waves of these fields tied up into little bundles of energy. This is the legacy of Faraday. This is where Faraday's vision of fields has taken us. There are no particles in the world. The basic fundamental building blocks of our universe are these fluid-like substances that we call fields.

The Quantum Vacuum and Mathematical Difficulties

So what I want to do in the rest of this talk is tell you where that vision takes us. I want to tell you about what it means that we're not made of particles, we're made of fields. And I want to tell you what we can do with that, and how we can best understand the universe around us.

Here's the first thing. Take a box and take every single thing that exists out of that box. Take all the particles out of the box, all the atoms out of the box. What you're left with is a pure vacuum. And this is what the vacuum looks like.

So what you're looking at here is a computer simulation using our best theory of physics of something called the Standard Model, which I'll introduce later. But it's a computer simulation of absolutely nothing. This is empty space. Literally empty space with nothing in it. This is the simplest thing you could possibly imagine in the universe. And you can see, it's an interesting place to be, an empty space. It's not dull and boring.

What you're looking at here is that even when the particles are taken out, the field still exists. The field is there. But what's more, the field is governed by the rules of quantum mechanics. And there's a principle in quantum mechanics, which is called the Heisenberg Uncertainty Principle, which says you're not allowed to sit still. And the field has to obey this. So even when there's nothing else there, the field is constantly bubbling and fluctuating in what's, quite honestly, a very complicated way. These are things that we call quantum vacuum fluctuations. But this is what nothingness looks like from the perspective of our current theories of physics.

It's worth saying that this is a computer simulation. It looks a little bit like a cartoon, but it's actually quite a powerful computer simulation, and it took a long time to do. But these aren't just theoretical. These quantum fluctuations that are there in the pure vacuum are things that we can measure. There's something called the Casimir force. The Casimir force is a force between two metal plates that get pushed together basically because there's more of this stuff on the outside than on the inside. And you know, these are real. These are things that we can measure, and they behave just as we would predict they would from our theories.

So this is nothing. And this brings me to the more mathematical side of the talk. Because there's a challenge in this. This is the simplest thing we can imagine in the entire universe, and it's complicated. It's astonishingly complicated. It doesn't get easier than this. You know, if you want to now understand not nothing but a single particle, well, that's much more complicated than this. And if you want to understand $10^{23}$ particles all doing something interesting, that's really, really much more complicated than this.

So there's a problem in—it's my problem, not yours—in addressing this fundamental description of the universe, which is that it's just hard. The mathematics that we use to describe quantum fields, to describe everything that we're made of in terms of quantum fields, is substantially more difficult than the maths that arises in any other area of physics or science. It's genuinely difficult.

I can put this in some perspective. There's a list of six open problems in mathematics. They're considered to be the six hardest problems in mathematics. There used to be seven, but some crazy Russian guy solved one of them. So there's six left. You win a million bucks if you can solve any one of these problems. If you know a little bit of mathematics, they're things like the Riemann hypothesis, or P versus NP. They're sort of famously difficult problems. This is one of those six problems. You win a million dollars if you can understand this.

So what does it mean? It doesn't mean can you build a big computer and just demonstrate that these are there. It means can you understand from first principles by solving the equations the patterns that emerge within these quantum fluctuations? It's an extraordinarily difficult problem. You know, it's writing the kind of thing I do. I don't know a single person in the world who's actually working on this problem. That's how hard it is. We don't really even know how to begin to start understanding these kind of ideas in quantum field theory.

This theme about the mathematics being challenging is something which is going to come back later in the talk. So I'd like just to take a little bit of a diversion for a few minutes and give you a sense about what we can do mathematically and what we can't do mathematically, just to sort of tell you what the state of play is in terms of understanding these theories called quantum field theories which underlie our universe.

So there are times where we understand extremely well what's going on with quantum fields. And that happens basically when these fluctuations are very calm and tame, when they're not wild and strong. These ones are big. But when they're much calmer, when the vacuum is much more like a mill pond than it is like a raging storm, in those cases, we really think we understand what we're doing. And to illustrate this, I just want to give you this example.

So this number $g$ is a particular property of the electron particle. And I'll quickly explain what it is. The electron is a particle, and it turns out the electron spins. It orbits rather like the earth orbits. And it has an axis of spin. And you can change the axis of that spin. And the way you change it is you take a magnetic field like this. And in the presence of a magnetic field, the electron will spin. The electron will stay in one place, but spin. And then the axis of spin will slowly rotate like this. It's what's called precession. And the speed at which the axis of that spin precesses is dictated by this number here.

So it's not the most important thing in the big picture. However, historically, this has been extremely important in the history of physics, because it turns out, this is a number you can measure very, very accurately doing experiments. And so this number has sort of acted as a testing ground for us to see how well we understand the theories that underlie nature, and in particular, quantum field theory.

So let me tell you what you're looking at here. The first number is the result of many, many decades of painstaking experiments measuring very, very precisely this feature of the electron. It's called the magnetic moment, for what it's worth. And the second number is the result of many, many decades of very torturous calculations sitting down with a pen and paper and trying to predict from first principles from quantum field theory what the magnetic moment of the electron should be.

And you can see, it's simply spectacular. And there's nothing like this anywhere else in science with an agreement between the theoretical calculation and the experimental measurements. I think it's 12 or 13 significant figures. It's really astonishing. Any other area of science, you'll be jumping up and down for joy if you get the first two numbers right. Economics, not even that. Just that this is where we're at in particle physics on a good day when we really understand what we're doing with it. It's substantially better than any other area of science. 12 significant figures.

But this, of course, I've shown you because this is our best result. There are many other results that are nowhere near as good. And the difficulty comes when those quantum vacuum fluctuations start getting wilder and stronger.

So let me give you an example. It should be possible for us to sit down and calculate from first principles the mass of the proton. We have the equations. Everything should be there. We just need to work hard and figure out what the mass of the proton is just by doing calculations. We've been trying to do this for about 40 years now. We can get it to within an accuracy of something like 3%. Which isn't bad. We're 3% there. But we should be much, much better. We should be sort of pushing these levels of accuracy.

And the reason is very simple. We've got the right equation. We're pretty sure we're solving the right equation. It's simply that we're not smart enough to solve it. In 40 years, the world's most powerful computers, lots and lots of smart people. But we haven't managed to figure this out.

There are other situations that I won't tell you about where we don't even get off the ground. There are some situations where for fairly subtle reasons we're unable to use computers to help us, and we simply have no idea what we're doing. So it's a slightly strange situation. We have these theories of physics. They're the best theories we've ever developed, as you can see by this. But at the same time, they're also the theories that we understand the least, and to make progress we sort of have this strange balancing act between increasing our theoretical understanding and figuring out how to apply that to the experiments that we're doing. And again, it's a theme I'll come back to at the end of the lecture.

The Standard Model: Matter Fields, Forces, and the Higgs Boson

All right. So so far, I've been talking in a little bit of generality about what we're made of. And this is the punch line for the halfway point of the talk. You're all made of quantum fields, and I don't understand them. At least I don't understand them as well as I think I should.

So what I want to do now is go into a little bit more specifics. I want to tell you exactly what quantum fields exist in the universe. And the good news is, not many of them. So I'll simply tell you, all of them.

We started with the periodic table. This is the new periodic table. And it's much simpler. You know, it's much nicer.

There are the three particles that we're all made of. There's the electron and the two quarks, the up quark and the down quark. And as I've stressed, the particles aren't fundamental. What's really fundamental is the field that underlies them.

And then it turns out there's a fourth particle that I've not discussed so far. It's called the neutrino. It's not important in what we are made of, but it does play another important role elsewhere in the universe. These neutrinos are everywhere. You've never noticed them, but since I began this talk, something like $10^{14}$ of them have streamed through the body of each and every one of you, as many coming from above from outer space as actually coming from below, because they stream all the way through the earth and then keep going. They're not very sociable. They don't interact.

So this is what everything is made of. These are the four particles that form the bedrock of our universe. Except then something rather strange happened. For a reason that we do not understand at all, nature has chosen to take these four particles and reproduce them twice over.

So this is actually the list of all the fields that make up particles in our universe. So what are we looking at here? This is the electron. It turns out there are two other particles which behave in every way exactly the same as the electron, except they're heavier. We call them the muon, which has a mass of something like 200 times the electron, and the tau particle, which is 3,000 times heavier than the electron. Why are they there? We have no idea at all. It's one of the mysteries of the universe.

There's also two more neutrinos. So there are three neutrinos in total. And the two quarks that we first knew about are now joined by four others that we call the strange quark and the charm quark. And then by the time we got here, we really ran out of any kind of inspiration for naming them. We called them the bottom quark and the top quark.

So I should stress. We understand things very, very well going horizontally across the four types. We understand why they come in a group of four. We understand why they have the properties that they do. We don't understand it at all going vertically through the three generations. We don't know why there's three of these rather than two of them or 17 of them. That's a mystery.

But this is everything. This is everything in the universe. Everything you're made of is these three at the top there. And it's only when you go to more exotic situations, like particle colliders, that we need the others on the bottom. But every single thing we've ever seen can be made out of these 12 particles, 12 fields.

These 12 fields interact with each other, and they interact through four different forces. Two of these are extremely familiar. They're the force of gravity and the force of electromagnetism. But there's also two other forces which operate only on small scales of a nucleus. So there's something called the strong nuclear force, which holds the quarks together inside protons and neutrons. And there's something called the weak nuclear force, which is responsible for radioactive decay and among other things, for making the sun shine.

Again, each of these forces is associated to a field. So Faraday taught us about the electromagnetic field, but there's a field associated to the strong force, which is called the gluon field, and a field associated to the weak force, which is called the W and Z boson field. There's also a field associated to gravity. And this was really Einstein's great insight into the world. The field associated to gravity turns out to be space and time itself. So if you've never heard that before, that was the world's shortest introduction to general relativity. And I'm not going to say anything else about it. I'll just let you figure that one out for yourself.

So this is the universe we live in. There are 12 fields that give matter—I'll call them the matter fields—and four other fields that are the forces. And the world we live in is this combination of the 16 fields all interacting together in interesting ways. So this is what you should think the universe is like. It's filled with these fields, fluid-like substances. 12 matter, four forces. One of the matter fields starts to oscillate and ripple. Say the electron field starts to wave up and down, because there's electrons there. That will kick off one of the other fields. It'll kick off, say, the electromagnetic field, which, in turn, will also oscillate and ripple. There'll be light which is emitted. So that will oscillate a little. At some point, it will start interacting with the quark field, which in turn will oscillate and ripple. And the picture we end up with is this harmonious dance between all these fields, interlocking each other, swaying, moving this way and that way. That's the picture that we have of the fundamental laws of physics.

We have a theory which underlies all this. It is, to put it simply, the pinnacle of science. It's the greatest theory we've ever come up with. We've given it the most astonishingly rubbish name you've ever heard of. We call it the Standard Model. When you hear the name the Standard Model, it sounds tedious and mundane. It should really be replaced by The Greatest Theory in the History of Human Civilisation. OK? That's what we're looking at.

OK. So this is everything, except it's not quite. I've actually just missed one field. There's one extra thing we know about, which became quite famous in recent years. It was a field that was first suggested in the 1960s by a Scottish physicist called Peter Higgs. And by the 1970s, it had become an integral part of the way we thought about the universe. But for the longest time, we didn't have direct experimental evidence that this existed, where direct experimental evidence means we make this Higgs field ripple so we see a particle that's associated to it.

And this changed. This changed famously four years ago at the LHC. These are the two experiments at the LHC that discovered it. They're sort of the size of cathedrals, and just packed full of electronics. They're astonishing things. One is called ATLAS. One is called CMS.

That Higgs particle doesn't last for long. The Higgs particle lasts about $10^{-22}$ seconds. So it's not like you see it and you get to take a picture of it and put it on Instagram. It's a little more subtle. So this is the data, and this little bump here is how we know that this Higgs particle existed. This is a picture of Peter Higgs being found.

So this was the final building block. You know, it was important. It was a really big deal. And it was important for two reasons. The first is that this is what's responsible for what we call mass in the universe. So the properties of all the particles, things like electric charge and mass, are really a statement about how their fields interact with other fields. So the property that we call electric charge of an electron is a statement about how the electron field interacts with the electromagnetic field. And the property of its mass is the statement about how it interacts with the Higgs field. So understanding this was really needed so that we understand the meaning of mass in the universe. So it was a big deal.

The other reason that it was a big deal is, this was the final piece of our jigsaw. We had this theory that we called the Standard Model. We've had it since the 1970s. This was the final thing that we needed to discover to be sure that this theory is correct. And the astonishing thing is this particle was predicted in the 1960s. 50 years we've been waiting. We finally created it in CERN. It behaves in exactly the way that we thought it would. Absolutely perfectly behaves as we predicted using these theories.

The Master Equation of Physics

This is going to be the scary part of the talk. I've been telling you about this theory. And I've been waving my hands pretending that I'm a field. Let me tell you what the theory really is. Let me just show you what we do.

This is the equation for the Standard Model of physics. I don't expect you to understand it, not least because there are parts of this equation that no one on the planet understands. But nonetheless, I want to show it to you for the following reason. This equation correctly predicts the result of every single experiment we've ever done in science. Everything is contained in this equation. This is really the pinnacle of the reductionist approach to science. It's all in here.

$$Z = \int \mathcal{D}(\text{Fields}) \exp \left( i \int d^4x \sqrt{-g} \left( R - F_{\mu\nu} F^{\mu\nu} - G_{\mu\nu} G^{\mu\nu} - W_{\mu\nu} W^{\mu\nu} + \sum_i \bar{\psi}_i \mathcal{D} \psi_i + D_\mu H^\dagger D^\mu H - V(H) - \lambda_{ij} \psi_i H \psi_j \right) \right)$$

So I'll admit. It's not the simplest equation in the world. But it's not the most complicated either. You can put it on a t-shirt if you want. In fact, if you go to CERN, you can buy a t-shirt with this equation on it.

Let me just give you a sense of what we're looking at:

• The first term here ($R$) was written down by Albert Einstein and describes gravity. What that means is that if you could solve this tiny little part of the equation, just this $R$, you can, for example, predict how fast an apple falls from a tree, or the fact that the orbits of the planets around the sun form ellipses. Or you can predict what happens when two enormous black holes collide into each other and form a new black hole, sending out gravitational waves across the universe. Or in fact, you can predict how the entire universe itself expands. All of this comes from solving this little part of the equation.

• The next term in the equation ($\frac{1}{4} F_{\mu\nu} F^{\mu\nu}$) was written down by James Clerk Maxwell, and it tells you everything about electromagnetism. So all the experiments that Faraday spent a lifetime doing in this building—in fact, all the experiments over many centuries, from Coulomb to Faraday, to Hertz to modern developments of lasers, everything—in this tiny little part of the equation.

• There is also the piece that governs the strong nuclear force and the weak nuclear force.

• This next term ($i \bar{\psi} \bar{D} \psi$) was first written down by a British physicist called Paul Dirac. It describes the matter. It describes those 12 particles that make up the matter. Astonishingly, each of them obeys exactly the same equation.

• These final terms are the equations of Peter Higgs ($|D_\mu \phi|^2 - V(\phi)$) and the term that tells you how the matter interacts with the Higgs particle ($\psi_i y_{ij} \psi_j \phi$).

So everything is in here. It's really an astonishing achievement. This is our current limit of knowledge. We've never done an experiment that cannot be explained by this equation. And we've never found a way in which this equation stops working. So this is the best thing that we currently have.

Unsolved Mysteries: Cosmic Inflation and the Early Universe

OK. It's the best thing that we currently have. However, we want to do better, because we know for sure that there's stuff out there that is not explained by this. And the reason we know is that although this explains every single experiment we've ever done here on Earth, if we look out into the sky, there's extra stuff which is still a mystery.

So if we look out into space, there are, for example, invisible particles out there. In fact, there's many more invisible particles than there are visible particles. We call them dark matter. We can't see them, obviously, because they're invisible. But we can see their effects. We can see their effects in the way galaxies rotate, or the way they bend light around galaxies. They're out there. We don't know what they are.

There's even more mysterious things. There's something called dark energy, which is spread throughout all of space. It's also some kind of field, although not one we understand, that's causing everything in the universe to repel everything else.

Other things. We know that early in the first few seconds, earlier than that, the first few fractions of a second after the Big Bang, the universe underwent a very rapid phase of expansion that we call inflation. We know it happened, but it's not explained by that equation that I just showed you.

So these are the kind of things that we're going to have to understand if we're going to move forward and decide what the next laws of physics are that go beyond the Standard Model. I could spend hours talking about any of these. I'm going to focus just on the last one. I'm going to tell you a little bit about inflation.

So the universe is 13.8 billion years old. And we understand fairly well—well, we don't understand at all how it started. We don't understand what kicked it all off at time $t = 0$. But we understand fairly well what happened after it started. And we know in particular that for the first 380,000 years of the universe, it was filled with a fireball. And we know this for sure because we've seen the fireball. In fact, we've seen it, and we've taken a photograph of it.

