Theories of Everything with Curt Jaimungal - The Death of Inflation. The Rise of Simplicity | Latham Boyle

Episode Date: September 18, 2024

Welcome to Theories of Everything's "Rethinking the Foundations of the Physics: What is Unification?" series featuring Latham Boyle. Latham Boyle is a theoretical physicist known for his work on cosmology, quantum gravity, and the early universe, particularly in the context of exploring new models of the Big Bang and time symmetry. SPONSOR: As a listener of TOE, you can now enjoy full digital access to The Economist. Get a 20% off discount by visiting: https://www.economist.com/toe LINKS MENTIONED: - Neil Turok on TOE: https://www.youtube.com/watch?v=ZUp9x44N3uE - Neil Turok’s lecture on TOE: https://www.youtube.com/watch?v=-gwhqmPqRl4 - Latham's paper on the Primordial Power Spectrum: https://arxiv.org/pdf/2302.00344v1 - A Model of Leptons (paper): https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.19.1264 - The dominant model of the universe is creaking (article): https://www.economist.com/science-and-technology/2024/06/19/the-dominant-model-of-the-universe-is-creaking - Introduction to Axiomatic Quantum Field Theory (book): https://archive.org/details/introductiontoax0000nnbo/mode/2up TOE'S TOP LINKS: - Support TOE on Patreon: https://patreon.com/curtjaimungal (early access to ad-free audio episodes!) - Listen to TOE on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Become a YouTube Member Here: https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Join TOE's Newsletter 'TOEmail' at https://www.curtjaimungal.org SPONSORS (please check them out to support TOE): - THE ECONOMIST: As a listener of TOE, you can now enjoy full digital access to The Economist. Get a 20% off discount by visiting: https://www.economist.com/toe - INDEED: Get your jobs more visibility at https://indeed.com/theories ($75 credit to book your job visibility) - HELLOFRESH: For FREE breakfast for life go to https://www.HelloFresh.com/freetheoriesofeverything Other Links: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 - Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything #science #physics #podcast #universe #theoreticalphysics #theory #bigbang #singularity Learn more about your ad choices. Visit megaphone.fm/adchoices

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We want to take a very minimal approach. We would like to see where we can get by just taking the solutions of the known laws of physics seriously. It's a very elegant, minimal proposal. Inflation is assumed by virtually every model of physics, from string theory to the standard model of cosmology. What if it's wrong? What if the laws of physics are already staring us in the face?

By taking the known laws of physics to their logical extreme, Professor Latham Boyle and Neil Turok have uncovered a CPT symmetric universe that could explain dark matter. The dark matter has this particular mass of 4.8 x 10 to the 8 giga electron volts. The arrow of 4.8 times 10 to the 8 giga electron volts. The arrow of time. In other words, there's just one space in the space-time where you're really pinned to a state of low entropy. And even the properties of the most elusive particle, the neutrino. It has to be the case that the lightest neutrino is massless.

All without invoking cosmic inflation. Boyle, from the Perimeter Institute Institute has been developing this revolutionary minimalistic model of the early universe with Neil Turok, who's been interviewed on this podcast before and links in the description, a model that completely violates the consensus of ever more mathematical complexity in physics, angering his peers. The consequence? What if the Big Bang was actually a cosmic mirror? Professor, welcome. I appreciate you coming on and sharing your potential theory of everything.

Thanks so much, Kurt, for having me. And yeah, I appreciate you having me on. I appreciate you having me on. So, okay, so yeah, I wanted to explain this cosmological model, a new picture of the early universe or really of the universe as a whole that I've been working on for the past several years with my collaborator, Neil Turok. The more we've understood it, we stumbled upon it about six years ago now, and then the more we've understood it, we stumbled upon it about six years ago now, and then the more

we've understood it over the subsequent years, the more excited we've gotten about it. And I want to sketch the reasons. So my title is the CPT Symmetric Universe, or in other words, what is the simplicity of the early universe trying to tell us? And okay, I'll just dive in. Well, maybe actually before I dive in, I'll just say somehow that the gist of the idea is that we want to take a very minimal approach, a very minimalistic philosophy where we really take seriously the laws of physics as we currently understand them, the standard model of particle physics and Einstein's theory of gravity gravity and not try to speculate about new particles or forces or extra dimensions or anything like that. If we're not forced to, we would like to see as far as possible where we can get by

just taking the solutions of the known laws of physics seriously and by taking them seriously, as seriously as possible, I mean thinking about their so-called maximal analytic extensions and then if those maximal analytic extensions have interesting properties, if they have interesting symmetries, if they have interesting structures of other sort that I'll explain, that instead of disregarding those as maybe mathematical artifacts, we want to take them seriously as potential hints about the universe and the laws of physics. And then by following those hints, we'll see that we get led down a sort of exciting path. We end up coming across new explanations for a number of observed phenomena in the universe

and new predictions for future experiments. That's what I will try to explain. Great. And I have a quick question, if you don't mind. Yeah. You mentioned that you want to try to not introduce any particles unnecessarily or add dimensions unnecessarily unless you're forced to. Now people who are string theorists or other theorists who do introduce new particles or new dimensions or even reduce the amount of dimensions, they tend to claim that it is a constraint of the theory

itself. So they're forced to. So can you please explain why is it that they're led down an alleyway that they feel as if they're being led by gunpoint? And you also are saying I'm only going to go down this alleyway if it's by gunpoint, but there is no gun. Yeah, good question. So I mean, I guess in the case of string theory, it is the case that if you take as a fundamental principle that the basic objects in the universe are strings, described by the super string action, that does force you, it seems, to ten dimensions of space-time. And let's say that they are forced to that because they think the theory has many beautiful properties, beautiful mathematical properties let's

say, and bears some very intriguing relationships to the known world. You know it provides a quantum theory of gravity, that's the idea that from the string theorist standpoint. But as an explanation of the world as the way it is, a reality as we observe it, I'm not sure that it has really borne out very successfully in terms of the number of things that it explains about the real world versus the number of problematic things that it introduces in terms of explaining the real world. So I personally am more inclined to, I heard, this is a bit of a tangent here, but I mean I remember hearing several years ago, a number of years ago now, on the 50th anniversary

of the discovery of the Standard Model, so I guess the Standard Model was discovered in 19, maybe Weinberg's paper was 1967, and I heard a talk by Weinberg on the 50th anniversary, and he gave a summary of the history of the discovery of the model. And it was this great sort of detective story where a large number of people over decades were sort of putting together, they were presented with a whole bunch of different bits of evidence, and were sort of putting together, they were presented with a whole bunch of different bits of evidence and were sort of gradually putting together a more and more elegant and explanatory account of the evidence at hand until they finally arrived at the full-fledged standard model sometime in the mid-1970s, I guess. And so I like that kind of detective story, detective putting together an account of the facts about the universe as they are feeling.

And so I'm kind of going for that rather than getting particularly wedded to one particular theory from the get-go. So I guess my philosophy is that the known laws of physics are very well tested at the moment, so I don't want to diverge from them until... I want to see how much I can explain with them, let's put it that way. I think we haven't gotten to the bottom of figuring out how much of the universe can be explained just with the known laws of physics without having to introduce new stuff. And then once we've decided that we've gotten to the end of that process, maybe we'll be forced to add all sorts of new ingredients,

lots of new particles or extra dimensions. Maybe ultimately we'll be forced to that, and maybe I'm just been slow to see it basically. But to me it doesn't look like we are forced to. And yeah, so I'm hoping to find a kind of more explanatory path by following this more minimalistic trajectory. Maybe that brings me to the next point, which is I was going to say it another way. Great. Which is that you had suggested that I

should touch on the question of what is unification. And I wanted to modify that question slightly by discussing what is a good explanation. I think of an explanation as a kind of machine here. So it's a kind of machine that takes some inputs at the bottom here, the green inputs, some assumptions. Here I'm feeding in three assumptions, A1, A2, and A3.

And then those assumptions, combined with whatever principles make up the explanation, the explanation is some combination of ideas and principles, let's say. the explanation is some combination of ideas and principles, let's say, those things together yield some outputs, some facts that some explanations about the world. And if some of those outputs just contradict what we know about the world, well, we throw away that explanation. But if they all agree with the world that we see, okay, then that's a good candidate explanation. But then there's a sort of constant competition between different competing explanations of the world

and basically the name of the game in science or just more generally in any kind of process of improving understanding, you know, not just in science, in all branches of human inquiry. You know, new explanations can come along. People are constantly guessing new explanations, and when a new competing explanation comes along, you can check, is it better or worse than the previous one. And so, you know, I've just illustrated the two basic ways

that an explanation could be better. You know, here I have a new explanation E prime over here, which is better in the sense that, well, it requires the same basic assumptions A1, A2, and A3 about the world, but it successfully explains now more output explanations. It successfully explains five things about the world instead of the previous three.

So that's what one would call a broader explanation, more outputs for the same number of inputs. And then on the right, there's another explanation, which I've called E double prime, and that has a different sort of inputs. And then on the right, there's another explanation, which I've called E double prime, and that has a different sort of advantage. It still only explains the same three outputs as the original explanation, but it now only requires one of the original three assumptions. So it does so based on fewer assumptions. And so that's what I would call a deeper explanation. Now for people who know what partially ordered sets are, this looks like an anti-click, the broader and deeper,

because you can't compare them. So to be more specific, what I'm saying is that the top one is less wanted than the bottom deeper or the bottom broader one. That the deeper one is better than the top and the broader one is better than the top. But can you compare the broader versus the deeper in this instant? I think you're right. Yes. No, I don't. Well, according to the way I've set it up,

no, I think you're absolutely right that it's more like a judgment call. If you just had two explanations, one of which was broader and one of which was deeper, then I'm not sure how you would choose between them. Interesting. Now in practice, I know of no reason that this has to be the case, but in practice, I mean, I guess what often happens is that I was trying to sort of separate these into two different ways an explanation could improve. But in practice, typically progress involves moving to an explanation that is both broader and deeper. So somehow if you end up finding a new explanation that explains all the previous outputs, but with fewer assumptions.

Well, that usually means that it's actually closer to reality. And as a consequence of being closer to reality, often it explains more about reality than the previous guy did. So maybe in some sense, you might expect broadness and deepness to sort of come part and parcel

with one another, but I don't know any general reason that has to be the case. Yeah, good point. Not sure how much seriousness or rigor, with how much rigor one should view this picture, but I'm just trying to convey schematically what one could hope for in a better explanation. Understood.

And yeah, and then I think of unification as a particular type of improvement of this sort where you might have two explanations, E1 and E2, and then find a, you know, where E1 explains, E1 might be, you know, the theory of electricity, and it explains certain things about the world, and then E2 might be the theory of magnetism, and it explains other things about the world.

But then the more unified theory of electromagnetism ends up explaining everything that both of those previous theories did. And so is a much broader theory than either of the previous ones were. So what I want to try to explain is a theory of the early universe that I think that we increasingly think seems to explain,

have a lot more explanatory power than the standard or conventional theory of the early universe, the inflationary model, which I learned as a student and was learned by decades of students before me and since. Yeah, so I want to basically introduce you to the picture and then try to go through some of the things, some of the explanations that it gives for different aspects of the observed universe. Great.

OK, so let's start with this cartoon picture of the observed universe. Great. Okay, so let's start with this cartoon picture of the expanding universe. So this is, so here time is running up the page in this cartoon. Actually, it's the conformal time tau. It's a cosmologists like to use this time coordinate, the conformal time coordinate. Okay, and so the idea is that at the top of the,

this is supposed to show the universe expanding. So as the universe gets older, as we go further up the slide, the circle, the cross-section of this cone or lampshade grows. So that's showing that the universe is expanding. And so then we're here at the top. This is the late universe, the top of the lampshade, and we look back with our telescopes back toward the beginning and we can see very, very far back, but not all the way back to the bang.