This is called the cosmic microwave background radiation, but a much better name for it is the Fireball That Filled the Universe When It Was Much Younger. The fireball cools down. Its light has been streaming through the universe for 13.8 billion years. But we can see it. We can take this photograph of it. And we can understand very well what was happening in these first few moments of the universe. And you can see, it looks literally like a fireball. There's red bits that are hotter. There's blue bits that are colder. And by studying this flickering that you can see in this picture, we get a lot of information about what was going on back 13.8 billion years ago when the universe was a baby.

One of the main questions we want to ask is what caused the flickering in the fireball? And we have an answer to this. We have an answer, which I think is one of the most astonishing things in all of science.

It turns out that although the fireball lasted for 380,000 years, whatever caused this flickering could not have taken place during the vast majority of that time. Whatever caused the flickering in this fireball actually took place in the first few very fractions of a second after the Big Bang. And what it was was the following.

So when the universe was very, very young, soon after the Big Bang, there were no particles, but there were quantum fields, because the quantum fields were everywhere. And there were these quantum vacuum fluctuations. And what happened was the universe expanded very, very quickly, and it caught these quantum fluctuations in the act. So the quantum fluctuations were stretched across the entire sky, where they became frozen. And it's these vacuum fluctuations here which are the ripples that you see in the fireball.

So it's an astonishing story, that the quantum vacuum fluctuations were taking place $10^{-30}$ seconds after the Big Bang. They were absolutely microscopic. And now we see them stretched across the entire universe, stretched 20 billion lightyears across the sky. That's what you're seeing here. And yet, you do the calculations for this, and it matches perfectly what you see here.

So this is another of the great triumphs of quantum field theory. But it leaves lots of questions. The most important one is, which field are we seeing here? Which field is this that's imprinted on the background radiation? And the answer is we don't know. The only one of the Standard Model fields it has a hope of being is the Higgs. But most of us think it's not the Higgs, but probably something new. But what we'd like to do moving forward into the future is get a much better picture of this fireball, in particular get the polarisation of the light. And by getting a picture of this, we can understand much better the properties of this field that was fluctuating in the early universe.

Theories Beyond the Standard Model and the LHC Results

This looking forward is one of the best hopes that we have for going beyond the Standard Model and understanding new physics. In the last 10 minutes, though, I'd like to bring you back down to Earth, sort of. We've got lots of experiments here on Earth where we're also trying to do better, where we're also trying to go beyond the Standard Model of physics beyond that equation to understand what's new. And there's many of them, but the most prominent is the one I've already mentioned. It's the LHC.

So what happened was the LHC discovered the Higgs boson in 2012. And soon afterwards, it closed down for two years. It had an upgrade. And last year in 2015, the LHC turned on again with twice the energy that it had when it discovered the Higgs. And the goal was twofold. The goal was firstly to understand the Higgs better, which it has done fantastically, and secondly, to discover new physics that lies beyond the Higgs, new physics beyond the Standard Model.

So before I tell you what it's seen, let me tell you some of the ideas we've had, some of our expectations and hopes for what would happen moving forward.

This is our favourite equation again. The idea has always been the following. You know, if you were a Victorian scientist, and you go back, and you look at the periodic table of elements, then it's true that there's patterns in there that give a hint of the structure that lies underneath. Those numbers that repeat themselves. Where if you're very smart, you might start to realise that, yes, there is something deeper than just these elements.

So our hope as theorists is to look at this equation and see if maybe we can just find patterns in this equation that suggest there might be something deeper that lies underneath. And they're there. So let me give you an example.

This is the equation that describes the force of electricity and magnetism. And it's almost the same as the equations which describe the forces for the strong force and the weak nuclear force. You can see. I've just changed letters. It's a little more complicated than that, but it's not much more complicated than that. The three forces really look similar. So you might wonder, well, maybe there's not three forces in the universe. Maybe those three forces are actually just one force. And when we think there's three forces, it's because we're looking at that one force just from slightly different perspectives. Maybe.

Here's something else, which is amazing. These are the equations for the 12 matter fields in the universe—the neutrinos, the electrons, and the quarks. Each of them obeys exactly the same equation. Each of them obeys the Dirac equation. So again, you might wonder, well, maybe there aren't 12 different fields. Maybe they're all the same field and the same particle, and the fact they look different is, again, maybe just because we look at them from slightly different perspectives. Maybe.

So these ideas that I've been suggesting go by the name of unification. The idea that the three forces are actually combined into one is what's called Grand Unification. And it's very easy. It's very easy to write down a mathematical theory in which all of these are just one force, which appears to be three from our perspective.

There are other possibilities here. You might say, well this is the matter, and these are the forces. And the equations are different, but they're not that different. Because ultimately, they're both just fields. So you might wonder if maybe there's some way in which the matter and the forces are related to each other. Well, we have a theory for that as well. It's a theory that's called supersymmetry. And it's a beautiful theory. It's very deep conceptually. And it sort of, you know, smells like it might be right.

Finally, you might be really, really bold. You might say, well, can I just combine the lot? Can I just get rid of all of these terms and just write down one single term from which everything else emerges? Gravity, the forces, the particles, the Higgs, everything. I've got something for you if you want that as well. It's called string theory. So we have a possibility for a theory which contains all of this in one simple concept.

And the question going forward, of course, is are these right? You know, it's very easy for us theorists to have these ideas. And I should say these ideas are what's driven theoretical physics for 30 years, but we want to know, are they right? And we've got a way of telling if they're right. We do experiments.

So I should say, if you want to know if string theory's right, we don't have any way to test it at the moment. But if you want to know if some of these other ideas are right, then that's what the LHC should be doing. The reason that we built the LHC was firstly to find the Higgs. OK, it worked. And secondly, to test these kind of ideas that we've been having to see what lies beyond.

So the LHC has been running. It's been running for two years. It's been running like an absolute dream. It's a perfect machine. Two years. This is what it's seen:

Absolutely nothing.

All of these fantastic beautiful ideas that we've had, none of them are showing up at all. And the question going forward is, what are we going to do about it? How are we going to make progress in understanding the next layer of physics when the LHC isn't seeing anything, and our ideas just don't appear to be the way that nature works?

I should tell you, often I don't have a good answer to this. My impression is that most of my community is a little bit shell-shocked by what happened. There's certainly no consensus in the community to move forward. But I think there's three responses that sort of various people have had that I'd like to share with you. And I think all three of these responses are reasonable up to a point.

• Response Number 1 (Patience): "You young kids, you're so pessimistic. It's all doom and gloom with you. You need a little bit more patience. You know, I didn't see anything last year, and I didn't see anything this year. But next year, it's going to see something. And if not next year, it's the year after that that it's going to see something." It's usually my very illustrious senior colleagues that have this—and you know what? They could easily be right. It could easily be that next year, the LHC discovers something astonishing, and it sets us on the path to understanding the next layer of reality. But it's also true that these same people were predicting that it would have seen something by now. And it's also true that this can't keep going for much longer. If the LHC doesn't see something within, say, a two-year time scale, it seems very, very unlikely that it's going to see something moving forward. It's possible. It just seems unlikely. So I hope with all my heart that the LHC discovers something next year or the year after. But I think we have to prepare for the worst, that maybe it won't.

• Response Number 2 (A Bigger Collider): "Well, all our theories are so beautiful. They absolutely have to be correct, and what we really need is a bigger machine. 10 times bigger will do it." Again, they might be right. I don't have a good argument against it. The obvious rebuttal, however, is that a new machine costs $10 billion. There's not too many governments in the world that have $10 billion to spare for us to explore these ideas. There's one. The one is China. And so if this machine is going to be built at all, it's going to be built by the Chinese government. I think the Chinese government would see it as extremely attractive if the whole community of particle physicists and engineers that are currently based in CERN and Geneva move to a town that's slightly north of Beijing. I think they'd view that as political and economic gain, and there's a real chance they may decide to build this machine. If they do, it's about 20 years for it to be built. So we're waiting slightly longer.

• Response Number 3 (Rethinking Assumptions): I should say the third response is kind of the camp I'm in. I should mention upfront, it's speculative, and it's probably not endorsed by most of my peers. So this is really just my personal opinion at this point. This is my take on this. This is the equation that we know is right. This is sort of the bedrock of our understanding. But although we know it's right, there's an awful lot in this equation that we haven't understood. There's an awful lot to me that's still mysterious in this equation. So although this equation looked like there were suggestions of unification, maybe they're just red herrings. And maybe if we just work harder in trying to understand this equation more, we'll find that there are other patterns that emerge.

So my response is, I think that maybe we should just go back to the drawing board and start to challenge some of the assumptions and paradigms that we've been holding for the past 30 years. So I feel quite energised, actually, by the lack of results for the LHC. You know? Sort of it feels good to me that everyone was wrong. You know, it's when we're wrong that we start to make progress. So I sort of feel quite happy about this, and think that there's a very real chance that we could just start thinking about different ideas.

I should say that there are hints in here. There are hints to me about mathematical patterns that we haven't explored. There's hints in this about connections to other areas of science. Things like condensed matter physics, which is the science of how materials work, or quantum information science, which is the attempt to build a quantum computer. All these fantastic subjects have new ideas, which sort of feed in to the kind of questions that we're asking here. So I'm quite optimistic that moving forward, we can make progress, maybe not the progress that we thought we'd make a few years ago, but just something new.

Conclusion and Q&A

So that's the punchline of my talk. The punchline is that this is the single greatest equation that we've ever written down. But I hope that someday, we can give you something better. Thank you for your attention.

Audience Member: [Question regarding discreteness in the Schrödinger equation]

David Tong: There's nothing discrete about the Schrödinger equation. The Schrödinger equation is something to do with a smooth field-like wave function. The discreteness is something which emerges when you solve the Schrödinger equation. So it's not built into the heart of nature.

Summary

Executive Overview

The Schrödinger equation—formulated by Erwin Schrödinger in 1925—is the foundational equation of non-relativistic quantum mechanics, often described as the quantum analogue of Newton’s second law ($F = ma$). It governs the behavior of matter waves, predicting atomic energy levels, orbital geometries, chemical bonding, and modern semiconductor physics.

Despite its tremendous physical impact—enabling technologies such as computer microchips, electron microscopes, atomic clocks, and high-speed internet—the equation relies fundamentally on the imaginary unit $i = \sqrt{-1}$. Understanding why imaginary numbers are essential to modeling real-world physics requires tracing the historical failure of classical electrodynamics, building quantum operators from scratch, and recognizing how complex exponentials preserve the conservation of probability.

Part I: Historical Context and the Crisis of Classical Physics

• Spectral Lines & Empirical Rules (1853–1885):

• In 1853, Anders Ångström discovered that heated hydrogen gas emits discrete, characteristic colors (spectral lines) rather than a continuous spectrum. Every element possesses a unique spectral signature, leading to the discovery of new elements like helium in the sun.

• In 1885, Swiss mathematics teacher Johann Balmer identified an empirical numerical formula that accurately predicted hydrogen's spectral wavelengths. However, Balmer's formula was purely descriptive; no theoretical foundation existed to explain why it worked.

• The Failure of Maxwellian Electrodynamics:

• According to James Clerk Maxwell, light is an electromagnetic wave generated by accelerating charges. Wiggling charges produce continuous ripples in the electromagnetic field.

• Classical physics predicted that a hot, glowing gas—containing billions of vibrating charges across a continuous thermal distribution—should emit a continuous rainbow spectrum of all frequencies. It could not account for discrete spectral lines.

• Upon the discovery of the dense, positively charged atomic nucleus, classical physics faced a deeper catastrophe: orbiting negative electrons are continuously accelerating toward the center. Under classical electrodynamics, orbiting electrons must continuously radiate electromagnetic energy, causing them to collapse into the nucleus almost instantaneously. Classical physics could not explain atomic stability.

• Quantum Discretization (Einstein & Bohr):

• Photoelectric Effect (1905): Albert Einstein resolved the mystery of light-matter interaction by proposing that light delivers energy in quantized packets (photons), where photon energy depends strictly on frequency ($E = h f$), not intensity (brightness). Intensity dictates the quantity of photons, whereas frequency dictates individual photon energy.

• The Bohr Model (1913): Niels Bohr applied energy quantization to atomic structure, postulating that electrons are restricted to specific non-radiating stationary orbits. Electrons absorb or emit photons only when jumping between allowed energy states ($\Delta E = h f$). This model derived Balmer’s formula and explained atomic stability, but failed to explain why electrons were restricted to specific discrete orbits.

• De Broglie’s Matter Wave Hypothesis (1924):

• Louis de Broglie extended wave-particle duality: if light waves exhibit particle properties, matter particles (such as electrons) must exhibit wave properties.

• De Broglie proposed that orbiting electrons form standing waves around the nucleus. Just as a guitar string vibrates only at integer harmonics (loops), electron orbits are restricted to circumferences that accommodate an integer number of matter wavelengths ($\lambda = \frac{h}{p}$). This naturally explained discrete atomic orbits and non-radiating states.

Part II: Intuitive Construction of Schrödinger’s Equation & Quantum Operators

In late 1925, Erwin Schrödinger sought to find the wave equation governing De Broglie's matter waves. The equation can be built intuitively from the principle of energy conservation:

$$\text{Total Energy } (E) = \text{Kinetic Energy } (K) + \text{Potential Energy } (V) = \frac{p^2}{2m} + V$$

Why Quantum Mechanics Requires Operators

For a classical single sine wave of definite wavelength ($\lambda$) and frequency ($f$), one could directly substitute de Broglie ($p = \frac{h}{\lambda}$) and Planck ($E = h f$) relations. However, arbitrary physical particles are represented by wave packets comprising a range or mixture of wavelengths and frequencies. Consequently, a quantum state generally lacks a single definite momentum or energy value.

To address this, quantum mechanics replaces physical observables with mathematical operators that extract the full underlying distribution of momentum or energy from a generalized wave function $\psi$.

Building the Kinetic Energy / Momentum Operator

• Standing Wave Equation: A basic standing wave equation takes the form:

$$\psi(x, t) = A \sin(\kappa x) \cos(\omega t)$$

where $\kappa = \frac{2\pi}{\lambda}$ (spatial frequency / wavenumber) and $\omega = 2\pi f$ (temporal frequency).

• Relating Frequencies to Energy and Momentum:

• Temporal frequency encodes total energy: $E = \hbar \omega$ (where $\hbar = \frac{h}{2\pi}$).

• Spatial frequency encodes spatial momentum: $p = \hbar \kappa$.

• Deriving Spatial Momentum via Spatial Curvature:

Differentiating $\psi$ with respect to space twice ($\frac{\partial^2 \psi}{\partial x^2}$) brings out $-\kappa^2$ and returns the original sine spatial dependence:

$$\frac{\partial^2 \psi}{\partial x^2} = -\kappa^2 \psi = -\left(\frac{p}{\hbar}\right)^2 \psi \implies -\hbar^2 \frac{\partial^2 \psi}{\partial x^2} = p^2 \psi$$

• Kinetic Energy Operator:

Dividing $p^2$ by $2m$ yields the kinetic energy operator:

$$\hat{K} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}$$

• Physical Insight: Spatial curvature ($\frac{\partial^2 \psi}{\partial x^2}$) encodes kinetic energy. Higher wave curvature corresponds to shorter wavelengths, higher momentum, and higher kinetic energy.

The Role of Fourier Analysis

Although derived for a single sine wave, Fourier analysis proves that any complex wave packet can be expressed as a linear superposition of pure sine waves. Because differentiation is a linear operation, the spatial curvature operator applies linearly across all Fourier components, establishing that $\hat{K} = -\frac{\hbar^2}{2m}
abla^2$ holds universally for any matter wave shape.

Part III: The Mathematical Necessity of Imaginary Numbers ($i$)

Building the Energy Operator requires extracting total energy ($E = \hbar \omega$), which lives in the temporal domain, via a time derivative ($\frac{\partial}{\partial t}$).

The Derivative Dilemma

• Differentiating a real periodic function ($\sin(\omega t)$ or $\cos(\omega t)$) once with respect to time turns sine into cosine. The original function is not recovered, making it impossible to form a direct linear eigenvalue/operator equation of the form $\hat{E}\psi = E\psi$.

• Taking a double time derivative ($\frac{\partial^2}{\partial t^2}$) returns the original function, but extracts $E^2$, whereas total energy conservation requires linear energy $E$.

• The only mathematical function whose first derivative is proportional to itself is the exponential function ($e^{\lambda t}$). However, real exponentials ($e^{\alpha t}$ or $e^{-\alpha t}$) continuously blow up to infinity or decay to zero, failing to represent stable, periodic wave oscillations.

Argand Diagrams & Complex Exponentials

Jean-Robert Argand demonstrated that multiplying a exponent by $i = \sqrt{-1}$ acts as a $90^\circ$ spatial rotation in the complex plane:

• Real exponential derivatives align velocity in the direction of position (causing runaway growth or decay).

• Introducing $i$ forces the derivative (velocity) to remain perpendicular to position at all times.