At some point further down past where we can see, if we continue to extrapolate, the cone would come down to a point, which is the big bang. But before that happens, the universe becomes so dense that it becomes opaque and we just can't see any further. And so that surface is the surface shown at the bottom here, which is the...

I've put in a little shrunken down picture of the cosmic microwave background as taken by the 2018 Planck satellite data. What we learn from these observations, so we look back as far as we can toward the Big Bang, can't see all the way back, but the basic story that we learn from our observations is that the further back we look toward the Big Bang, the simpler the universe got. And in particular, by the time we get back to a few hundred thousand years after the Big Bang, when the cosmic microwave background was released, which we can see very directly, or more indirectly, we know from the cosmic microwave background and another type of observations about the relative,

called involving the relative abundances of the light nuclei in the periodic table. We know that as we go all the way back to a fraction of a second after the Big Bang, that the universe was incredibly simple, basically simple, as simple as can be in every way that we have been able to imagine. Okay, so here I've sort of summarized on the right-hand side of the slide all the

different ways in which this initial slice, this initial snapshot, fraction of a second after the bang, was ultra simple. So first of all, to a first approximation, the universe was maximally symmetric, completely homogeneous and isotropic. So that means it was basically, if you went to any different point along this three-dimensional snapshot in time, any point in space at this moment in time, they all looked identical to one another. And if you stood at any point in space at this moment in time, they all looked identical to one another, and if you stood at any point in space and turned and looked in any direction, all the different directions looked identical to one another.

So that's called being homogeneous and isotropic. And then, so the universe was maximally symmetric in that sense. And then on top of that, actually there were tiny little fluctuations, tiny little perturbations in the temperature and density as you went from point to point. They're small, they're only about a few parts in 100,000. So these are these tiny perturbations. They appear to be completely random.

But they're random, again, in the simplest possible way. So they're drawn from a probability distribution that, first of all, respects the maximal symmetry of the background. So although the perturbations fluctuate from point to point, they appear to have been drawn from a probability distribution that's again exactly homogeneous and isotropic, as far as we can tell.

And they're purely scalar, or in other words, purely density type perturbations. We don't see any evidence for vorticity or primordial gravitational waves, which are the other two types of more complicated perturbations that could have been present. As I mentioned, as I just mentioned, they're statistically symmetric. They also are statistically Gaussian or basically drawn from a normal distribution, so the simplest

type of probability distribution. They have another simple property called adiabaticity, which maybe I won't get into the details of what that means. And their correlations, the correlations between different points are described as far as we can tell by a pure power law. It's a nearly scale invariant power law. I'll explain more of that later.

But, and then finally, a very important point for us is that another fact is that the density perturbations, they oscillate, the density waves all have their phases synchronized and they're synchronized to a very particular value so that if you follow them, if you imagine following them all back to the Big Bang, they would all be right at the maximum of their oscillation, right, as you got back to the Big Bang, they would all be right at the maximum of their oscillation right as you got back to the Big Bang.

So that's exactly how they would be if the Big Bang was a mirror. And well, we are going to, that's not usually how things, that's not usually how they're described, we'll get there. But this is, as we'll see, this is one of the key hints that lead us to a picture in which we do take the big bang

to be a mirror. Okay, so just, can you go back to the slide that you were on? Yes. So you don't see my mouse, but I'm highlighting currently. So to summarize, we're this guy here at the top slice that was drawn in blue. And as you go more backward in time with conformal time, you get to the CMB, the Cosmic Microwave

Background Radiation. All these points to the right hand side are describing the Cosmic Microwave Background Radiation, not us now. Now this maximally symmetric space here and all of these criteria outside of the last one of special phases being synchronized as if Big Bang is a mirror, all of the rest are described in a lecture series on Lecture 2, I believe, by Latham Boyle, by Professor Latham Boyle called Cosmology. Yeah.

So, somehow this is the key, this is the crux of the matter. This is our most fundamental hint about the early universe that for some reason, as we look further back to the Big Bang, as far back as we can look, the further back we look, the simpler things get in every way we can think of, every way anyone has thought to test.

And so the question is, what is that trying to tell us? What is the right interpretation of that fundamental fact? So the conventional interpretation, the most popular interpretation is a theory called inflation. That the idea is that, the idea in inflation is that, okay, so if you, as far back as we've looked,

things seem to get simpler the further back we look. But if we could look back even further, if we could look back deep into the first fraction of a set, tiny fraction of a second after the Big Bang, we would see that that observed trend is actually an illusion. If we could look back even further, we would see that the universe even further back became a big mess. Okay? And according to that picture, the role of a theory of the early universe is to explain how the initial mess got cleaned up, got turned into the neat and tidy initial state we actually see. And so in inflation, the idea is that there is a period of accelerated expansion, the universe

supposedly in prior to the epic we can see. The idea is that before the epic we can see the is that before the epic we can see the universe, according to inflation underwent a period of very rapid stretching and expansion, which kind of, if you want, zoomed in on a little patch of the mess, kind of stretched out the mess so that the part we see in our past is just a tiny little fragment of the mess.

And such a tiny fragment that supposedly if we've zoomed in so much that it looks tidy. Okay, it looks like the simple initial state we can actually see. So, you know, the only thing is that we don't actually have any observational or theoretical evidence that this early big mess period actually existed. So anyway, moreover, I've sort of become over the years dissatisfied with inflation because I think it doesn't do well as an explanation in the previous sense that I described in the sense that if you want the fundamental principle insofar as there is one in inflation is the idea that in the very early universe there was a period of slowly decaying positive energy

that dominated the universe and that led the universe to accelerate. But if you ask what are the predictions that follow from that principle, well, it turns out they're all over the map. You can come up with many, many, you know, over the years as people have explored the idea, they've just come up with tons of different inflation models

that implement that principle. And, you know, for every, so for every prediction that there's a certain large number of inflation models in the literature that predict one thing, like that the universe has a scale-invariant power spectrum, there's another big set of models in the literature, inflationary models, that predict the opposite. Or for every big class that predict that the primordial perturbations were adiabatic, there's another gigantic class of published models of inflation that predict that they're non-adiabatic.

So basically, for every property predicted by some inflation models, there's other inflation models that predict the opposite now just a moment some of the motivation behind inflation is because You want to explain why is it that two parts which are causally disconnected have something in common something akin to that? Can you explain the motivation behind people? suggesting inflation The the idea is that it's supposed to explain that the universe is homogeneous, isotropic,

spatially flat, and with a primordial power spectrum that is, with primordial density perturbations that have correlations on very long wavelengths and that initially had their phases in sync in the way that I described in the early universe. As a creator, I understand the importance of having the right tools to support your business growth. Prior to using Shopify, it was far more complicated and convoluted.

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your selling today that's Shopify.com slash theories. And the idea is that because when the universe accelerates, I wish I had drawn some pictures here that I could point to, but when the universe accelerates, co-moving length scales so that basically the wavelengths of the density perturbations grow faster than a characteristic size of the universe which is called the Hubble size and so they get stretched outside the horizon and get stretched outside the they get stretched to be longer than the Hubble radius, so-called.

Yeah, and so that has the effect that if they were correlated, if these density perturbations previously were correlated on shorter scales than the Hubble length scale. Then during a period of accelerated expansion, those perturbations could be then dragged outside the horizon. And if the dragging is right, they'll be correlated then on, they'll inherit correlations on scales larger than the Hubble horizon, although exactly what those correlations then will look like depends

a lot on the particular inflation model that you use to implement that. So for some, I mean the motivation for inflation was that there are some inflation models that do create perturbations that look similar to the ones that we observe in the CMB. And the problem is that there's also an infinite number of other inflation models that produce perturbations that look all sorts of different ways. And so one is left with the question of why, yeah, so one is left with the question of what is the evidence that this period of acceleration in the early universe actually took place?

What is the irreducible prediction of a period of accelerated expansion in the early universe that tells us that it must have happened because we, or that gives us evidence that it happened because if acceleration took place, the universe has to be this way and not that way. I don't know of a prediction like that. Got it. So it just, so yeah, that was all a kind of,

that was more going into more detail, I must say about inflation that I had intended to do, but it just began to not smell right to me. Like, I, you know, a good scientific theory is somehow like a key sliding into a lock or something like that. You know, it's rigid. It has a kind of clicks into place and then, and you can, you know, you've made progress because you have less wiggle room. You can't wiggle in ways that

you used to be able to. You can't have it this way or that way anymore about certain questions. The sign of progress is that you have to, you know, the answer has to be such and such and not the other option. I basically just don't believe it anymore. But so anyway, so I think, so Neil and I got interested in trying to, you know, ask, is there a more, try to see if there's a, where we get by trying to take the observations more at face value. So, it seems to be the case,

all observations seem to indicate that as far back as we can see, there's this very striking pattern that the closer you get to the bang, the simpler things get. So let's just try to take that at face value. Maybe that's telling, maybe that just really is the way the universe works as you get closer to the Big Bang

and it's telling us that for whatever reason, the Big Bang was an extremely simple initial state for the universe and see where that assumption leads us. Okay, so if the early universe really was as simple as it seems, then well, here's what it was like. I mean, maybe I know the equations aren't crucial here, but then we know what the metric describing space time

was like near the Big Bang. It was basically just the metric of flat space, a very simple, nice metric, times an overall conformal factor in front called the scale factor, which basically describes how the universe was, how it stretches as time goes forward, or in other words, how it was shrinking as we went back to the bang. And so as we continue to follow the, and near the bang, moreover,

the solution, the scale factor had this particularly simple form. It was just proportional to the conformal time. So if we follow it back to conformal time equals zero, there is a singularity, the so-called Big Bang singularity, when the scale factor A of tau becomes equal to zero. But it's the simplest sort of singularity you could imagine. The metric is completely non-singular apart from a simple zero in the conformal factor in front momentarily at tau equals zero. And so unlike most singularities in general relativity, and typically if you follow some space time metric

to a singularity, you can't follow the solution past the singularity. It becomes non-analytic there, it becomes a nightmare there, and you just can't follow the predictions of GR past that. But this singularity has a very different character. You can unambiguously analytically extend the metric. You simply let tau become, let's say, for the time being, we just, instead of restricting to positive values of tau,

like you would normally do, you just let tau become negative. And then later on in the talk, we'll let it become, we'll really genuinely analytically extend it and consider complex tau. And that'll be a whole interesting story too. But for now, let's just let tau become negative. Has anyone ever noticed prior to you

that the singularity at the Big Bang was an analytic one? Yes, so Tolman in the 1930s at Caltech, he noticed that if you took certain simple models that if you took certain simple models of the Big Bang with just pure radiation or pure non-relativistic matter, that they were described by simple functions like sine waves and cosines, and which would describe the universe, you know, growing from a Big Bang and then re-collapsing to a big crunch.

But he said, oh, why not just extend it and let the scale factor, you know, let this oscillating scale factor represent a cyclic universe that oscillates and that grows and re-collapses again and again without end. So that's very different than what we'll be doing. But that's, I think, was, that might be the first example of analytically extending the solution through the bang. You know, more recently, you know,

we've been very influenced by Roger Penrose, who has since the seventies emphasized this fact that I just mentioned that the big bang singularity is so unusual as a singularity in general relativity. This fact that it is, uh, a purely conformal and analytic singularity that, uh, so that the metric apart from the conformal factor going to zero, the overall size squishing to zero, that the rest of the metric, the so-called conformal metric, remains completely non-singular there.

That he, yeah, he's emphasized that this is a, you know, very striking fact about the universe that should sort of be one of the main things that any cosmological theory tries to contend with. So here I just have analytically extended the solution by letting tau become negative. And so you see instead of just having the top half of the cone and having it cut off, we now have this full double cone. And the first thing you notice about the maximally extended solution is that it has a new symmetry, basically a time reversal symmetry. Tau goes to minus tau, basically flipping over the hourglass.