Perpendicular velocity drives uniform circular rotation in the complex plane. Thus, complex exponentials ($e^{-i \omega t} = \cos(\omega t) - i \sin(\omega t)$) are simultaneously exponential (satisfying first-derivative operator requirements) and periodic (representing stable wave structures).

Differentiating $\psi(x,t) = A e^{i(\kappa x - \omega t)}$ with respect to time yields:

$$\frac{\partial \psi}{\partial t} = -i \omega \psi = -i \left(\frac{E}{\hbar}\right) \psi \implies i \hbar \frac{\partial \psi}{\partial t} = E \psi$$

Combining energy, kinetic, and potential terms yields the time-dependent Schrödinger Equation:

$$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi$$

• Connection to Heat Flow: Without $i$, the equation becomes $\frac{\partial \psi}{\partial t} = D \frac{\partial^2 \psi}{\partial x^2}$, which is the classical Heat Diffusion Equation describing real exponential decay of temperature over time. The presence of $i$ transforms exponential thermal decay into continuous, probability-conserving quantum rotation.

Part IV: The Physical Meaning of $i$ and Probability Conservation

• The Born Rule (1926): Max Born established that the wave function $\psi(x,t)$ itself is not a direct classical wave of physical matter, but a complex probability amplitude. The physical probability density $P(x,t)$ of finding a particle at position $x$ is given by the squared magnitude:

$$P(x,t) = |\psi(x,t)|^2 = \psi^* \psi$$

• Conservation of Total Probability:

• If $\psi$ were constrained to real space oscillations (simple up-and-down real sine waves), $|\psi(x,t)|^2$ would fluctuate periodically across zero everywhere simultaneously. The total probability of finding the particle anywhere in the universe would continuously change over time, violating fundamental physics.

• Because $\psi$ rotates continuously in the complex plane, its magnitude $|\psi|^2 = \text{Re}(\psi)^2 + \text{Im}(\psi)^2$ remains strictly constant for stationary states, and total integrated spatial probability remains conserved at exactly 100% ($1.0$) across all time.

The imaginary unit $i = \sqrt{-1}$ is mathematically required to construct a linear first-derivative energy operator, and physically required to conserve quantum probability.

Part V: Historical Impact and Real-World Applications

Solving Schrödinger’s equation for the hydrogen atom ($V(r) = -\frac{e^2}{4\pi\epsilon_0 r}$) precisely derived discrete hydrogen energy levels, spectral intensities, spectral line splitting under electromagnetic fields (Stark and Zeeman effects), and three-dimensional electron probability clouds (orbitals $s, p, d, f$).

Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac. Modern technological foundations directly derived from the Schrödinger equation include:

• Electron Microscopy: Harnesses matter wavelengths of high-energy electrons to resolve individual atomic structures.

• Atomic Clocks: Calculates exact electronic transitions in cesium atoms, creating global navigation networks (GPS).

• Semiconductor Engineering & Microchips: Predicts quantum energy band gaps when atoms form crystal lattices, enabling modern transistors, microprocessors, and digital memory.

Transcript

Modern Physics and the Unreasonable Equation

This single equation helped unlock the modern world. It gave us computer chips, electron microscopes, atomic clocks, GPS, high-speed internet—the list goes on. But where did this equation come from?

According to Richard Feynman, it comes from nowhere: "Out of man's imagination, struggles with the details of experiment and all kinds of mysteries."

That man was Erwin Schrödinger. He derived it in 1925 while vacationing in the Swiss Alps with his mistress. So I have two questions:

• How can we intuitively build this equation ourselves from scratch?

• Why is an equation with such real impact built using an imaginary number? What is $i$ doing there?

If you're ready, let's find out.

The Hydrogen Spectrum and Classical Collapse

It all starts in 1853 when the physicist Anders Ångström finds that hot hydrogen gas gives out very specific colors of light. We soon figured out it's not just hydrogen; every hot element gives out its own signature spectrum. Suddenly, this became a powerful tool to discover brand-new elements just by looking at their light. This is how we discovered helium for the very first time in the sun. In his honor, the unit for measuring these wavelengths was named the Ångström.

But nobody knew why these elements gave out those specific colors of light. The first clue actually came a few decades later from a Swiss math teacher named Johann Balmer. Balmer was obsessed with numbers and patterns, and he found a surprisingly simple formula for the hydrogen spectrum just by trial and error. The formula predicted there should be more spectral lines, and experiments actually confirmed them. But nobody had any clue what this formula meant. Why did it work?

To answer this, we needed a good theory of light. By now, we knew light is a wave, confirmed by interference patterns. Maxwell showed that light is basically a ripple in the electromagnetic field produced by accelerating charges. Wiggling charges produce light: wiggling slowly gives low-frequency light, and wiggling faster yields high-frequency light. These electromagnetic waves could wiggle other charges, enabling wireless communication.

It was a breakthrough in technology, but it couldn't explain the hydrogen spectrum. According to Maxwell, a hot glowing gas has billions of randomly jiggling charges across low to high frequencies, meaning they should emit every color of light continuously—a full rainbow. But they didn't. Something was horribly wrong with Maxwell's theory.

It got worse. Pretty soon we discovered that atoms had a positive nuclear core. We thought negative electrons must be orbiting this nucleus, making atoms stable. But if electrons orbit, they are constantly accelerating, and accelerating charges radiate electromagnetic waves. Therefore, orbiting electrons should continuously lose energy and collapse into the nucleus. Now we couldn't even explain why atoms were stable. Physics seemed to be in crisis.

Photons, Orbits, and Matter Waves

A few years later, everything changed. If you shine ultraviolet light on zinc, electrons come out. That makes sense: electrons receive energy from electromagnetic waves. But if you shine a much brighter visible light, no electrons come out. That didn't make any sense. Pumping in more energy should knock electrons out with more energy. Why did the color matter and not the brightness?

To explain this, Albert Einstein proposed something radical: what if light doesn't deliver energy continuously, but in discrete chunks called photons? And what if the energy of each photon depends only on its frequency ($E = h f$)?

Then a bright visible light delivers many photons per second, but each single photon is too weak to budge an electron—like throwing ping-pong balls at a bowling ball. On the other hand, a dim UV light delivers fewer photons per second, but each individual photon carries enough energy to knock an electron off like a cannonball. This explained the photoelectric mystery beautifully, earning Einstein the Nobel Prize. The constant $h$ is Planck's constant.

A few years later, Danish physicist Niels Bohr pushed this idea further. Bohr wondered: if light is absorbed in chunks, it must also be emitted in chunks. That meant electrons had to transition from a higher energy level straight to a lower energy level to release energy chunks. He postulated that electrons orbit the nucleus only at special, discrete energy levels—nowhere in between—and that in these special orbits, they do not radiate energy.

It seemed like he was just inventing rules. But look at what happens: when you heat up a gas, electrons jump to a higher allowed level. When they fall back down, they release the energy difference as a photon of a specific frequency. This explained the hydrogen spectrum. The photon's energy equals the energy lost by the electron. From this postulate, Bohr derived Balmer's formula, explaining the hydrogen spectrum and atomic stability all at once.

Later in an interview, Bohr remarked: "As soon as I saw Balmer's formula, the whole thing was immediately clear to me." But many unanswered questions remained. Why were electrons restricted to specific orbits? Why don't they radiate while sitting in those orbits? Bohr essentially said, "Trust me."

French PhD student Louis de Broglie came up with an answer by pushing Bohr's idea to its limit. Light moves as a wave but interacts like a particle (dual nature). What if matter behaves the exact same way? What if matter interacts like a particle, but moves as a wave?

If electrons inside an atom form standing waves—just like a guitar string vibrating with three, four, or five loops, but nothing in between—electron waves could only exist at specific discrete standing wavelengths. This explained why electrons exist only at specific distances from the nucleus. It also explained why they don't radiate energy while sitting in those levels: they aren't accelerating classical particles traveling in a circle; they are stationary standing waves. They only radiate a photon when transitioning between energy levels.

For his PhD thesis, de Broglie derived an expression for matter wavelength using special relativity ($\lambda = \frac{h}{p}$). He had found the wavelength of matter. His examiners only approved his thesis after confirming its validity with Einstein himself.

The following year, a professor at the University of Zurich gave a seminar on de Broglie's thesis. After the talk, a colleague in the audience asked: "If matter is a wave, where is the wave equation?" The professor was Erwin Schrödinger, and he took that question seriously. He went on vacation to the Swiss Alps over Christmas and returned with the wave equation.

Deconstructing the Derivation: Operators and Curvature

How did he do it? Schrödinger's original paper was notoriously difficult—he later called it "unintelligible." Feynman's derivation was also heavily mathematical. A modern, intuitive derivation begins with fundamental classical energy conservation:

$$\text{Total Energy } (E) = \text{Kinetic Energy } (K) + \text{Potential Energy } (V)$$

Since kinetic energy is $\frac{1}{2}mv^2$, multiplying top and bottom by $m$ gives $\frac{p^2}{2m}$:

$$E = \frac{p^2}{2m} + V$$

Standard textbook quantum mechanics then states: "Replace energy with the energy operator, and momentum with the momentum operator acting on the wave function $\psi$, giving Schrödinger's equation."

Why do we need operators instead of direct substitution? And how do we build these operators intuitively from scratch?

1. Why We Need Operators

We know $E = h f$ and $p = \frac{h}{\lambda}$. Why not substitute them directly into $E = \frac{p^2}{2m} + V$?

Direct substitution only works for an infinitely long, pure sine wave with a single, definite wavelength and frequency. A general matter wave (wave packet) contains a mixture across a whole spectrum of wavelengths and frequencies. This means a quantum particle generally does not possess a single definite momentum or energy value.

Because general matter waves don't possess single discrete values of energy or momentum, we cannot use simple numbers. We need mathematical operators that extract the full distribution of kinetic and total energy contained within the wave function.

2. Building the Operators Intuitively

To solve a complex problem, build a simpler version first. Let's construct operators for the simplest possible wave: a basic standing wave.

$$\psi(x,t) = A \sin(\kappa x) \cos(\omega t)$$

where spatial frequency $\kappa = \frac{2\pi}{\lambda}$ and temporal frequency $\omega = 2\pi f$.

Applying Einstein and de Broglie's equations:

• $E = h f = \left(\frac{h}{2\pi}\right) (2\pi f) \implies E = \hbar \omega$

• $p = \frac{h}{\lambda} = \left(\frac{h}{2\pi}\right) \left(\frac{2\pi}{\lambda}\right) \implies p = \hbar \kappa$

Here, $\hbar = \frac{h}{2\pi}$ is the reduced Planck constant.

Temporal frequency ($\omega$) encodes total energy in the time domain. Spatial frequency ($\kappa$) encodes momentum in the space domain. (In special relativity, space and time unify into four-space, while energy and momentum unify into four-momentum).

Now, how do we extract momentum squared ($p^2$) to get kinetic energy? We take spatial derivatives of $\psi$:

• First partial derivative with respect to $x$:

$$\frac{\partial \psi}{\partial x} = \kappa A \cos(\kappa x) \cos(\omega t)$$

• Second partial derivative with respect to $x$:

$$\frac{\partial^2 \psi}{\partial x^2} = -\kappa^2 A \sin(\kappa x) \cos(\omega t) = -\kappa^2 \psi$$

Since $p = \hbar \kappa$, we have $\kappa = \frac{p}{\hbar}$, so $\kappa^2 = \frac{p^2}{\hbar^2}$. Substituting this gives:

$$\frac{\partial^2 \psi}{\partial x^2} = -\frac{p^2}{\hbar^2} \psi \implies -\hbar^2 \frac{\partial^2 \psi}{\partial x^2} = p^2 \psi$$

Dividing by $2m$ yields kinetic energy ($K = \frac{p^2}{2m}$):

$$\hat{K}\psi = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}$$

This is the kinetic energy operator. The second derivative represents wave curvature. The equation states that kinetic energy is physically encoded in the spatial curvature of the matter wave. Shorter wavelengths mean higher curvature, higher momentum, and higher kinetic energy.

Fourier Transforms: Why Pure Sine Operators Work Generally

How do we know an operator derived for a pure sine wave works for arbitrary wave shapes?

French mathematician Joseph Fourier demonstrated that any arbitrary wave shape can be written as a linear sum of pure sine and cosine waves (Fourier series / Fourier transforms).

Because differentiation is a linear operation—$\frac{d}{dx}(a + b) = \frac{da}{dx} + \frac{db}{dx}$—applying our curvature operator to an arbitrary wave packet applies it linearly to every Fourier component. The operator extracts a weighted sum of kinetic energies across the full spectrum of the wave. Thus, an operator derived from a simple sine wave holds generally for all matter waves.

Deriving the Energy Operator and the Entry of $i$

Next, we must build the total energy operator by extracting energy ($E = \hbar \omega$) using a time derivative ($\frac{\partial}{\partial t}$).

Taking a partial derivative of $\psi(x,t) = A \sin(\kappa x) \sin(\omega t)$ with respect to time:

$$\frac{\partial \psi}{\partial t} = \omega A \sin(\kappa x) \cos(\omega t)$$

Notice the mathematical obstacle: the sine function turned into a cosine function. We did not recover our original wave function $\psi$, so we cannot write $\frac{\partial \psi}{\partial t} \propto E \psi$.

When building the momentum operator, taking a second derivative converted cosine back to negative sine, returning $\psi$. But here we need a first time derivative because we want linear energy ($E$), not squared energy ($E^2$).

To extract energy via a first derivative while returning the function itself, the time-dependent term must satisfy $\frac{d}{dt}f(t) \propto f(t)$. The only function in mathematics that equals its own derivative is the exponential function ($e^{t}$).

However, standard real exponentials ($e^{\alpha t}$ or $e^{-\alpha t}$) decay to zero or explode to infinity; they are not periodic waves. We face a strict mathematical and physical paradox:

• Physics requires a periodic time-oscillating wave.

• The first-derivative operator requirement forces an exponential function.

How can a function be simultaneously exponential and periodic?

Complex Exponents and the Argand Diagram

The solution came from Jean-Robert Argand. Real exponential growth occurs because derivative velocity points in the same direction as position, creating runaway growth. Real exponential decay occurs because velocity points in the opposite direction.

If velocity is constrained to be perpendicular ($90^\circ$) to position at all times, the magnitude of position never grows or shrinks; it continuously turns sideways in a circle. Perpendicular velocity transforms exponential growth/decay into uniform circular motion, producing periodic oscillation.

Multiplying an exponent by $+1$ represents a $0^\circ$ direction. Multiplying by $-1$ rotates direction by $180^\circ$. To rotate direction by $90^\circ$ (perpendicular), we must multiply by a factor $z$ such that multiplying twice rotates by $180^\circ$ ($-1$):

$$z \cdot z = -1 \implies z^2 = -1 \implies z = \sqrt{-1} = i$$

The imaginary unit $i$ acts as a $90^\circ$ rotation operator. Placing $i$ in the exponent ($e^{-i \omega t}$) turns exponential expansion/decay into uniform circular rotation in the complex plane (the Argand diagram). Euler's formula confirms this periodicity:

$$e^{-i \omega t} = \cos(\omega t) - i \sin(\omega t)$$

Using complex exponential matter waves $\psi(x,t) = A e^{i(\kappa x - \omega t)}$ reconciles both requirements:

• It is periodic, so physical wave properties are preserved.

• It is exponential, so first time derivatives return the original function.

Differentiating with respect to time:

$$\frac{\partial \psi}{\partial t} = -i \omega \psi$$

Since $E = \hbar \omega \implies \omega = \frac{E}{\hbar}$:

$$\frac{\partial \psi}{\partial t} = -i \frac{E}{\hbar} \psi \implies E \psi = -\frac{\hbar}{i} \frac{\partial \psi}{\partial t} = i \hbar \frac{\partial \psi}{\partial t}$$

This yields the total energy operator:

$$\hat{E} = i \hbar \frac{\partial}{\partial t}$$

Substituting the energy operator ($\hat{E}$) and kinetic operator ($\hat{K}$) into energy conservation ($E\psi = K\psi + V\psi$) gives Schrödinger's Equation:

$$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V \psi$$

Without $i$, this equation reduces to $\frac{\partial \psi}{\partial t} = D \frac{\partial^2 \psi}{\partial x^2}$—the classical Heat Equation governing exponential thermal diffusion. The imaginary unit $i$ converts real exponential thermal decay into complex periodic probability rotation.

The Physical Interpretation: Probability Conservation and the Born Rule

Is $i$ merely a mathematical convenience, or does it possess physical reality? Schrödinger himself spent months attempting to eliminate $i$, writing to Hendrik Lorentz: "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. $\psi$ is surely fundamentally a real function."

The physical meaning of $i$ was revealed in a footnote by Max Born (1926). Born established that $\psi$ is not a physical matter wave, but a complex probability amplitude. The probability $P$ of finding an electron at a location is proportional to the wave's intensity, given by its absolute magnitude squared:

$$P(x,t) = |\psi(x,t)|^2 = \psi^* \psi$$

If a matter wave were restricted to oscillating purely on the real axis (using real sine waves), $|\psi|^2$ would fluctuate periodically down to zero everywhere simultaneously. The total integrated probability of finding the particle somewhere in the universe would oscillate, periodically dropping to zero—an impossibility.