And so as I mentioned at the beginning of the talk, we were struck by that. We, you might say, oh, that maybe that's just a mathematical artifact associated with following the solution maybe further than you should. And well, maybe that's the case, but we wanted to try to take it seriously. You know, it seems to be a consequence of the basic laws.

And so let's try to take it seriously and see where it leads us. Well, if you do take this extended space time seriously with its extra symmetry, then one interesting consequence we noticed right at the beginning was that that extended space time now has a preferred vacuum state for fields living on it. So this is something, an interesting point in that in curved space time, if you put quantum fields on a curved space time background, that in general the vacuum state, the zero

particle state of those fields becomes ambiguous. So for example, one observer in one state of motion, somewhere in the space time might define what to them seems like the natural vacuum state of the quantum fields in that space time. And so according to that vacuum state, they would have one, that would be their definition of what it means to have no particles present. And a different observer in a different part

might have a completely, might be led to define a completely different vacuum. So one person's vacuum would look full of particles to the other and vice versa. So normally we don't think about this in flat spacetime because flat spacetime, in other words Minkowski space, has so much symmetry that actually there is a unique preferred vacuum that respects all of that symmetry. But in a curved space time and more generally, and in particular in a cosmological space time

in a so-called FRW space time, Friedman-Robertson-Walker, there's less symmetry. And so normally there's just not enough to pick out a preferred vacuum. And so there's this ambiguity. But now with this extra time symmetry, there is enough symmetry. And so there's this interesting fact.

There's this preferred vacuum state for fields in the spacetime. It has, in particular, in addition to the other symmetries of the Friedman-Robertson-Walker metric, it respects CPT symmetries. CPT is this very fundamental symmetry of nature. C stands for charge conjugation. That's the operation where you swap every particle

with its antiparticle. And P stands for parity. That's where you reflect the whole universe in a mirror, basically. Reflect space in a mirror. And T stands for time reversal. So that's where you imagine running everything backward in time.

So if you do any one or two of those operations on their own, in general, that will not be a symmetry of the laws of nature. But if you do all three of those operations together, it turns out that's a very fundamental symmetry that essentially any theory that we think is reasonable should obey. It basically follows from the laws of relativity

and quantum mechanics. And here I've just included the figure from this great 1941 paper by Ernst Stuckelberg where he first realized that an essential component of this idea is that an antiparticle is really nothing but a particle traveling back in time. And so this is this great, great paper that, where Stuckelberg first realized this. So it wasn't Feynman who said that first? No, that's a good, that's a, that's a, it's very interesting actually. No, Stuckelberg, indeed that, that idea is often credited to Feynman. In fact,

that's, there's another idea, which is often credited to Feynman. In fact, there's another idea which is often credited to Feynman, which is even more famous, which is Feynman diagrams, which also were actually introduced really by Stuckelberg. So Gelman always called them Stuckelberg diagrams. Yeah. Gelman was not a fan of Feynman though. That's true.

That's true. Yeah. So anyway, Stuckelberg was, but in this case, yeah, Feynman's paper on this was in the late 1940s, so and site Stokelberg, so he definitely got the idea from Stokelberg. Cool. But yeah, it's very, Stokelberg was a real genius. He had many of the great ideas of 20th century physics.

He sort of came up with first renormalization, the renormalization group, but he was not a professional academic. He was, I think he was independently wealthy. He was like a Swiss nobleman basically. I think he was a Baron or something like that. And he was just, so I don't know, he wasn't very, I think he was a little bit eccentric. He didn't really work very hard publishing his papers. So he had a lot of great ideas, but

then a lot of them didn't, weren't credited to him for a long time. And yeah, there's a famous thing that ideas are named not after the person who discovered them first, but the person who discovered them last. And I think that's also the case with- Yes. Stokelberg. Okay.

Okay, so yeah, so we want to take this fact seriously Yeah, so we want to take this fact seriously and basically imagine that really, so you see, if we were just looking at the, when we were just looking at the top half of the space time here, it looked like, well, maybe the laws of nature satisfy CPT symmetry. We believe they do. But somehow the universe as a whole doesn't seem to satisfy it.

I mean, it seems to be that there's some preferred direction in time in which the universe is expanding and things are getting cooler. But yeah, when we look at this extended picture, actually we see, no, wait a minute, with this broader perspective, actually the whole thing can be CPT symmetric. And so we're going to hypothesize that it is CPT symmetric and see, we're going to basically see if that hypothesis leads, you know, is fruitful as an explanatory hypothesis. See where it leads us.

So here's the first thing that it explains. So here I've done a trick. I've used Roger Penrose's trick that he invented in the 1960s for displaying spacetimes. These are called Penrose diagrams where you do a conformal rescaling of the spacetime to bring infinity into a finite boundary so you can see infinity and to stretch out the singularities so you can see their causal structure more clearly, basically so you can see the causal structure

of the space-time more clearly. So if you want, here I've taken the previous double cone picture and kind of the conformal transformation basically untwists it. So if you remember on my previous picture. Ever been lost deep in thought and then realized you don't have any dinner plans?

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free. Previous double cone picture, I was, I don't think I mentioned them, but there were these two, these two points are related by the symmetry, by the CPT symmetry. And now when I untwist the space time, the point of the double cone has been expanded out into this circle.

So would you characterize it as an un-pinching or an un-twisting? It's kind of, you can think of it as an un-twisting. Basically, yeah, the conformal factor on one side is positive and on the other side is negative and it goes through zero at the bang. So it has this kind of unusual topological property that it, you could say it untwists

the space time in this way. So it's not like you're just blowing up the point to expand it? That's right. That's right. That's right. Exactly. right. You that's right. Exactly. Because that would really be doing topological violence to the to the space time that is a sort of we think that we think there's a there's a there's a fundamental a fundamental

fact about about about the relationship between the two sides is that is this fact that the conformal factor goes through zero. Okay, goes from positive to negative. Okay, we don't want violence. is this fact that the conformal factor goes through zero. It goes from positive to negative. Okay, we don't want violence. This is a safe space for space time diagrams. Yeah, that's right. Yeah.

So in this picture, okay, so it's just a different way of drawing the same space time, but again, it has the property that it makes the causal structure of the bang clearer. And now we see that the bang is, we see something important about the bang, which is that it's not really a point from the standpoint of the causal diagram.

It's a surface and that surface is a mirror. So it's a lot like an ordinary mirror, except in an ordinary mirror, the direction perpendicular to the mirror is a spatial direction. You're looking at the mirror and you're looking at it along a spatial direction. You're standing in front of it separated

by a spatial direction. Whereas here, just the only difference really is that the direction or perpendicular to the mirror is the time direction. But otherwise it's very similar. And in particular, just like the waves in the electromagnetic field have to satisfy a special condition, a special so-called reflecting boundary

condition at the surface of a mirror, which is why it looks like there is a copy of you on the other side of the mirror. Here, the fields in the universe also have to satisfy a special condition at the mirror, which I've shown here, so-called reflecting boundary condition, or Neumann boundary condition, which makes the wave

be symmetric on both sides. And so I want to emphasize first that this, if you look at the density perturbations, this mirror reflecting boundary condition explains the ringing, they should say ringing, of the CMB power spectrum. Okay, so this is an observed fact about the early universe. So here, this is a famous analysis of the, so previously I showed an actual picture of the cosmic microwave background,

but here they've done a statistical analysis to extract the so-called power spectrum of that picture. And anyway, the details are not crucial except to say that you see this famous ringing pattern, this kind of oscillating pattern that's beautifully fit by the data, that ringing pattern is precisely saying that the perturbations in the universe obey this condition, this reflecting boundary

condition at the Big Bang. That's not how it's usually described. You usually describe it as in some other way, in the inflationary context, you usually say that at early times, as the modes leave the horizon during inflation, they freeze and so that at the beginning of the radiation dominated era,

they were frozen, they were constant. But let's just, if we just re-describe it in a universe where we don't have an inflation epic, it was just the radiation dominated era we see in the early universe, we just follow that back to the bang, like we were proposing doing, then it's just, it's literally just saying

that the modes satisfy this reflecting boundary condition at the Big Bang. So that's a kind of very direct clue that the Big Bang is a kind of mirror if you just take the radiation-dominated early period seriously. In addition, this same boundary condition turns out to kill, to rule out, any potential primordial vorticity perturbations, which is good because we don't see any primordial vorticity perturbations, and it also kills the possible singular perturbations in the early universe. There's a singular mode in the density perturbations, and there's a potential singular mode in the gravitational waves.

And if either of those were present, the early universe would look very different than it does, and the Big Bang singularity would be much more of a mess than it is. Inflationary models have an explanation for number one, but not two and three? No, no, they have, for all three of these things, they have a different explanation. Yeah. Now, again, you can come up with inflation models that Now, again, you can come up with inflation models that probably violate all three of these features, but generically, these are also explained by inflation, but just in a different way that involves invoking a period of primordial inflation. And what I'm trying to emphasize is that the you know, the first point I'm just trying to make here

is that there's a simpler explanation for these phenomena, which is just that there's no, that it's just radiation denomination all the way back to the bang, but that the bang is a mirror. When you say inflation has a different explanation for all three, do you mean to say there's a different explanation for one as for number two as for number three? Like in yours, one fact gives rise to all three of these in your model. Yeah, that's pretty much true in inflation

too. In inflation, you would, you would, in the, you know, a typical model, a typical inflation model does explain these three facts. And they're basically all explained by the fact that the explanation there would be that the modes that we don't see are the decaying modes, which get decay away as modes get stretched outside the horizon. And the remaining ringing mode that we do see of the density perturbations is the one that doesn't decay away.

Okay. Now, inflation will also, I mean, we'll get to this. Inflation will also typically predict a corresponding ringing mode for the gravitational waves, which has not been seen, whereas we predict no such mode. So we predict no primordial gravitational waves. Inflation, as always, has some models that do predict gravitational waves and some that don't. But famously, the simplest models of inflation

do predict gravitational waves with a ringing type spectrum, and which have not been seen. Got it. Okay, so the next thing I wanted to say was that this mirror picture, the CPT symmetry picture also gives a nice account of what the dark matter is. Okay, so we seem to need dark matter

So we seem to need dark matter to explain all sorts of things about the observed universe, the rotation curves of galaxies, the detailed structure of the cosmic, the detailed properties of the cosmic microwave background and of galaxy correlation functions, large scale structure. But what is it? There's a nearly infinite number of different proposals

that have been made for what the dark matter is, many different proposals for different types of particles beyond the standard model, many different types of proposals. But what we want to argue is that actually there is a particle, basically the last conceivable dark matter candidate in the standard model becomes a good dark matter candidate in the sense I'll explain when the universe is in this CPT symmetric state. Okay, so let me just unpack that a little bit.

So here I have a diagram of the standard model as it was originally proposed in the 60s and 70s. So back then, so the particles and the theory, well, we have six quarks, up charm, top, down, strange, and bottom. Those are the purple guys. And then in addition, there's six leptons, the electron, muon, tauon, and the corresponding electron neutrino, mu neutrino, and tau neutrino. So those are all the fermions. And then there are the corresponding bosons, the basically force carrying particles in the standard model, the gluon, the photon,

the Z boson, the W boson, and the Higgs boson. But when the model was originally proposed, neutrinos were thought to be massless and no one had ever seen a right-handed neutrino. And so in its original form, the standard model had just left-handed neutrinos. So that's why I've just left the left-hand side of each of the neutrino boxes and I've covered up the right-hand side to indicate that in the original version of the Standard Model, there were no right-handed parts of the neutrino, in contrast to all the other fermions in the Standard Model, which came in both right- and left-handed parts. So now if you look at this model and you ask, is there any dark matter

candidate in this model? The answer is no. There's just nothing, there's no particle left in the model that has on the one hand not been detected. So, you know, dark matter, it has to be a particle that we have not detected except by its gravitational influence, so we can't have detected it any other way. And it has to be stable. It has to live for the age of the universe.