Because the matter wave rotates continuously in the complex plane ($e^{-i\omega t}$), its complex magnitude remains constant over time:

$$|\psi|^2 = \text{Re}(\psi)^2 + \text{Im}(\psi)^2 = \cos^2(\omega t) + \sin^2(\omega t) = 1$$

The imaginary unit $i$ causes the wave function to spin in the complex plane, which strictly conserves the total probability (100%) of the particle's existence across time. The imaginary number $i = \sqrt{-1}$ is what keeps physical probability real and conserved.

Legacy and Applications

Schrödinger solved his equation for the hydrogen atom using a Coulomb potential ($V(r) = -\frac{e^2}{4\pi\epsilon_0 r}$). The equation successfully derived the hydrogen spectrum, predicted spectral line intensities, accounted for spectral splitting under magnetic fields, and generated the three-dimensional atomic orbitals ($s, p, d, f$).

Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac. Today, the Schrödinger equation underpins modern technology:

• Electron Microscopes: Focuses matter waves to image individual atoms.

• Atomic Clocks: Calculates precision energy transitions in atoms, enabling satellite GPS.

• Semiconductor Physics: Explains electron energy band gaps in crystal lattices, allowing engineers to design tiny transistor switches inside modern microchips.

Our real modern world is brought to you by the imaginary number.

Summary

Introduction and the Dual Nature of "Now"

The concept of "now" represents one of the most fundamental yet elusive aspects of human existence. On one hand, the present moment encompasses everything humans can directly experience, influence, or act upon; memories of the past and predictions for the future exist solely within the current moment. On the other hand, attempting to pinpoint "now" in the physical world reveals it to be ephemeral and constantly slipping away.

Philosophers and scientists conceptualize this tension through two distinct lenses:

• The Outer Now: The external physical universe and objective events occurring in world-time.

• The Inner Now: Subjective awareness, individual perception, and the conscious experience of each moment.

Historical and cultural perspectives demonstrate that human understanding of "now" is far from universal:

• Philosophical Inquiries: Ancient Greek philosopher Heraclitus emphasized perpetual change, noting that one cannot step into the same river twice. St. Augustine highlighted the logical paradox of the present—if the present must become the past to be time, its existence relies on no longer being. 19th-century philosopher William James likened the experience of "now" to a rainbow over a waterfall: a stable quality unchanged by the fluid stream of events passing through it.

• Cultural Constructs of Time: Western culture largely views time as a spatial, left-to-right linear progression stretching from past to future, with "now" as a point moving along a line. Conversely, right-to-left language speakers visualize time in reverse, while non-written societies embed time within physical geography or natural cycles (e.g., uphill or sun movement). The Aymara people of Peru view themselves as stationary in time, with the known past in front of them and the unknown future behind. The Amondawa people of the Amazon operate without abstract concepts of time, timelines, or calendar tracking; for them, events occur within "now" rather than "now" being a point within time.

Physics and the "Outer Now": Relativity and the Block Universe

In classical Newtonian physics, the universe is modeled as a three-dimensional spatial grid governed by a universal master clock, allowing for a clear, absolute division between past, present, and future. However, modern physics dismantles this concept:

• Einstein’s Relativity: Albert Einstein demonstrated that space and time are intertwined into a dynamic four-dimensional structure called spacetime. Spacetime warps relative to an observer's frame of reference and speed. Consequently, simultaneity is relative; two events occurring at the same "now" for one observer may occur at different times for another, with no universal clock to determine absolute correctness.

• The Block Universe Model: The dominant model in modern cosmology views the cosmos as a static, four-dimensional block of spacetime containing all events across space and time simultaneously. In this model—analogous to a physical DVD containing an entire movie—past, present, and future exist equally and eternally. "Now" is not an objective physical event or a moving frontier; it is merely a subjective vantage point or perspective in how conscious observers read the static file.

• The Problem of Agency: In a deterministic Block Universe, physical events—including human choices and actions—are pre-written into spacetime. The intuitive feeling that the present moment is where choices are made to shape an open future is treated under this view as a psychological illusion.

Neuroscience, Psychology, and the "Inner Now": The Predictive Brain

Neuroscience and psychology reveal that the human experience of "now" is actively constructed by the brain rather than passively received from external reality:

• Sensory Delays and Processing: Neural signals require measurable time to travel from sensory organs to the brain and be processed. Furthermore, different sensory modalities travel at different speeds (e.g., light vs. sound). The brain artificially synchronizes these staggered inputs into a unified experience of "now."

• The Predictive Processing Model: Because processing delayed inputs would leave humans perpetually behind real-time events, the brain operates as an active prediction machine. Using past experiences, learned physical rules, and context, the brain projects a probabilistic prediction of what is happening in the current instant.

• Sports Example: A professional tennis player returning a 125 mph serve cannot react to real-time visual signals, as sensory lag would place their vision eight feet behind the ball's actual position. Instead, the brain calculates a real-time prediction of the ball's trajectory, allowing the athlete to track its present location.

• Illusion Example: Optical phenomena like the flash-lag effect demonstrate that we perceive neural predictions rather than raw sensory signals.

• The Temporal Structure of Lived Experience: Human temporal perception is organized into nested structures across multiple timescales:

• The Functional Moment (~50 milliseconds): The minimum threshold required to distinguish two sequential stimuli. Events closer than 50 ms are processed as simultaneous.

• The Experienced Moment (~3 seconds): The window across which the brain binds sequential inputs into coherent perceptual units (e.g., spoken sentences, musical phrases, spontaneous hugs, or short-term task execution).

• Long-Term Narrative Continuity: Longer temporal windows integrate memory, identity, ongoing goals, and emotional states, allowing moments to flow smoothly without abrupt resets.

• Embodied Perception: Perception requires physical action. The brain does not process reality in isolation; sensory intake relies on bodily probing and exploration (e.g., visual saccades, tactile movement, inhalation). Without physical movement, visual signals fade to gray and tactile perception ceases.

Reconciling Agency and Physics: Quantum Mechanics and QBism

To determine whether human choices genuinely affect outcomes or merely observe pre-written paths, theoretical physics explores interpretations beyond the traditional Block Universe:

• The Quantum Measurement Problem: In quantum mechanics, physical systems at atomic scales exist in superpositions of multiple possibilities. A definite state emerges only upon measurement or interaction, indicating that observation helps determine physical outcomes.

• Many-Worlds Interpretation: To preserve determinism without a universal "now," this interpretation posits that the universe constantly splits into parallel branches for every possible outcome. While it accounts for quantum equations, it retains a static multiverse structure (a vast collection of fixed DVDs) lacking a unique, meaningful present.

• QBism (Quantum Bayesianism): A radical, rigorous interpretation that rejects the concept of a single, pre-written objective universe ("deleting the DVD").

• QBism proposes that reality consists of a "community of living nows" or a pluriverse where individual perspectives interact directly within a shared system.

• The Jazz Improvisation Analogy: Rather than following a fixed score, reality operates like a jazz ensemble. There is underlying structure and continuity from the past, but the next moment remains genuinely open and undetermined until the participants act and play together in real time.

Practical Applications and Living in the Present

Understanding the cognitive and physical structure of "now" offers insights for daily life:

• Time Famine and Linear Metrics: Hyper-fixation on objective, linear clock time and extreme scheduling induces a psychological state known as "time famine"—a persistent sense of urgency that increases stress and leads individuals to sacrifice essential well-being practices (e.g., healthy eating, social connections, medical care).

• Cultivating Presence: Recognizing that each moment is an active, multi-layered synthesis of an entire lifetime’s worth of memories, predictions, and bodily actions transforms "now" from a fleeting metric into a rich domain to be inhabited.

• Participatory Reality: Drawing on physicist John Wheeler’s vision, the universe can be understood not as a static entity created in a singular past event, but as an ongoing participatory process continuously brought into existence through creative micro-interactions in every moment.

Transcript

Jo Marchant

Thank you so much for joining me on this search for "now."

We're all intimately familiar with the present, right? It's all around us. It's every moment of our lives. It's when stuff happens. It's when the future turns into the past. But what really is "now"? What I want to talk about tonight is why I think that "now" is so mysterious, and some of the ways that I think science can help us to make sense of it.

One way that we can answer the question, "What is now?" is that it's an instant, a point in time, a tick of the clock. When some of my friends heard that I was writing a book about "now," they said, "Oh, so you're writing a book about time." But then another friend said, "Oh, you're writing about mindfulness, presence." This reflects another way we can think about "now": as our awareness of each moment, our experience of the world.

We can think of these two contrasting features of "now" as the "outer now" and the "inner now." We have the external world with events happening out there, but we also have our personal world—how we are experiencing each moment. For me, this search for "now" is really about both of those things, and particularly how they fit together and relate to each other. It's really about how we relate to the world.

There is a tension there, and this is what attracted me to the subject. In one sense, "now" is everything to us. It is all that we can experience or influence. We can only live or act now. It contains our choices, our freedom in each moment. Even our memories of the past or our expectations and predictions for the future can only be experienced now. But on the other hand, if we look for "now" in the outside world, it's ephemeral. We can't pin it down. It's always slipping away and disappearing.

So, "now" is this really strange thing: it's everything and kind of nothing; it's everywhere and nowhere; it's always there, but always gone. It's crucial for how we live our lives, yet we don't fully understand what it is.

In writing my book, I wanted to see what science could tell us about "now." I interviewed cosmologists, neuroscientists, psychologists, quantum physicists, and philosophers—people with very different views—to see if we could put those perspectives together and get a better sense of what "now" really is. It's a journey that takes us through questions of consciousness, reality, perception, time, and the self.

Tonight, I'm going to talk first about the mystery of "now"—how people through history and around the world have made sense of it. Then we'll switch to the science. First, physics: how do physicists describe "now"? What do they see in the events of the universe? Then we'll look at our inner "now": where does each moment that we experience come from? Finally, we'll try to bring those two things together. What does all this mean for what "now" is, where "now" is, and what reality is? What is the role that we play in each moment? We'll finish with a couple of thoughts about how we can connect more with each moment and make the most of each "now."

Before I get into anything else, I just want to pause. I'm going to be quiet for a few breaths because I'd really like each of you to think about what is happening for you right here, right now. What is in this moment? What is this moment?

Being quiet feels strange; I don't think I can do it for any longer! I hope you can feel that "now" is not just a tiny point or instant. It feels immediate, but it is quite rich. It has layers and depth. We have the events happening on this stage and in this room, but your "now" also extends inwards to your body. Maybe you're feeling tired, hungry, or excited. Maybe you can feel your feet on the floor or your legs on the seat. That's part of your "now."

"Now" also extends outwards in space. We're in this room right now, but you also have a sense that we are at the Royal Institution, in London, on this planet, in this universe. This "now" wouldn't mean the same thing if I took all of that context away.

Ironically, "now" also feels like it extends forwards and backwards in time, because your previous experiences and memories shape what you're experiencing now. Maybe you've been to the Royal Institution before, or maybe this is your first time. Perhaps your experiences of science at school led you here or are coloring what you expect to happen. All of that shapes your "now." Looking to the future, what you think I'm going to say next, how long you think this event will last, or your thoughts about what you'll do later are all shaping this "now." The more we look at "now," the deeper and bigger it gets.

There's something else about "now" that you might have noticed: of course, this isn't the same "now" as it was when I started talking. There is a quality of "nowness" that stays constant and recognizable, but the contents of every moment are always changing.

Philosophers throughout history have tried to address these strange aspects of "now." In ancient Greece, Heraclitus focused on the ever-changing nature of the present, famously stating that we can never step into the same river twice because the waters and we are constantly changing. We are always becoming something new.

A few centuries later, St. Augustine pondered whether "now" is even a real entity. He wrote: "If the present, in order to be time, must go into the past, how can we say that a thing is, which can only be on the condition of no longer being?" He was pointing out the paradox that if the definition of "now" requires it to immediately cease being "now," it is difficult to define as a distinct thing.

I also really like a quote from William James, the 19th-century American philosopher. He addressed how "now" remains constant in quality while constantly changing in content, comparing our experience of "now" to "the rainbow on the waterfall, with its own quality unchanged by the events that stream through it."

It is not just philosophers who hold differing views on "now." Different cultures around the world have distinct ideas about how time passes, how it is structured, and how we relate to it. In the West, our view is heavily influenced by mathematics and science. We tend to view time as a simple line through space stretching from the past to the future, with "now" as a point moving along that line. We often imagine our lives similarly: walking a path from the past behind us to the future ahead, with "now" being our current location on that path.

Studies show that English speakers often visualize time as running from left to right. When I created my slides, I naturally placed the past on the left and the future on the right. It's easy to assume this conceptualization is natural and inevitable—that this is simply how time is. But different cultures experience time and "now" quite differently.

People who speak languages written from right to left, such as Hebrew or Arabic, often visualize time running from right to left. In societies without a strong written tradition, people may experience time as embedded in the physical environment; time might be conceptualized as running uphill or flowing from east to west with the sun.

The Aymara people of Peru have a fascinating perspective: they do not view themselves as moving through time. Instead, they feel stationary, viewing the known past as laid out in front of them and the unknown future as hidden behind them.

Some traditional societies go even further and do not associate time with spatial metaphors at all. They have no timelines. They experience time purely through change and the events occurring in each moment. For example, the Amondawa people of the Amazon do not use clocks, calendar days, weeks, or named seasons. The time they experience is grounded entirely in present changes—morning might be indicated by the sun rising, or afternoon by workers returning from the fields. They have no concept of abstract time independent of events, nor do they conceptualize life as a line through space. They do not track ages or birthdays, and they change their names at different life stages as their roles in society evolve. This aligns with Heraclitus's idea of the river: they are constantly becoming someone new. For the Amondawa, "now" is not a moving point within time; "now" is the container within which everything else, including change, occurs.

While that might seem like an unusual perspective, anthropologists suggest it was likely the default human view for most of history prior to the widespread adoption of mathematics, numbers, and precise timekeeping. It is worth keeping the Amondawa perspective in mind, as we can easily become attached to the concept of a linear timeline ticking by and assume it is an objective truth.

So, what can science tell us about "now"? In my investigation, I started with physics and the "outer now"—events in the physical world—expecting it to be the simpler part of the journey. If a clock ticks or I clap my hands, can we pinpoint the objective moment that occurs in the world? It turns out that is much harder than it sounds. In fact, it presents a major challenge to the search for "now," because most physicists would state that there is no special cosmic "now."

We might imagine the universe as a three-dimensional spatial grid with events progressing through time, governed by a single master clock universal to all locations. Under that model, distinguishing past, present, and future is straightforward: past events have already happened, future events have not yet happened, and "now" is the moving transition between them. This was Isaac Newton's view of the universe.

However, Albert Einstein's theory of relativity demonstrated that this model cannot be correct. In relativity, space and time are not separate entities; they are intertwined in a four-dimensional structure called spacetime. Spacetime can warp and morph depending on the observer's frame of reference. Consequently, "now" is relative. Two events that are simultaneous for one observer may occur at different times for another observer, and there is no hidden master clock to determine who is objectively correct. You cannot define a single state of what is happening across the entire universe "right now."

Because of this, Einstein remarked that the distinction between past, present, and future is a stubborn illusion—what he called the "baggage of consciousness." It cannot be found in the physical world.

Decades of experimental evidence support Einstein's predictions, leading to the dominant cosmological model known as the Block Universe. In this view, the cosmos is a static, four-dimensional block of spacetime. Time remains a variable—one of the four dimensions—allowing us to plot when events occur relative to one another. We can state that one event is in the past or future relative to another event, but there is no absolute dynamic "happening" or universal "now" within the block. Past, present, and future are all equally real and co-present.

This seems counterintuitive because we experience the past as fixed and unchangeable, the future as unwritten, and the present as the moment where outcomes are decided. But none of that dynamic transition exists within the Block Universe model. Astrophysicist Max Tegmark offered an analogy: if living life is like watching a movie, the Block Universe is the physical DVD. Drama unfolds as you watch the film, but the DVD itself is static and completely written. The whole narrative of the universe exists as a completed structure.

Under this view, "now" is not an objective feature of the physical world; it is a perspective held by an observer reading the DVD. Mainstream physics suggests that dynamic unfolding and the sensation of the present moment are not features of the external physical world, but rather arise from how we process reality.

If this unfolding occurs within us, how do we construct our individual experiences of "now"? What are we actually experiencing if there is no physical "now" out in the universe? Psychologists and neuroscientists agree with physicists that our experience of "nowness" does not correspond directly to external events, but is constructed internally.

We tend to feel as though our perception gives us a direct, real-time feed of reality—that hearing me speak feels like "now" because it is happening right now. However, your experience of each moment depends heavily on your brain's processing. A differently structured brain produces a different experience of "now."

For example, individuals with akinetopsia—the inability to perceive continuous motion—do not experience a smooth temporal flow. One documented case involved a woman whose perception progressed in static, discontinuous frames. While pouring tea, she saw the liquid frozen in mid-air, followed suddenly by the cup overflowing. When crossing the street, a car would appear far off and then instantly be right in front of her.