It has to be able to live, you know, for 10 billion years at least. So there's just no particle in this table that has those properties. Okay, and now you're going to explain dark matter, I'm sure. So there's a but here. But in the late 1990s, people... experimentalists measured that neutrinos do have small masses and that they oscillate. And the simplest explanation for those facts is that actually the neutrinos are just like the other particles in the standard model. They do have a left-handed part and a right-handed part. And so ever since then, the most plausible explanation of those facts is that we add three more particles to the theory, three right-handed neutrinos to the theory,

basically to complete the picture, so that to complete the pattern. So the neutrinos to the theory, basically to complete the picture, so that to complete the pattern, so the neutrinos are just like everybody else. Now, if you ask, are there any dark matter candidates in this model? Well, actually, yes, there's exactly one. So if you look at these three new particles we added, well, actually two of the right-handed neutrinos,

we know that two of them have to be unstable. They, so they can't be the dark matter. We know that from neutrino oscillation experiments. But one of them could be, the third one could be perfectly stable as far as we know. So that is the last conceivable dark matter candidate in the standard model. And we'd like to use that.

That would clearly be the most elegant, simplest explanation for the dark matter if it was just something in the standard model we know and love. But the problem is that if it's stable, it turns out that in the same, that it turns out that because of the structure of the standard model,

if one of the right-handed neutrinos is stable, so it lives for the whole age of the universe, then it also is completely decoupled. The same parameters that make it stable turn out to mean that it doesn't talk to the rest of the standard model. It only talks to gravity. Not even to itself.

Not even to itself, that's right. So it just freely propagates like a free particle moving through space-time. Now, usually the heaviest particle is the least stable. Is there a reason here that that's not the case? Well, yeah. So if you, it is true that if you have two particles that have the same coupling constant

that would allow them to decay, but one is heavier than the other, that the heavier one will decay more quickly. But here we have a particle which, as we will see, it will turn out to be quite heavy, but the coupling that would allow it to decay we know has to be zero in order for it to be stable. So these are, you know, the standard model has a bunch of free parameters in it. And those parameters are, unfortunately, we

don't know a priori how to determine their values. We just, we don't know how to predict their values. We can only determine them from experiment so far. That's the level of our understanding of the standard model. So they're all determined from experiment. And some are big, some are tiny. They're all over the place.

So here, the parameters that would allow this particle to decay would have to be 0. There's actually a symmetry which would set them to zero, which is just that if the Lagrangian is just symmetric under that neutrino goes to minus itself, if it has a minus one parity, then that kills the term that would allow it to decay, but it also at the same time kills all of its couplings

to anything except for gravity. So that's actually great from the standpoint of dark matter on the one hand, I mean, that's what you want from dark matter, you want a dark matter particle to be dark in the sense that it only feels and causes gravity, but not any of the other forces. But the usual downside of that would be that you have to also make a certain amount of it somehow.

You have to explain, you know, the universe today in order to, in the standard model of cosmology, the universe today is full of this dark matter. So how did it get produced in the first place if it doesn't talk to the rest of the particles in the standard model? So that's where this CPT symmetric idea comes in. The idea is, okay, so we're going to imagine that this last remaining dark matter candidate in the Standard Model is the dark matter, and we're going to imagine that it, like the other particles, start in the CPT symmetric vacuum state that I described before, this preferred vacuum state on the

symmetric double cone spacetime. In this diagram, this rectangle with the crossing out, that is not to indicate that we are not taking into account the right-handed tau neutrino. You are just focusing in on it, because before that was an exclusion symbol. That's a good point. That's a confusing... You're right. I should fix these slides. You're exactly right. I'm trying to just say that that little bit of the diagram is the proposed dark matter candidate.

Okay, okay. And then, now it turns out that because the CPT symmetric vacuum is different than the vacuum state that we in the late universe would define, different than the vacuum state that I would define in my office here, that that means that if the universe is really in the CPT symmetric vacuum state, it'll look to me, a late universe observer, like it's actually full of these dark matter particles.

And how much of the dark matter is present depends on what its mass is. So it turns out that basically I can turn the... So I should say, what I say at the bottom of the slide here is that this idea that there is a distinction between... So basically the effect that I just described, because of the difference between the CPT symmetric vacuum state that the particle is really in and the vacuum state that I, an

observer far from the Big Bang define, it looks to me like the universe is full of this dark matter particle, is exactly analogous to this phenomenon of Hawking radiation that Stephen Hawking discovered in the 1970s. The idea is the same, that the black hole is really in one state, but a distant observer defines a different vacuum state, and so it appears to them that there is a flux of particles emerging from the black hole. So it's very closely analogous calculation idea.

It's basically the same idea, but applied in a different space time. And so it turns out that in order to get the right abundance of the dark matter today, to get to match the density of dark matter that we observe that you get a measurement of one of the previously unmeasured parameters in the standard in the standard model, namely the mass of that dark matter particle. So that's the first prediction here is that if this model is right, so we get a bunch of predictions here, which I'm going to just quickly list, and which if any of them are wrong, then this model is wrong, which of course could very well be.

So we'll see. They'll be tested. So the first prediction is that the dark matter has this particular mass of 4.8 times 10 to the 8 giga electron volts. So that's, in other words, to put that in perspective, that's about 500 million times the mass of the proton, so quite heavy. The second prediction is that the dark matter is completely cold.

So I said there's all these millions of different dark matter models that have been proposed, but in terms of their observational consequences, they often can be grouped into a they often can be grouped into it according to their observational effect in cosmology. And so there's cold dark matter models, but there's also warm dark matter models, fuzzy dark matter models like axions. And so anyway, so at the moment that the data is compatible

with cold dark matter, but if in the future, as it improves, it favors warm or fuzzy dark matter or anything else like that, anything besides completely cold, collisionless dark matter, then that would also rule this particle out. And just briefly, what is cold? Cold means that the velocity of the particle has essentially been completely negligible,

can be completely neglected in the late universe. You can basically think of it as basically being at rest. I see. Okay. So the third prediction is that the, okay. Now, remember that in the standard model now, there's three left-handed neutrinos and three right-handed neutrinos. Now, it's not quite the, okay, now remember that in the standard model now there's three left-handed neutrinos and three right-handed neutrinos.