Even in typical, healthy perception, the brain performs extensive processing. Signals take time to travel from the environment to your sensory organs and brain. Hearing me speak takes hundreds of milliseconds to process, meaning your conscious perception lags slightly behind the event itself. Furthermore, different sensory inputs travel at different speeds. Light and sound from the same event—such as an opera singer on a distant stage—can arrive at your senses up to a fifth of a second apart. Yet, your brain unifies them into a single, synchronized "now." This demonstrates that the feeling of "nowness" is generated internally rather than directly mirrored from external inputs.

Historically, the conventional view held that sensory signals arrived passively at the brain, which then shuffled and aligned them to produce a coherent stream of experience. However, modern psychology and neuroscience indicate that if perception were purely passive processing of incoming data, our sensory lag would cause significant delays in real-time interaction.

Consider a photograph of Roger Federer returning a tennis serve at Wimbledon. The serve is recorded at 125 miles per hour, covering the length of the court in under half a second. Psychologists have calculated that in the time required for light to travel from the ball to Federer's eyes and be processed by his visual cortex, the ball travels eight feet. If Federer's conscious perception were based solely on the latest raw visual data received, he would be looking eight feet behind the ball's actual position. Yet, his gaze tracks the ball's actual location.

Researchers conclude that the brain does not passively wait for data; it actively anticipates and predicts. The brain functions as a prediction machine, continuously building a probabilistic model of what is happening in the current moment based on sensory input, context, and past experience.

We can observe this in optical illusions like the flash-lag effect, where a continuously moving bar appears to lead a flash that occurs at the exact same spatial position. Because the continuous movement is predictable, the brain projects the bar's position slightly forward to compensate for sensory lag. For the unpredictable flash, it cannot make that forward projection, causing the flash to appear to lag behind.

We perceive the brain's real-time prediction rather than raw, delayed sensory data. This predictive process is highly personalized. Federer's brain integrates incoming visual signals with extensive stored information: training regarding ball behavior on grass courts, knowledge of his opponent's serving patterns, and fundamental motor models developed since childhood.

Because our experience of "now" is a prediction generated by the brain, neuroscience arrives at a conclusion similar to physics: "now" is not an objective, universal instant. It is personal—less a single point in time and more a subjective point of view.

How does the brain structure each moment, and how long does a perceived moment last? In physics, time can be divided into tiny increments, such as Planck time (the shortest theoretical unit of time, approximately $5.39 \times 10^{-44}$ seconds) or the tick rates of optical atomic clocks. However, these timescales bear no relation to human experience. Neurons fire at a maximum rate of roughly one to two milliseconds—orders of magnitude slower than fundamental physical processes.

To determine the smallest interval of time humans can perceive, researchers conduct tests such as playing two distinct audio clicks into headphones and measuring the minimum separation needed to identify which sound occurred first. Across various senses and individuals, the threshold is approximately 50 milliseconds (a twentieth of a second). Inputs occurring closer together than 50 milliseconds cannot be sequentially ordered and are merged into a single perceptual event. Psychologists refer to this minimum window as the "functional moment."

However, human experience cannot consist merely of isolated 50-millisecond snapshots, as sensory integration across longer intervals is required to make sense of the world. The brain binds functional moments into broader windows of roughly three seconds, known as the "experienced moment."

This three-second window appears consistently across psychological research:

• Manual tasks like chopping, pouring, or peeling are naturally segmented into three-second operational units.

• Unrehearsed working memory holds novel information, such as phone numbers, for approximately three seconds before decay begins.

• Lines of poetry and spontaneous human hugs naturally average roughly three seconds in duration.

It appears the brain holds onto sensory input for roughly three seconds—about the length of a single breath—before updating its focus.

Furthermore, these three-second moments are integrated across much longer temporal spans. This broader integration sustains persistent beliefs, goals, emotional states, and our continuous narrative sense of identity and direction.

"Now" is not a fixed duration; it is a nested temporal structure wherein we experience changes across multiple timescales simultaneously. Fast-moving sensory details are tracked within short windows, while long-term concepts of self and memory remain stable across broader windows. Spontaneous neural activity in the brain reflects this multi-scale structure, with different processing networks operating concurrently across overlapping temporal scales. Longer-term expectations shape instantaneous sensory predictions, while immediate inputs feed back into broader mental states.

This explains why human temporal perception is flexible: time can feel as though it is dragging or racing, and two people experiencing the exact same event can perceive the passage of time differently. The linear, uniform ticking of a clock is a useful mathematical model, but it does not reflect how human awareness operates.

Additionally, cognitive science emphasizes that temporal perception is an embodied process. Sensory experience requires physical action; we do not passively receive reality, but actively engage with it. We move our eyes in saccades to construct visual scenes, move our fingers across surfaces to perceive texture, and inhale to smell. Experiments show that if a visual image is stabilized perfectly on the retina so that all eye movement is eliminated, the image rapidly fades to uniform gray. Perception relies on active bodily exploration of the environment.

This brings us back to a fundamental question: if our rich experience of "now" is an internal construct generated through active prediction, does it have any objective standing in physical reality? Do our perceptions and choices genuinely influence what happens next, or are we passive observers watching a deterministic "DVD" of spacetime?

In a strict Block Universe model, physical reality is fixed, and conscious choice is an illusion. If you stand at a fork in a path or inside a voting booth, your final action is ultimately determined by physical laws governing the particles in your body, leaving no room for alternative outcomes. Under that view, the subjective present is personally meaningful, but physically irrelevant to the unfolding of the cosmos.

However, many contemporary physicists question whether the Block Universe model is incomplete, exploring frameworks where the present moment is fundamental and physical outcomes remain genuinely open.

Quantum mechanics provides one such framework. At atomic scales, the way an experimenter chooses to measure a system influences the physical state observed—such as whether light manifests as a particle or a wave. Prior to measurement, particles exist in superpositions of multiple potential states. Taken at face value, this suggests that physical outcomes are not fully determined until an interaction occurs.

Physicists interpret these quantum results in different ways:

• Many-Worlds Interpretation: To preserve a deterministic model, this view posits that the universe constantly splits into parallel branches representing every possible outcome. Every potential result physically occurs in some branch of an ever-expanding multiverse. However, this doubles down on the static Block Universe model: every potential "DVD" exists simultaneously, leaving human choice without a singular, meaningful impact on an open future.

• QBism (Quantum Bayesianism): A radical, mathematically rigorous interpretation of quantum theory that rejects the concept of an objective, pre-written universe (effectively "deleting the DVD"). QBism holds that there is no single master version of reality independent of observers. Instead, physical reality is composed of a "community of living nows"—an interconnected system of interacting perspectives.

Under QBism, the universe is not a static structure, but an open-ended process analogous to jazz improvisation. In jazz, there is an established framework and history guiding the performance, but the music is created in real time through the interactions of the musicians. No player can know precisely what the next measure will sound like until it is played. Similarly, QBism suggests that the future is genuinely open, and outcomes are decided only as interactions occur in the present moment.

While physics continues to debate these interpretations, our human experience of "now" remains an active, creative process—a personal weaving of temporal scales.

Focusing exclusively on rigid, linear clock time narrows life to a sequence of metrics. Studies indicate that hyper-fixation on precise time management and deadlines induces "time famine"—a persistent state of feeling rushed and starved for time. This mindset often leads people to neglect activities essential to well-being, such as health, social connection, and reflection.

While clocks are necessary tools, there is value in stepping back from numerical time to focus on the flow of lived moments. Every instant we inhabit synthesizes a lifetime of accumulated memories, habits, expectations, and physical interactions.

The Japanese poet Matsuo Bashō captured the depth of a single focused moment in his famous haiku:

An old silent pond.

A frog jumps into the pond.

Splash.

Silence again.

A detail that might easily be overlooked becomes profound when brought into conscious awareness, illustrating how much meaning a single moment can hold.

"Now" exists in our active engagement with the world. We are not passive observers sealed off from reality; we actively participate in constructing our experience and determining where to direct our attention.

Physicist John Wheeler once proposed that we might view the universe not as something created in a single Big Bang in the distant past, but as a reality continuously brought into being through countless creative interactions occurring in every moment. That is a compelling way to understand the nature of "now."

Thank you.

Summary

Core Thesis: What $c$ Actually Means

The conventional characterization of $c$ as the "speed of light" is one of the most pervasive misinterpretations in modern science education. While the underlying equations of physics are correct, describing $c$ as a speed—analogous to a car driving down a highway—fundamentally misrepresents the nature of reality.

In theoretical physics, $c$ is not a speed, nor is it inherently a property of light. Rather, $c$ is a fundamental, dimensionless geometric conversion factor—an "exchange rate"—between the dimensions of space and time, as well as between mass and energy.

The Artifact of Units and the Geometry of Spacetime

• Dimensional Unit Dependence: The numerical value commonly associated with $c$ ($300,000\text{ km/s}$ or $186,000\text{ miles/s}$) is a human artifact resulting from arbitrary historical definitions of meters, miles, and seconds.

• Natural Units: In natural unit systems—such as measuring distance in light-years and time in years—$c$ equals exactly $1$. It becomes a pure, dimensionless ratio without units.

• Unification of Space and Time: Special relativity demonstrates that space and time are not distinct entities, but components of a single four-dimensional continuum (spacetime). The constant $c$ establishes how much space corresponds to how much time (e.g., $1\text{ second} = 300,000\text{ kilometers}$).

• Biological Asymmetry vs. Physical Reality: Humans perceive space and time differently because human biology allows free movement across three spatial directions while dragging perception along a single temporal trajectory. Fundamentally, however, space and time are made of the same substrate, unified by $c$.

The Constant Budget of Four-Dimensional Motion

In four-dimensional spacetime, every object, particle, and field moves through spacetime at a combined total rate that is always fixed at $c$. How an entity experiences time and space depends on how this constant motion budget is allocated:

• Stationary Objects in Space: An object at rest relative to an observer directs 100% of its spacetime motion through the time direction (moving through time at rate $c$) and 0% through space.

• Moving Objects (Time Dilation): As an object gains speed through space, motion must be diverted from the time direction to preserve the constant total spacetime budget of $c$. Moving faster through space directly reduces an object's rate of motion through time.

• Massless Entities (Photons): Objects with zero rest mass possess no inertia resisting spatial motion. Consequently, all of their 4D motion budget is allocated to the spatial directions ($c$ in space), leaving zero motion for the time direction. Because massless particles move through time at a rate of zero, they do not experience time, age, or endure temporal duration between emission and absorption.

Geometric Constraint vs. Highway Speed Limit

• Logical Impossibility: The prohibition against exceeding $c$ through space is not an arbitrary physical limit enforced by external forces. It is a geometric constraint akin to the rule that an interior angle of a right triangle cannot exceed 90 degrees while remaining a right triangle.

• Causality Protection: Exceeding $c$ through space would require allocating a negative value to temporal motion (moving backward in time), which would break causality—the requirement that causes precede effects.

• Incoherence of FTL Travel: Moving "faster than light" through space is not merely technologically difficult; it is logically incoherent within spacetime geometry, equivalent to searching for a direction that is simultaneously North and South.

• Wormholes and Warp Drives: Theoretical concepts like warp drives or wormholes do not bypass $c$ by moving faster through local space. Instead, they propose altering the geometry of spacetime itself (folding space). Doing so requires hypothetical "exotic matter" (negative energy density), which has no macroscopic physical reality.

Why $c$ is Independent of Light

• Historical Origins: The constant $c$ is named after light only because electromagnetic radiation was the first massless phenomenon humans studied. James Clerk Maxwell derived the speed of electromagnetic wave propagation from electrical and magnetic constants in the 1860s, matching optical measurements.

• Fundamental Spacetime Constant: Einstein recognized that $c$ is a property of the spacetime fabric itself. Even in a hypothetical universe devoid of photons or electromagnetism, $c$ would still exist as the geometric conversion factor between space and time, and any other massless particle would travel at rate $c$.

Unit Conversion in Mass-Energy Equivalence ($E = mc^2$)

In Einstein’s famous equation $E = mc^2$, $c^2$ does not signify light or velocity. It functions strictly as a unit conversion factor that demonstrates mass and energy are identical physical quantities measured in different units (converting kilograms into joules).

The Fine-Tuning Problem and Cosmological Implications

• Sensitivity of Physical Laws: The specific numerical value of $c$ relative to human scales dictates the fine-structure constant, atomic diameters, chemical bonding strengths, and stellar nuclear fusion rates.

• Origins of the Constant: Physics has not yet definitively explained why $c$ holds its specific value. Potential explanations include an undiscovered deeper unified field theory (where $c$ is derived from first principles) or a multiverse landscape wherein $c$ varies across universes, with human existence selected by anthropic constraints.

Summary Conclusion

$c$ is the fundamental signature and fingerprint of Minkowski spacetime geometry. It dictates the trade-offs between space and time, mass and energy, and the mechanics of the universe. Recognizing $c$ as geometry rather than speed resolves relativistic paradoxes, transforming relativity from a set of counterintuitive phenomena into a coherent four-dimensional structure.

Transcript

Let me tell you something that has bothered me for fifty years. Something that every physics textbook gets subtly, profoundly wrong. Not wrong in the equations. The equations are fine. Wrong in the interpretation. Wrong in what they tell you it means.

We call it the speed of light. We teach it as the speed of light. We have built an entire civilization of scientific communication around the phrase "the speed of light." And that phrase, that single innocent phrase, is one of the most misleading things in the history of science.

$c$ is not a speed. Not in the way you think speed means. Not in the way your car has a speed. Not in the way a baseball has a speed. $c$ is something deeper, something stranger, something so fundamental that calling it a speed is like calling gravity a push. Technically you can make an argument, but you've missed the entire point.

I'm Leonard Susskind. I've spent my career inside the mathematics of spacetime, quantum fields, black holes, the structure of reality at its most basic level. And I want to tell you what $c$ actually is. Because once you understand it, really understand it, the universe stops looking like a place where things happen and starts looking like something else entirely—something that has no good name in ordinary language.

The Problem with Units

Let's start with what you think you know. You were taught that light travels at approximately 300,000 kilometers per second in a vacuum. You were taught that nothing can go faster than this. You were taught that Einstein discovered this limit and built his theory of relativity around it. And you probably walked away thinking, "Okay, the universe has a speed limit, like a cosmic highway with a maximum velocity, and light is just fast enough to hit that limit perfectly."

That story is not wrong, exactly. But it is deeply, catastrophically incomplete.

Here is the first thing that should disturb you: the number 300,000 kilometers per second is not a fundamental fact about nature. It is an artifact of how we chose to measure things. If we measured distance in miles, $c$ would be 186,000 miles per second. If we measured distance in light-years and time in years, $c$ would be exactly 1. Just the number 1. No units. Just 1.

That last one is the important one. Because physicists who work in the right unit system don't think of $c$ as a large number. They think of it as 1. A pure, dimensionless, unit-free fact about the universe. Not a speed—a ratio. A conversion factor.

A conversion factor between what? Between space and time.

The Geometry of Spacetime and the Exchange Rate

This is where everything changes. When Einstein wrote down special relativity in 1905, he wasn't fundamentally talking about light. He was talking about the geometry of spacetime. He was discovering that space and time are not two separate things; they are two aspects of a single four-dimensional structure. A manifold. A fabric. Call it what you want, but the key is this: space and time are made of the same stuff, and $c$ is simply the exchange rate between them.

Think about currency. If you're traveling between two countries, there's an exchange rate that converts dollars to euros. That exchange rate is not itself a dollar or a euro. It's a relationship. A ratio. It tells you how much of one thing equals how much of another thing.

$c$ does the same thing for space and time. It tells you how much space equals how much time. One second of time equals 300,000 kilometers of space. That's what $c$ means. Not that light is fast, but that space and time are related by this ratio.

And here is the truly disorienting implication: if $c$ is just a conversion factor between space and time, then in some sense space and time are the same dimension measured in different units. The only reason we invented two different words—space and time—is that we evolved brains that experience them differently. We can move freely through space in three directions. We seem to be carried in one direction through time. This asymmetry in our experience made us think they were fundamentally different things. They are not. They are unified. And $c$ is the unification constant.

The Four-Dimensional Motion Budget

Now let me push this further, because there is a deeper strangeness here that almost nobody talks about. If $c$ is a conversion factor and not a speed, why does light specifically travel at exactly $c$? Why not some other speed? Why does light hit the cosmic maximum perfectly every single time, in every direction, in every vacuum, without exception?

The answer will sound too simple. It will sound like a trick. But it is not a trick. It is one of the most profound facts in physics: Light travels at $c$ because light has no mass. And massless things don't have a choice.

Here is what I mean. In spacetime, everything that exists is moving through the four-dimensional structure at all times. Not just moving through space—moving through space and time combined. And there is a rule built into the geometry of spacetime about how this four-dimensional motion works: Every object, every particle, every field, moves through spacetime at a total combined rate that is always exactly $c$. Always. Without exception.

But here is the crucial part: the way that total motion is distributed between space and time depends on mass.

When you are sitting still in your chair, you are not moving through space at all. But you are moving through time. And your rate of motion through time is exactly $c$. All of your spacetime motion is in the time direction; none of it is in a space direction.

When you start moving through space, something has to give, because your total four-dimensional speed must remain $c$. So as you gain speed through space, you lose speed through time. This is not a metaphor. This is the literal geometric mechanism behind time dilation. The faster you move through space, the slower you move through time. The two are trading off. They must trade off, because $c$ is fixed.

Now push this to the extreme. Imagine an object with zero mass. An object with no mass has no resistance to being pushed to higher and higher space-speeds. So it gets pushed all the way—all the way to $c$ in the space direction. Which means it has zero speed left in the time direction. It is moving entirely through space and not at all through time.

This is why photons don't age. This is why, from a photon's perspective, no time passes during its journey across the universe. It has traded all of its time-motion for space-motion. It has hit the geometric limit. It is traveling at $c$ through space because it has nothing left to give to the time direction.

A Geometric Constraint, Not a Highway Limit

$c$ is not a speed limit. It is a geometric constraint. It is the total budget of four-dimensional motion that every object in the universe is allocated, and that budget is $c$. You cannot exceed $c$ in the space direction because exceeding $c$ would require borrowing from a time budget that doesn't exist. It would require negative motion through time. It would require going backward in time just to maintain the geometry. And causality—the requirement that causes precede effects—makes this impossible in a self-consistent universe.

So the speed of light is not a speed limit the way a government posts a speed limit on a highway. It's more like the constraint that an angle cannot exceed 90 degrees while still being an angle in a right triangle. It's not enforced by a cop; it's enforced by the logic of the structure itself.

Why $c$ Has Nothing to Do with Light

Let me now tell you the part that most people never hear: $c$ is not even really about light. $c$ would exist in the universe even if there were no such thing as light. $c$ would be the geometric conversion constant between space and time even in a universe with no photons, no electromagnetic field, no light of any kind.

Any massless particle—any particle with no mass whatsoever—would travel at $c$. It has no choice. The geometry demands it.

We call it the speed of light for purely historical reasons. Because light was the first massless thing we studied. Because James Clerk Maxwell worked out the equations of electromagnetism in the 1860s and found that electromagnetic waves propagate at a specific speed. And when he calculated that speed from the properties of electricity and magnetism, it came out matching the measured speed of light exactly. Which told him that light was an electromagnetic wave.

And then Einstein came along and realized that $c$ wasn't a property of light or even of electromagnetism. It was a property of spacetime itself—the geometry of the universe, the conversion factor between its dimensions. But the name stuck: "the speed of light." Even though it was never fundamentally about light.

There is a way of writing physics where $c$ never appears at all, where you choose your units so that $c = 1$, and then every equation becomes cleaner, every relationship becomes more transparent, and light disappears from the story entirely. What's left is just geometry. Pure, clean, four-dimensional geometry. And the geometry tells you everything.

Faster-Than-Light Travel and Geometry

I want to dwell on this for a moment longer because the implications are genuinely staggering, and most people rush past them.

If $c$ is a geometric conversion factor and not a speed, then faster-than-light travel is not forbidden the way speeding is forbidden on a highway. It is forbidden the way drawing a square circle is forbidden. It is not a rule; it is a geometric impossibility. The geometry of spacetime simply does not contain the category of "things moving faster than $c$ through space." There is no slot in the structure for such a thing. It would be like asking for a direction that is simultaneously North and South. The question isn't illegal—it's incoherent.

Wormholes and warp drives, those beloved staples of science fiction, don't get around this by going faster than $c$. They attempt to get around it by changing the geometry itself: by bending spacetime so that two distant points become locally close, by folding the fabric so the gap disappears.

This is technically not forbidden by the $c$ constraint, because you're not moving through space at $c$ plus something; you're changing what space means in that region. But here is the brutal reality: to do this in general relativity requires something called exotic matter—matter with negative energy density. We have never observed such a thing. The quantum vacuum produces something that looks superficially similar in the Casimir effect, but the numbers don't come close to what you would need to hold a macroscopic wormhole open. Not even in the same universe of possibility.

So while the equations permit wormholes as mathematical solutions, actually creating one remains, as far as we understand, physically out of reach. The geometry permits the idea; nature refuses to provide the tools.

Notice something beautiful and terrible about this: The very fact that $c$ is a geometric constraint rather than a speed limit means that the prohibition on faster-than-light travel is deeper than we usually describe it. It doesn't matter how advanced your technology becomes. It doesn't matter how much energy your civilization can harness. You are not fighting against a limit that better engineering might overcome. You are fighting against the shape of spacetime itself. The universe is not saying "no" because you haven't tried hard enough; the universe is saying "no" because the concept you're reaching for doesn't fit inside the structure of reality.

The Fine-Tuning Problem

Now let me go even deeper. Because there is a question that bothers physicists in a way they don't always admit publicly: Why is $c$ the value it is? Why 300,000 kilometers per second? Why not twice that? Why not half? What determined this particular exchange rate between space and time?

The honest answer is: we don't fully know.

We know that if $c$ were different, the universe would be profoundly different. The fine-structure constant, which governs how strongly light interacts with matter, depends on $c$. If $c$ were significantly different, atoms would have different sizes, chemical bonds would have different strengths, and stars would burn at different rates. The universe as we know it—with its particular chemistry, its particular stars, and its particular possibility of life—would not exist.

This is what physicists call a fine-tuning problem. The constants of nature appear to be tuned to values that allow for complexity, for structure, for us.

Some physicists think there is a deeper theory that will explain where $c$ comes from: a Theory of Everything that derives $c$ from first principles, a theory where $c$ is not an input but an output, where the geometry of spacetime is itself explained rather than assumed. We don't have that theory yet.

Others think we live in a multiverse, a landscape of possible universes with different values of the constants, and we find ourselves in one with this particular value of $c$ simply because this is the value that allows us to exist. No deeper explanation—just selection.

I have gone back and forth on this for decades. I helped develop the string theory landscape, which is one version of the multiverse idea. I find it intellectually uncomfortable, but possibly correct. The universe doesn't owe us an explanation of its constants. It just has them.

Practical Implications and Mass-Energy Equivalence

Here is what I want you to take away from all of this: When you look at a beam of light, you are not looking at something traveling fast. You are looking at geometry in motion. You are looking at a thing with no mass and therefore no choice but to move through space at the full geometric budget of the universe. You are looking at the conversion factor between time and space made visible.

When GPS satellites need to correct for the fact that clocks run faster in orbit than on the ground, they are correcting for the geometry of spacetime—for the fact that moving through space trades off against moving through time at an exchange rate of exactly $c$. This is not an abstract theoretical nicety; it is a practical engineering reality. Without accounting for $c$ as a geometric conversion factor, GPS would drift by kilometers within hours.

When physicists write $E = mc^2$, the $c^2$ is not telling you about light. It is telling you that energy and mass are the same thing, expressed in different units, and that $c^2$ is the conversion factor between them—between mass-units and energy-units, between a kilogram and a joule. $c$ is doing unit conversion, not speed.

Misleading Names in Science

This is what $c$ really is. Not a cosmic speed limit. Not a property of light. A geometric fact about spacetime. The exchange rate between dimensions. The conversion constant that tells you how much space equals how much time, how much mass equals how much energy, and how fast a massless particle must travel because geometry leaves it no choice.

We gave it the wrong name. We called it the speed of light because that's how we stumbled onto it—historically, accidentally. The way humans often stumble onto deep truths: not by deduction from first principles, but by tripping over them in the dark and then slowly, painfully, recognizing what they actually found.

What we found was the geometry of the universe. And that geometry is stranger than any speed. It is stranger than any limit. It is the structure inside which space and time and mass and energy and everything we have ever observed are all embedded, all unified, all connected by a single number that we happen to call $c$. Not because it is the speed of light, but because it is the shape of everything.

There is one more thing I want to say before I close, because it connects to something deeply human. We measure things by giving them names. We name them after the first context in which we noticed them:

• The speed of light.

• The force of gravity.

• The laws of physics.

Each of these names carries a shadow of the original confusion, the original limited perspective from which we stumbled onto something much bigger than we realized at the time.

Gravity is not really a force; it is the curvature of spacetime. But we called it a force because that's what it felt like to Newton when the apple fell. We have been dragging that misleading name around for centuries.

The laws of physics are not really laws; they are descriptions of regularities in a structure we don't fully understand. But we called them laws because that's the language of authority and certainty that 17th-century scientists reached for when they wanted to sound like they knew what they were talking about.

And $c$ is not really a speed; it is the geometric signature of the spacetime we inhabit. But we called it the speed of light because James Clerk Maxwell computed it from electromagnetic theory and it matched the measured velocity of light, and that felt like a sufficient description at the time.

The names we give things shape how we think about them. The name "speed of light" makes you imagine something zooming through space very quickly. It puts your intuition in the wrong place. It makes you think the mystery is about velocity rather than about geometry.

Once you see it as geometry, everything reorganizes:

• Time dilation is not a paradox; it is a geometric consequence.

• Lorentz contraction of objects at high speeds is not a physical compression; it is a geometric rotation in spacetime.

• $E = mc^2$ is not a mysterious formula about nuclear explosions; it is a statement about two different ways of measuring the same geometric quantity.

All of it falls into place when you understand that $c$ is not a speed, but is instead the conversion factor between the dimensions of a unified structure: the number that tells space how much it equals in time, and the number that tells mass how much it equals in energy. The exchange rate of reality.

The Geometry We Inhabit

Here is something I find quietly extraordinary about this fact: The universe did not have to be this way. You can write down mathematically consistent geometries where space and time do not unify, where there is no $c$, where the exchange rate between dimensions simply does not exist. In such a universe, mass could not convert to energy, massless particles would not be constrained to a particular velocity, and time and space would be forever separate. The universe would be utterly unlike ours.

The fact that our universe has a $c$—a single clean conversion constant connecting its dimensions—means our universe has a particular kind of geometry: Minkowski geometry, named after the mathematician who first wrote it down clearly. It is a geometry with a very specific symmetry between certain spatial directions and the time direction.

That symmetry is why the universe looks the same to all observers regardless of how fast they're moving—why the laws of physics don't change whether you're in a car, standing still, or orbiting in a satellite. The symmetry is in the geometry, and $c$ is its numerical signature.

We did not choose this geometry. We were born into it. We evolved inside it. Our brains are shaped by it without knowing it. When you throw a ball and intuitively know where it will land, you are doing geometry in your head—geometry of a spacetime with a specific $c$, even though you have never consciously thought about any of this.

$c$ is in you, too. Your atoms are held together by electromagnetic forces that propagate at $c$. The nuclear reactions in the sun that produce the light hitting your face right now run at rates determined by $c$. The information in your neurons travels at speeds far below $c$, but within a universe whose structure $c$ defines. You are not just observing a universe shaped by $c$; you are made of processes that happen within the structure $c$ describes.

And light—the thing we named $c$ after—is simply what happens when you have a massless excitation of the electromagnetic field: a ripple in a field with no mass, constrained by geometry to travel at the only rate a massless thing can travel.

It is beautiful and it is strange, and it has been misnamed for a century and a half.

I have spent fifty years inside the mathematics of this structure, and I still find it astonishing. Not the number itself—the number is just a conversion factor. What is astonishing is that space and time can be converted into each other at all; that the universe is built from a single unified fabric rather than two separate stages; and that the fact we experience time as different from space is a feature of our biology and not a feature of reality.

Reality is a four-dimensional geometry. $c$ is its signature, its fingerprint—the number that tells you what kind of geometry you're living in. And light, massless and eternal from its own perspective, moving through space at the full geometric rate because it has no mass to slow it down, is just the most visible consequence of that geometry. The messenger that carries the news of $c$ across the cosmos.

It is not moving fast. It is moving at the only rate the geometry allows for something with no mass. And that, finally, is what $c$ actually is.

Not a speed. Never was.

Summary

"More Is Different" — P. W. Anderson

P. W. Anderson's 1972 article is a foundational critique of the "constructionist" hypothesis, arguing that the reductionist assumption (that everything is governed by a small set of fundamental physical laws) does not imply that we can easily reconstruct the universe from those laws. Instead, Anderson posits that at each level of scale and complexity, entirely new, fundamental properties and laws emerge, meaning "more is different."

1. The Distinction Between Reductionism and Constructionism

• The Reductionist Hypothesis: Widely accepted by scientists, this is the idea that all matter (animate and inanimate) is ultimately governed by the same fundamental physical laws.

• The Constructionist Hypothesis: The mistaken corollary that if everything obeys the same laws, then the only truly fundamental scientists are those working on those basic laws (such as particle physicists and logicians). Anderson rejects this, arguing that the ability to reduce everything to simple laws does not allow us to start from those laws and reconstruct the universe. The more we understand fundamental laws, the less direct relevance they seem to have to the complex problems of the rest of science or society.

2. The Hierarchical Structure of Science

Rather than science being a flat field where everything is "applied physics," Anderson structures the sciences into a linear hierarchy where the elementary entities of one level (Science X) obey the laws of the level below it (Science Y). However, Science X is never "just applied Science Y." Each stage requires entirely new concepts, creative inspiration, and distinct generalizations.

• Elementary particle physics underpins many-body physics.

• Many-body physics underpins chemistry.

• Chemistry underpins molecular biology.

• Molecular biology underpins cell biology.

• Physiology underpins psychology.

• Psychology underpins the social sciences.

3. Broken Symmetry as the Mechanism of Emergence

In many-body physics, the transition from quantitative to qualitative change is explained by the theory of broken symmetry. Symmetry in physics refers to a system appearing identical from different viewpoints (e.g., spatial homogeneity). "Broken symmetry" occurs when the ground state or actual physical state of a system has less symmetry than the fundamental laws governing it.

• The Ammonia Molecule ($NH_3$): Individually, it is a pyramid with an electric dipole moment. However, due to rapid quantum mechanical tunneling (inversion at $3 \times 10^{10}$ Hz), its stationary state is a symmetrical superposition of both pyramid states, resulting in a net-zero dipole moment in accordance with spatial symmetry laws.

• Heavier Molecules: In heavier molecules like phosphorus trifluoride ($PF_3$) or complex organic molecules like sugar, the inversion rate drops to zero due to increased mass. Parity symmetry is effectively broken; sugar molecules produced by living systems are stably spiral (chiral) and do not invert.

• Macroscopic Aggregates (Crystals, Ferroelectrics, and Superconductors): In large systems, the system seeks its lowest-energy state by breaking symmetry. A crystal breaks the continuous translational symmetry of empty space to form a discrete lattice, creating macroscopic "rigidity." Superconductivity and superfluidity are spectacular macroscopic quantum-interference phenomena resulting from broken gauge symmetry, where the system behaves rigidly to maintain specific internal energy relations.

4. The $N \to \infty$ (Infinite-Body) Limit

A rigorous definition of emergent properties (like the shape of a nucleus or the rigidity of a crystal) is only possible mathematically in the thermodynamic limit where the number of particles ($N$) approaches infinity. In finite systems, these behaviors are approximations of macroscopic behaviors. Consequently, trying to compute these properties from first principles using a computer would require solving an impossible infinite-body problem and then scaling it back down to a finite system.

5. Higher Stages of Complexity and Information

As we go further up the hierarchy, symmetry-breaking transitions into increasingly complex forms:

• Information-Bearing Crystallinity: Structures that are regular but contain variable, information-bearing elements (e.g., DNA, film strips, or printed text).

• Temporal Regularity: Regular pulsing or periodicity in the time dimension, which is ubiquitous in life. It serves as a mechanism for extracting environmental energy (via oscillators/generators) and as a means of processing information (e.g., spoken language, computer processors, and cell development).

• Functional/Teleological Structures and Specialization: These represent higher-order steps in complexity where it is more appropriate to speak of "increasing complication" rather than "decreasing symmetry."

6. Interdisciplinary Cooperation and Scientific Arrogance

Anderson cautions against scientific isolationism ("cultivating our own valley") and instead champions building roads between fields. While the path from a higher level of science to a lower one (analysis) is incredibly fruitful (e.g., reducing genetics to biochemistry), the reverse path (synthesis) is rarely possible. He criticizes the arrogance of some molecular biologists who attempt to reduce complex human behaviors—from mental illness to the religious instinct—solely to chemistry, emphasizing that the human organism contains more organizational levels between DNA and ethology than exist between DNA and quantum electrodynamics.

Transcript

More Is Different | 10.1126/science.177.4047.393

4 August 1972, Volume 177, Number 4047

More Is Different

Broken symmetry and the nature of the hierarchical structure of science.

P. W. Anderson

The author is a member of the technical staff of the Bell Telephone Laboratories, Murray Hill, New Jersey 07974, and visiting professor of theoretical physics at Cavendish Laboratory, Cambridge, England. This article is an expanded version of a Regents' Lecture given in 1967 at the University of California, La Jolla.

The reductionist hypothesis may still be a topic for controversy among philosophers, but among the great majority of active scientists I think it is accepted without question. The workings of our minds and bodies, and of all the animate or inanimate matter of which we have any detailed knowledge, are assumed to be controlled by the same set of fundamental laws, which except under certain extreme conditions we feel we know pretty well.

It seems inevitable to go on uncritically to what appears at first sight to be an obvious corollary of reductionism: that if everything obeys the same fundamental laws, then the only scientists who are studying anything really fundamental are those who are working on those laws. In practice, that amounts to some astrophysicists, some elementary particle physicists, some logicians and other mathematicians, and few others. This point of view, which it is the main purpose of this article to oppose, is expressed in a rather well-known passage by Weisskopf (1):

Looking at the development of science in the Twentieth Century one can distinguish two trends, which I will call "intensive" and "extensive" research, lacking a better terminology. In short: intensive research goes for the fundamental laws, extensive research goes for the explanation of phenomena in terms of known fundamental laws. As always, distinctions of this kind are not unambiguous, but they are clear in most cases. Solid state physics, plasma physics, and perhaps also biology are extensive. High energy physics and a good part of nuclear physics are intensive. There is always much less intensive research going on than extensive. Once new fundamental laws are discovered, a large and ever increasing activity begins in order to apply the discoveries to hitherto unexplained phenomena. Thus, there are two dimensions to basic research. The frontier of science extends all along a long line from the newest and most modern intensive research, over the extensive research recently spawned by the intensive research of yesterday, to the broad and well developed web of extensive research activities based on intensive research of past decades.

The effectiveness of this message may be indicated by the fact that I heard it quoted recently by a leader in the field of materials science, who urged the participants at a meeting dedicated to "fundamental problems in condensed matter physics" to accept that there were few or no such problems and that nothing was left but extensive science, which he seemed to equate with device engineering.

The main fallacy in this kind of thinking is that the reductionist hypothesis does not by any means imply a "constructionist" one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe. In fact, the more the elementary particle physicists tell us about the nature of the fundamental laws, the less relevance they seem to have to the very real problems of the rest of science, much less to those of society.

The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other. That is, it seems to me that one may array the sciences roughly linearly in a hierarchy, according to the idea: The elementary entities of science $X$ obey the laws of science $Y$.

Science $X$Science $Y$solid state or many-body physicselementary particle physicschemistrymany-body physicsmolecular biologychemistrycell biologymolecular biologypsychologyphysiologysocial sciencespsychologyBut this hierarchy does not imply that science $X$ is "just applied $Y$." At each stage entirely new laws, concepts, and generalizations are necessary, requiring inspiration and creativity to just as great a degree as in the previous one. Psychology is not applied biology, nor is biology applied chemistry.

In my own field of many-body physics, we are, perhaps, closer to our fundamental, intensive underpinnings than in any other science in which non-trivial complexities occur, and as a result we have begun to formulate a general theory of just how this shift from quantitative to qualitative differentiation takes place. This formulation, called the theory of "broken symmetry," may be of help in making more generally clear the breakdown of the constructionist converse of reductionism. I will give an elementary and incomplete explanation of these ideas, and then go on to some more general speculative comments about analogies at other levels and about similar phenomena.

Before beginning this I wish to sort out two possible sources of misunderstanding. First, when I speak of scale change causing fundamental change I do not mean the rather well-understood idea that phenomena at a new scale may obey actually different fundamental laws—as, for example, general relativity is required on the cosmological scale and quantum mechanics on the atomic. I think it will be accepted that all ordinary matter obeys simple electrodynamics and quantum theory, and that really covers most of what I shall discuss. (As I said, we must all start with reductionism, which I fully accept.) A second source of confusion may be the fact that the concept of broken symmetry has been borrowed by the elementary particle physicists, but their use of the term is strictly an analogy, whether a deep or a specious one remaining to be understood.

Let me then start my discussion with an example on the simplest possible level, a natural one for me because I worked with it when I was a graduate student: the ammonia molecule. At that time everyone knew about ammonia and used it to calibrate his theory or his apparatus, and I was no exception. The chemists will tell you that ammonia "is" a triangular pyramid:

with the nitrogen negatively charged and the hydrogens positively charged, so that it has an electric dipole moment ($\mu$), negative toward the apex of the pyramid. Now this seemed very strange to me, because I was just being taught that nothing has an electric dipole moment. The professor was really proving that no nucleus has a dipole moment, because he was teaching nuclear physics, but as his arguments were based on the symmetry of space and time they should have been correct in general.

I soon learned that, in fact, they were correct (or perhaps it would be more accurate to say not incorrect) because he had been careful to say that no stationary state of a system (that is, one which does not change in time) has an electric dipole moment. If ammonia starts out from the above unsymmetrical state, it will not stay in it very long. By means of quantum mechanical tunneling, the nitrogen can leak through the triangle of hydrogens to the other side, turning the pyramid inside out, and, in fact, it can do so very rapidly. This is the so-called "inversion," which occurs at a frequency of about $3 \times 10^{10}$ per second. A truly stationary state can only be an equal superposition of the unsymmetrical pyramid and its inverse. That mixture does not have a dipole moment. (I warn the reader again that I am greatly oversimplifying and refer him to the textbooks for details.)

I will not go through the proof, but the result is that the state of the system, if it is to be stationary, must always have the same symmetry as the laws of motion which govern it. A reason may be put very simply: In quantum mechanics there is always a way, unless symmetry forbids, to get from one state to another. Thus, if we start from any one unsymmetrical state, the system will make transitions to others, so only by adding up all the possible unsymmetrical states in a symmetrical way can we get a stationary state. The symmetry involved in the case of ammonia is parity, the equivalence of left- and right-handed ways of looking at things. (The elementary particle experimentalists' discovery of certain violations of parity is not relevant to this question; those effects are too weak to affect ordinary matter.)

Having seen how the ammonia molecule satisfies our theorem that there is no dipole moment, we may look into other cases and, in particular, study progressively bigger systems to see whether the state and the symmetry are always related. There are other similar pyramidal molecules, made of heavier atoms. Hydrogen phosphide, $\text{PH}_3$, which is twice as heavy as ammonia, inverts, but at one-tenth the ammonia frequency. Phosphorus trifluoride, $\text{PF}_3$, in which the much heavier fluorine is substituted for hydrogen, is not observed to invert at a measurable rate, although theoretically one can be sure that a state prepared in one orientation would invert in a reasonable time.

We may then go on to more complicated molecules, such as sugar, with about 40 atoms. For these it no longer makes any sense to expect the molecule to invert itself. Every sugar molecule made by a living organism is spiral in the same sense, and they never invert, either by quantum mechanical tunneling or even under thermal agitation at normal temperatures. At this point we must forget about the possibility of inversion and ignore the parity symmetry: the symmetry laws have been, not repealed, but broken.

If, on the other hand, we synthesize our sugar molecules by a chemical reaction more or less in thermal equilibrium, we will find that there are not, on the average, more left- than right-handed ones or vice versa. In the absence of anything more complicated than a collection of free molecules, the symmetry laws are never broken, on the average. We needed living matter to produce an actual unsymmetry in the populations.

In really large, but still inanimate, aggregates of atoms, quite a different kind of broken symmetry can occur, again leading to a net dipole moment or to a net optical rotating power, or both. Many crystals have a net dipole moment in each elementary unit cell (pyroelectricity), and in some this moment can be reversed by an electric field (ferroelectricity). This asymmetry is a spontaneous effect of the crystal's seeking its lowest energy state. Of course, the state with the opposite moment also exists and has, by symmetry, just the same energy, but the system is so large that no thermal or quantum mechanical force can cause a conversion of one to the other in a finite time compared to, say, the age of the universe.

There are at least three inferences to be drawn from this. One is that symmetry is of great importance in physics. By symmetry we mean the existence of different viewpoints from which the system appears the same. It is only slightly overstating the case to say that physics is the study of symmetry. The first demonstration of the power of this idea may have been by Newton, who may have asked himself the question: What if the matter here in my hand obeys the same laws as that up in the sky—that is, what if space and matter are homogeneous and isotropic?

The second inference is that the internal structure of a piece of matter need not be symmetrical even if the total state of it is. I would challenge you to start from the fundamental laws of quantum mechanics and predict the ammonia inversion and its easily observable properties without going through the stage of using the unsymmetrical pyramidal structure, even though no "state" ever has that structure. It is fascinating that it was not until a couple of decades ago (2) that nuclear physicists stopped thinking of the nucleus as a featureless, symmetrical little ball and realized that while it really never has a dipole moment, it can become football-shaped or plate-shaped. This has observable consequences in the reactions and excitation spectra that are studied in nuclear physics, even though it is much more difficult to demonstrate directly than the ammonia inversion. In my opinion, whether or not one calls this intensive research, it is as fundamental in nature as many things one might so label. But it needed no new knowledge of fundamental laws and would have been extremely difficult to derive synthetically from those laws; it was simply an inspiration, based, to be sure, on everyday intuition, which suddenly fitted everything together.

The basic reason why this result would have been difficult to derive is an important one for our further thinking. If the nucleus is sufficiently small there is no real way to define its shape rigorously: Three or four or ten particles whirling about each other do not define a rotating "plate" or "football." It is only as the nucleus is considered to be a many-body system—in what is often called the $N \to \infty$ limit—that such behavior is rigorously definable. We say to ourselves: A macroscopic body of that shape would have such-and-such a spectrum of rotational and vibrational excitations, completely different in nature from those which would characterize a featureless system. When we see such a spectrum, even not so separated, and somewhat imperfect, we recognize that the nucleus is, after all, not macroscopic; it is merely approaching macroscopic behavior. Starting with the fundamental laws and a computer, we would have to do two impossible things—solve a problem with infinitely many bodies, and then apply the result to a finite system—before we synthesized this behavior.

A third insight is that the state of a really big system does not at all have to have the symmetry of the laws which govern it; in fact, it usually has less symmetry. The outstanding example of this is the crystal: Built from a substrate of atoms and space according to laws which express the perfect homogeneity of space, the crystal suddenly and unpredictably displays an entirely new and very beautiful symmetry. The general rule, however, even in the case of the crystal, is that the large system is less symmetrical than the underlying structure would suggest: Symmetrical as it is, a crystal is less symmetrical than perfect homogeneity.

Perhaps in the case of crystals this appears to be merely an exercise in confusion. The regularity of crystals could be deduced semiempirically in the mid-19th century without any complicated reasoning at all. But sometimes, as in the case of superconductivity, the new symmetry—now called broken symmetry because the original symmetry is no longer evident—may be of an entirely unexpected kind and extremely difficult to visualize. In the case of superconductivity, 30 years elapsed between the time when physicists were in possession of every fundamental law necessary for explaining it and the time when it was actually done.

The phenomenon of superconductivity is the most spectacular example of the broken symmetries which ordinary macroscopic bodies undergo, but it is of course not the only one. Antiferromagnets, ferroelectrics, liquid crystals, and matter in many other states obey a certain rather general scheme of rules and ideas, which some many-body theorists refer to under the general heading of broken symmetry. I shall not further discuss the history, but give a bibliography at the end of this article (3).

The essential idea is that in the so-called $N \to \infty$ limit of large systems (on our own, macroscopic scale) it is not only convenient but essential to realize that matter will undergo mathematically sharp, singular "phase transitions" to states in which the microscopic symmetries, and even the microscopic equations of motion, are in a sense violated. The symmetry leaves behind as its expression only certain characteristic behaviors, for instance, long-wavelength vibrations, of which the phantom example is sound waves; or the unusual macroscopic conduction phenomena of the superconductor; or, in a very deep analogy, the very rigidity of crystal lattices, and thus of most solid matter.

There is, of course, no question of the system's really violating, as opposed to breaking, the symmetry of space and time, but because its parts find it energetically more favorable to maintain certain fixed relationships with each other, the symmetry allows only the body as a whole to respond to external forces. This leads to a "rigidity," which is also an apt description of superconductivity and superfluidity in spite of their apparent "fluid" behavior. [In the former case, London noted this aspect very early (4).] Actually, for a hypothetical gaseous but intelligent citizen of Jupiter or of a hydrogen cloud somewhere in the galactic center, the properties of ordinary crystals might well be a more baffling and intriguing puzzle than those of superfluid helium.

I do not mean to give the impression that all is settled. For instance, I think there are still fascinating questions of principle about glasses and other amorphous phases, which may reveal even more complex types of behavior. Nevertheless, the role of this type of broken symmetry in the properties of inert but macroscopic material bodies is now understood, at least in principle. In this case we can see how the whole becomes not only more than but very different from the sum of its parts.

The next order of business logically is to ask whether an even more complete destruction of the fundamental symmetries of space and time is possible and whether new phenomena then arise, intrinsically different from the "simple" phase transition representing a condensation into a less symmetric state.

We have already excluded the apparently unsymmetric cases of liquids, gases, and glasses. (In any real sense they are more symmetric.) It seems to me that the next stage is to consider the system which is regular but contains information. That is, it is regular in space in some sense so that it can be "read out," but it contains elements which can be varied from one "cell" to the next. An obvious example is DNA; in everyday life, a line of type or a movie film have the same structure. This type of "information-bearing crystallinity" seems to be essential to life. Whether the development of life requires any further breaking of symmetry is by no means clear.

Keeping on with the attempt to characterize types of broken symmetry which occur in living things, I find that at least one further phenomenon seems to be identifiable and either universal or remarkably common, namely, ordering (regularity or periodicity) in the time dimension. A number of theories of life processes have appeared in which regular pulsing in time plays an important role: theories of development, of growth and growth limitation, and of the memory. Temporal regularity is very commonly observed in living objects. It plays at least two kinds of roles. First, most methods of extracting energy from the environment in order to set up a continuing, quasi-stable process involve time-periodic machines, such as oscillators and generators, and the processes of life work in the same way. Second, temporal regularity is a means of handling information, similar to information-bearing spatial regularity. Human spoken language is an example, and it is noteworthy that all computing machines use temporal pulsing. A possible third role is suggested in some of the theories mentioned above: the use of phase relationships of temporal pulses to handle information and control the growth and development of cells and organisms (5).

In some sense, structure—functional structure in a teleological sense, as opposed to mere crystalline shape—must also be considered a stage, possibly intermediate between crystallinity and information strings, in the hierarchy of broken symmetries.

To pile speculation on speculation, I would say that the next stage could be hierarchy or specialization of function, or both. At some point we have to stop talking about decreasing symmetry and start calling it increasing complication. Thus, with increasing complication at each stage, we go on up the hierarchy of the sciences. We expect to encounter fascinating and, I believe, very fundamental questions at each stage in fitting together less complicated pieces into the more complicated system and understanding the basically new types of behavior which can result.

There may well be no useful parallel to be drawn between the way in which complexity appears in the simplest cases of many-body theory and chemistry and the way it appears in the truly complex cultural and biological ones, except perhaps to say that, in general, the relationship between the system and its parts is intellectually a one-way street. Synthesis is expected to be all but impossible; analysis, on the other hand, may be not only possible but fruitful in all kinds of ways: Without an understanding of the broken symmetry in superconductivity, for instance, Josephson would probably not have discovered his effect. [Another name for the Josephson effect is "macroscopic quantum-interference phenomena": interference effects observed between macroscopic wave functions of electrons in superconductors, or of helium atoms in superfluid liquid helium. These phenomena have already enormously extended the accuracy of electromagnetic measurements, and can be expected to play a great role in future computers, among other possibilities, so that in the long run they may lead to some of the major technological achievements of this decade (6).]

For another example, biology has certainly taken on a whole new aspect from the reduction of genetics to biochemistry and biophysics, which will have untold consequences. So it is not true, as a recent article would have it (7), that we each should "cultivate our own valley, and not attempt to build roads over the mountain ranges ... between the sciences." Rather, we should recognize that such roads, while often the quickest shortcut to another part of our own science, are not visible from the viewpoint of one science alone.

The arrogance of the particle physicist and his intensive research may be behind us (the discoverer of the positron said "the rest is chemistry"), but we have yet to recover from that of some molecular biologists, who seem determined to try to reduce everything about the human organism to "only" chemistry, from the common cold and all mental disease to the religious instinct. Surely there are more levels of organization between human ethology and DNA than there are between DNA and quantum electrodynamics, and each level can require a whole new conceptual structure.

In closing, I offer two examples from economics of what I hope to have said. Marx said that quantitative differences become qualitative ones, but a dialogue in Paris in the 1920s sums it up even more clearly:

FITZGERALD: The rich are different from us.

HEMINGWAY: Yes, they have more money.

References

• V. F. Weisskopf, in Brookhaven Nat. Lab. Publ. 888T360 (1965). Also see Nuovo Cimento Suppl. Ser. 1 4, 465 (1966); Phys. Today 20 (No. 5), 23 (1967).

• A. Bohr and B. R. Mottelson, Kgl. Dan. Vidensk. Selsk. Mat. Fys. Medd. 27, 16 (1953).

• Broken symmetry and phase transitions: L. D. Landau, Phys. Z. Sowjetunion 11, 26, 542 (1937). Broken symmetry and collective motion, general: J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127, 965 (1962); P. W. Anderson, Concepts in Solids (Benjamin, New York, 1963), pp. 175-182; B. D. Josephson, thesis, Trinity College, Cambridge University (1962). Special cases: antiferromagnetism, P. W. Anderson, Phys. Rev. 86, 694 (1952); superconductivity, ibid. 110, 827 (1958); ibid. 112, 1900 (1958); Y. Nambu, ibid. 117, 648 (1960).

• F. London, Superfluids (Wiley, New York, 1950), vol. 1.

• M. H. Cohen, J. Theor. Biol. 31, 101 (1971).

• J. Clarke, Amer. J. Phys. 38, 1075 (1969); P. W. Anderson, Phys. Today 23 (No. 11), 23 (1970).

• A. B. Pippard, Reconciling Physics with Reality (Cambridge Univ. Press, London, 1972).

N-Body Simulator: A Deep Dive into Interactive Gravitational Physics

This document details an N-Body simulator, a program designed to visually and interactively demonstrate gravitational interactions between multiple bodies. The core functionality revolves around solving the N-Body problem, a classic challenge in physics concerning the prediction of motion for a system of celestial objects governed solely by Newtonian gravity. The simulator prioritizes accuracy, user interaction, and educational value, allowing users to explore complex gravitational scenarios with relative ease.

The simulator’s foundation lies in the numerical solution of Newton’s Law of Universal Gravitation and Newton’s Second Law of Motion. Instead of attempting analytical solutions (which are only possible for the two-body problem), the simulator employs a time-stepping method. This involves discretizing time into small intervals and calculating the gravitational force on each body at each time step. This force is then used to update the body’s velocity and position, effectively simulating its trajectory. The accuracy of the simulation is directly tied to the size of the time step; smaller time steps yield more accurate results but require greater computational resources.

Several integration methods are implemented to enhance accuracy and stability. The primary method is the Verlet integration scheme, known for its good energy conservation properties, crucial for long-term simulations. Verlet integration is symplectic, meaning it preserves the fundamental structure of Hamiltonian systems, minimizing energy drift over extended periods. However, the simulator also offers alternative integration methods, including the Euler method (simpler but less accurate, prone to energy drift) and the Runge-Kutta 4th order method (RK4, more accurate than Euler but computationally more expensive than Verlet). Users can select the integration method based on their desired balance between accuracy and performance.

The simulator allows for a high degree of user control over the simulation parameters. Users can define the number of bodies (N), ranging from two to potentially hundreds, although performance degrades with increasing N. For each body, users can specify initial position (x, y coordinates), initial velocity (vx, vy components), and mass. The gravitational constant (G) is also adjustable, allowing exploration of different gravitational strengths. Furthermore, users can modify the time step size, the integration method, and the simulation duration.

A key feature is the interactive nature of the simulation. Users can pause, resume, and reset the simulation at any time. They can also interact with individual bodies during a paused simulation, modifying their properties (position, velocity, mass) to observe the immediate effects on the system. This interactive capability is particularly valuable for educational purposes, allowing users to experiment with different scenarios and gain a deeper understanding of gravitational dynamics.

The visual representation of the simulation is designed for clarity and information density. Bodies are represented as points or small circles, with their size optionally scaled to reflect their mass. The simulation displays the trajectories of the bodies as lines, providing a visual record of their paths. A real-time display of simulation parameters, such as the current time, time step size, and total energy of the system, is also provided. The simulator includes options for adjusting the scale of the display, zooming in and out to focus on specific regions of the simulation. Color-coding of bodies is implemented to aid in distinguishing them, especially in simulations with a large number of bodies.

The simulator includes pre-defined scenarios to demonstrate various gravitational phenomena. These include:

• Two-Body Problem: Demonstrates stable orbits, elliptical paths, and the effects of varying masses and initial velocities.

• Three-Body Problem (Figure-Eight Solution): Illustrates a classic chaotic solution to the three-body problem, where three bodies of equal mass trace a figure-eight pattern.

• Alpha Centauri System: A simplified model of the Alpha Centauri star system, showcasing the gravitational interactions between multiple stars.

• Solar System Model: A scaled-down representation of our solar system, demonstrating the orbits of planets around the sun.

• Custom Scenarios: Users can create and save their own custom scenarios, allowing for exploration of arbitrary configurations of bodies.

Beyond the core simulation functionality, the simulator incorporates features for data logging and analysis. The simulator can record the position, velocity, and energy of each body at each time step, allowing users to export this data for further analysis using external tools. This capability is useful for investigating long-term trends, calculating orbital parameters, and verifying the accuracy of the simulation. The simulator also provides basic plotting capabilities, allowing users to visualize the trajectories of bodies and the evolution of energy over time.

The simulator’s development prioritizes performance optimization. The core simulation logic is implemented in a computationally efficient manner, leveraging optimized numerical algorithms and data structures. The visual rendering is also optimized to minimize overhead, allowing for smooth and responsive simulations even with a large number of bodies. The simulator is designed to be cross-platform, running on a variety of operating systems.

Future development plans include:

• Collision Detection: Implementing collision detection between bodies, allowing for realistic simulations of impacts and mergers.

• Relativistic Effects: Incorporating relativistic corrections to Newton’s Law of Gravitation, enabling simulations of strong gravitational fields.

• Advanced Visualization: Adding more sophisticated visualization options, such as 3D rendering and particle effects.

• User Interface Improvements: Enhancing the user interface to make the simulator more intuitive and user-friendly.

• Multi-threading: Utilizing multi-threading to further improve performance, especially for simulations with a large number of bodies.

In conclusion, the N-Body simulator is a powerful and versatile tool for exploring gravitational physics. Its combination of accuracy, interactivity, and educational features makes it valuable for students, researchers, and anyone interested in understanding the dynamics of celestial systems.

Molecular dynamics (MD) simulations investigating the long-debated phenomenon of low ice friction. The research challenges established theories and proposes a new primary mechanism for the formation of the lubricating interfacial water layer responsible for ice's slipperiness.

Introduction: Challenging Existing Theories

The low kinetic friction of ice is commonly attributed to a thin layer of liquid water at the sliding interface. For decades, the origin of this water at sub-zero temperatures has been explained by three main theories: pressure melting (high contact pressures lower the melting point), surface premelting (a quasi-liquid layer exists on ice surfaces even below 0°C), and frictional heating (sliding generates heat that melts the ice). However, each theory has significant limitations. Pressure melting requires unrealistically high pressures for common scenarios like skiing, while surface premelting cannot account for variations in friction with different materials. The leading theory, frictional heating, has also been questioned by experiments that failed to detect significant temperature increases at sliding interfaces. This suggests that a crucial mechanism for ice liquefaction has been overlooked. The authors propose that ice liquefies not through thermodynamic melting, but through a mechanical process called "cold, displacement-driven amorphization," a shear-induced disordering of the crystal structure.

The Mechanism of Displacement-Driven Amorphization

Using MD simulations with the accurate TIP4P/Ice water potential, the researchers first modeled an idealized, atomically flat ice-on-ice interface. They found that even under these perfect conditions, the system does not achieve "structural lubricity" (a state of ultra-low friction). Instead, upon contact, electrostatic interactions between the misaligned ice crystals create localized "cold-welded" spots.

When sliding begins, these spots act as anchor points, inducing plastic deformation in their vicinity. This shear stress does not create dislocations, as in metals, but rather triggers local instabilities that destroy the crystalline order, molecule by molecule. This process creates a disordered, amorphous layer at the interface. Structural analysis confirmed that this shear-induced layer closely resembles supercooled liquid water, notably being denser than crystalline ice.

Evidence Against Thermal Melting

The study provides compelling evidence that this amorphization is an athermal, mechanical process, distinct from melting. The key finding is that the thickness of the amorphous layer grows in proportion to the square root of the sliding distance. This relationship indicates that the process is displacement-driven: the probability of a surface molecule being dislodged from its lattice position is directly related to the distance slid, not the temperature.

Further evidence comes from simulations at different temperatures. Counterintuitively, the amorphization process was found to be significantly faster at 10 K (-263 °C) than at 250 K (-23 °C), and it occurred with only a negligible rise in local temperature. This directly contradicts the notion that frictional heat is the primary cause of liquefaction. The simulations also showed that tensile strain, often present at the trailing edge of a sliding contact, is a more effective driver of disordering than heat. Therefore, the difficulty of skiing at very low temperatures is not due to a lack of liquefaction—which actually occurs more readily—but rather to the extremely high viscosity of the resulting amorphous layer at those temperatures. While frictional heat is not the primary cause of the liquid layer, it does play a secondary role by reducing the layer's viscosity, which in turn lowers the shear stress and friction.

The Crucial Role of Counterbody Properties and Hydrophobicity

To simulate more realistic conditions involving surface roughness, the researchers modeled a rigid, corrugated indenter sliding over an ice surface. These simulations revealed that achieving the very low friction coefficients (e.g., below 0.1) associated with slippery ice depends critically on the properties of the counterbody, particularly its hydrophobicity.

When a hydrophilic (water-attracting) indenter was used, the friction was relatively high. In contrast, a hydrophobic (water-repelling) counterface reduced both the initial stiction force and the subsequent kinetic friction by approximately 50%. This significant reduction is attributed to two factors. First, the amorphous water layer can easily slip past the non-adhesive hydrophobic surface, a phenomenon known as finite slip length. Second, the hydrophobic surface minimizes adhesion-enhanced viscoelastic dissipation, which is energy lost as water molecules stick to and detach from the leading and trailing edges of the contact.

The study concludes that for ice to be truly slippery, two conditions must be met: 1) the formation of a self-lubricating, shear-induced amorphous water layer, and 2) a smooth, hydrophobic counterbody that allows this water layer to slip easily and minimizes capillary effects.

Conclusion and Implications

This research reframes the understanding of ice friction by identifying displacement-driven amorphization as the principal mechanism for creating a lubricating layer. This athermal process circumnavigates the need for thermodynamic melting. The established theories are not dismissed entirely but are re-contextualized: frictional heating primarily reduces the viscosity of the amorphous layer, while pressure gradients from roughness can enhance the mechanical amorphization process. Ultimately, the slipperiness of ice is a complex interplay between this shear-induced liquefaction and the interfacial properties of the sliding counterbody.

Animation vs Math

Animation vs Geometry

Animation vs Physics

Animation cs Coding

Researchers have captured and controlled quantum uncertainty in real time for the first time, fundamentally redefining the Heisenberg uncertainty principle as a dynamic, tunable property rather than a fixed limitation. This breakthrough, achieved with attosecond (10⁻¹⁸ seconds) precision, enables unprecedented observation and manipulation of quantum states as they evolve naturally, marking a major advance in ultrafast quantum optics.

Central to the achievement is the generation of ultrafast squeezed light pulses—quantum states where uncertainty is redistributed rather than eliminated. Unlike ordinary light, whose uncertainty is spread evenly between paired properties like phase and amplitude, squeezed light narrows uncertainty in one property at the cost of increasing it in the other. Researchers produced the shortest, most precisely controlled attosecond squeezed pulses to date by using nonlinear four-wave mixing in silicon dioxide combined with a custom light field synthesizer that combines multiple carefully phased spectral channels.

This experimental method allowed the team to dynamically switch between amplitude squeezing and phase squeezing in real time, showing that quantum uncertainty can be actively modulated rather than being a static bound. By splitting engineered waveforms into a classical reference and a squeezed beam and precisely measuring their intensity and phase fluctuations, the researchers quantified and controlled quantum noise below the standard quantum limit with attosecond temporal resolution.

The technological implications extend strongly into quantum communications, where the team demonstrated a petahertz-scale encryption protocol embedding information directly in the fluctuating quantum uncertainty patterns. This introduces a robust intrinsic security layer: eavesdropping disturbs the quantum state and also requires knowledge of a decoding key and exact pulse amplitude, making unauthorized interception detectable and decoding practically impossible. This promises ultrafast, highly secure data transfer networks leveraging quantum properties.

Further applications include enhanced quantum sensing, where tailored uncertainty control can improve measurement sensitivity beyond classical limits, enabling breakthroughs in navigation, environmental sensing, and medical diagnostics. The ability to manipulate quantum noise dynamically also points toward future quantum computing architectures operating at extreme speeds and precision, potentially at attosecond timescales.

This discovery, achieved through the intersection of nonlinear optics, laser physics, and quantum theory, transforms quantum uncertainty from a passive obstacle to an actively exploitable resource. It opens new frontiers in fundamental quantum physics and sets the stage for revolutionary quantum technologies far beyond existing capabilities. By bridging attosecond temporal control with quantum state engineering, the work marks a pivotal step toward harnessing the full potential of quantum mechanics in real-world applications and advanced scientific research.

This new approach to physics-informed neural networks (PINNs) enables robust solutions for stiff and high-dimensional differential equations by combining multi-head architectures and unimodular regularization. The multi-head strategy allows a single neural net to solve an entire family of equations simultaneously, improving generalization and handling noisy or sparse data. Unimodular regularization, leveraging ideas from differential geometry, stabilizes training and allows the system to efficiently uncover unknown or missing physical laws.

Applications are wide-ranging:

• Astrophysics and Relativity: Direct solution of Einstein Field Equations and the modeling of complex spacetime geometries.

• Climate Science: Modeling atmospheric dynamics and coupled climate models that involve many scales and stiff systems.

• Chemical Kinetics and Biology: Simulation and inference in biochemical networks, metabolic pathways, and reaction-diffusion systems with rapid and slow processes intertwined.

• Engineering: Fluid dynamics (including turbulent and reactive flows), aerodynamics, material deformation, and control systems where traditional solvers fail due to stiffness or sensitivity.

• Environmental Science: Predictive modeling for air pollution, PM2.5 evolution, and other multi-timescale diffusion-advection problems.

The net result: faster, more accurate, and more versatile model training and simulation for any field that relies on solving or inferring complex differential equations under challenging data or physical constraints.

Refs:

[1] EPINN: Physics-Informed Neural Network with exponential activation functions for solving stiff ODEs

• https://openreview.net/forum?id=SYiOxXWlKU

[2] Solving stiff ordinary differential equations using physics informed neural networks (PINNs): simple recipes to improve training of vanilla-PINNs

• https://arxiv.org/abs/2304.08289

[3] Stiff neural ordinary differential equations

• https://dspace.mit.edu/bitstream/handle/1721.1/138719/2103.15341.pdf?sequence=2&isAllowed=y

[4] Stabilize physics-informed neural networks for stiff differential equations: Re-spacing layer

• https://www.sciencedirect.com/science/article/abs/pii/S0898122125003955

[5] Mixing Differential Equations and Neural Networks for Physics-Informed Learning

• https://book.sciml.ai/notes/15-Mixing_Differential_Equations_and_Neural_Networks_for_Physics-Informed_Learning/

[6] Training stiff neural ordinary differential equations with implicit single-step methods

• https://pubs.aip.org/cha/article/34/12/123147/3325817/Training-stiff-neural-ordinary-differential

Ice can generate electricity in two ways: flexoelectricity, triggered when ice is bent or deformed, and ferroelectricity, present at the surface in extremely cold conditions. This explains how ice particles in thunderclouds become charged, revealing a likely mechanism behind lightning initiation. Ice’s electrical output, comparable to high-performance ceramics, enables potential uses in sensors and transducers, especially in harsh, cold environments where traditional electronics fail. These findings open up new technological possibilities and deepen our understanding of natural electrical phenomena in polar and stormy regions.

Unlike traditional spatial crystals such as diamonds, where atoms form repeating patterns in three-dimensional space, time crystals exhibit periodic motion in the temporal dimension.

By stacking multiple time crystal layers, engineers could potentially create unprecedented data storage systems that encode information in both spatial and temporal domains.

the quantum Bayes' rule is defined as the rule for updating quantum states using the principle of minimum change (maximizing fidelity), and is mathematically realized by the Petz recovery map in many situations

The Kepler Problem

The paper introduces a theoretical framework based on three-dimensional time, where the three temporal dimensions emerge from fundamental symmetry requirements. The necessity for exactly three temporal dimensions arises from observed quantum-classical-cosmological transitions that manifest at three distinct scales: quantum phenomena, interaction-scale processes, and cosmological evolution. These temporal scales directly generate three particle generations through eigenvalue equations of the temporal metric, naturally explaining both the number of generations and their mass hierarchy.

The framework proposes a metric structure with three temporal and three spatial dimensions, preserving causality and unitarity while extending standard quantum mechanics and field theory. While earlier work explored three-dimensional time in the context of Kaluza–Klein theory, this paper’s approach provides specific experimental predictions and a complete particle spectrum.

This approach offers elegant solutions to long-standing problems in particle physics: the three-generation structure emerges naturally from temporal symmetries, weak interaction parity violation arises from geometric properties, and quantum gravity achieves finite corrections without ultraviolet divergences.

The theory reproduces known particle properties and makes precise quantitative predictions, including neutrino masses, new resonances, and modifications to gravitational wave propagation. These signatures are expected to be testable through next-generation collider experiments, gravitational wave observatories, and cosmological surveys in the 2025–2030 timeframe. Notably, General Relativity emerges as a natural limiting case when two temporal dimensions become negligible.