Now it's not quite the same thing, but there's also three heavy neutrinos and three light neutrinos. They're not, the three heavy and three light neutrinos are, yeah, it's a bit of a complicated story, but because the light and heavy neutrinos are actually Majorana neutrinos, which are constructed by from putting together the left and right

guys with their antiparticles. But anyway, basically roughly speaking, you can think of the light guys as being made of the left-handed neutrinos and the heavy guys as being made of the right-handed neutrinos. So the three light neutrinos are the heavy guise is being made of the right-handed neutrinos. So the three light neutrinos are the ones that we have directly detected. So they're basically the same as the three left-handed neutrinos. Now we have measured the difference between the masses of the three light neutrinos, but we have not measured the actual masses, the masses themselves.

So in particular, we don't know the mass of the lightest of the light neutrinos. But if our model is correct, it turns out because of the structure of the standard model that it has to be the case that the lightest neutrino is massless. So that is something that will be tested actually just in the coming few years. That's the kind of test that will happen rather soon because cosmological galaxy surveys like Euclid and Desi by measuring the correlations between galaxies, will be able to extract a measurement of the sum of the masses of the three light neutrinos, and

so in particular the mass of the lightest neutrino. So we predict that that sum should be 0.06 electron volts. It could very well, which is still perfectly consistent with the data, but it could turn out that they'll get a value which is bigger than that. And if they do, then again, that'll rule out this theory. A little bit more further in the future, the theory predicts a particular value for the

rate of neutrino-less double beta decay, but that rate is quite low, so that's probably a decade or two away actually measuring that neutrino-less double beta decay rate. And then finally, I mention it here because it turns out that the same kind of calculation as we use to predict the dark matter abundance also gives a prediction for the primordial gravitational wave abundance. And in particular, we predict that there are no primordial gravitational waves, no detectable primordial gravitational waves on long wavelengths. And as I mentioned earlier, this is in contrast to many inflation models, but then there's many others that also predict negligible gravitational waves.

But many of the simplest inflation models have already been ruled out by one measures primordial gravitational waves on long wavelengths by looking for a certain pattern of polarization in the cosmic microwave background. So cosmic microwave background experiments have already measured this exquisitely and put a very tight upper bound on the amount of gravitational waves which have ruled out the simplest inflation models. But those bounds will continue to go down by another order of magnitude basically over the coming decade, hopefully,

if everything proceeds as quickly as hoped. So in The Economist, there's an article here called the dominant model of the universe's creaking, and it's about DESI, namely about the Lambda CDM model. Does any of that data from DESI violate what you've outlined here? Because there's the equation of state of Lambda, which is W equals minus 1, and DESI is indicating it's less than minus one.

That's the thing about the late universe. So there's the lambda CDM model which is our indeed the model, but plus dark matter, which is basically cold dark matter. That's the CDM and Lambda CDM and dark energy of a particular type, namely Lambda, just pure vacuum energy. Nothing fancier than vacuum energy. So vacuum energy has a fancier than vacuum energy.

So vacuum energy has a particular equation of state, which is W equals minus one. That's the ratio of its pressure to its energy density. So the ratio of its pressure to energy density is minus one for pure vacuum energy. Desi has, indeed has uh, has some evidence, uh, that, uh, for, for W less than minus one. Um, and yeah, if that, if that evidence holds up, uh, and gets stronger, they're not claiming a detection of that, uh, just to be clear.

They, they, they, they, they, they, they emphasize that they have not detected W less than minus one. Um, but they have a, I think they forget exactly how they phrased it, but they have a kind of weak preference for w less than minus 1, an intriguing preference for w less than minus 1. So I personally am betting that that will go away with more data, but I could be wrong.

And if that holds up, then that's a super exciting result that as far as I'm aware, we don't have any thing to say about that, but it would overturn the sort of standard model of cosmology, the lambda CDM model. Before I move on from dark matter, okay, so there are these various predictions. If any of them turn out to not be correct, then this model is ruled out.

But on the other hand, if this model holds up, it seems to me the simplest, most compelling explanation for what the dark matter is, it's just the last remaining possibility in the standard model itself, and with its abundance in the universe today, explained by this kind of elegant CPT symmetry-based hypothesis. Okay, another thing that we think we have a nice explanation for is the arrow of time. So in other words, it seems to be that we can only, you can move to the left or to the

right, you can move either direction along any direction of space, but for some reason in the time direction, you can only go forward. And so, why is it that when you shake a particle, it radiates to the future in time, but not to the past? Why is it that the entropy of the universe that the entropy of the universe is increasing as we get further from the Big Bang rather than as we got, you know, why isn't it the other way?

Why doesn't it get bigger as we get closer to the Big Bang? So this picture gives an interesting explanation for that. So it's basically, we just wanna again, take seriously this basic symmetry principle that space time has this reflection symmetry, CPT symmetry, reflecting, relating the top and the bottom. Ah, now naively speaking, it looks like this would complicate the issue

because you've introduced two arrows, one that's up and one that's down. Indeed, it will say that, yes, but it will have that consequence, but let me get there. It gives a definite prediction for what the arrow of time should be in any given region of space-time. But you're right, that it'll predict one arrow in the top half and one arrow in the bottom half. So yes, but then we also are going to take seriously, again, the analyticity of the solutions on the background.

So if you look at the solutions of the field equations for the different type of fields that might be living on this background, it turns out they're perfectly analytic and smooth near the Big bang, so you can follow them across, and it turns out that demanding that you take seriously the analytic extension of the fields from one side to another, and also that the fields have to be symmetric have to be symmetric under swapping the top and bottom of the diagram, that ends up enforcing, as I mentioned before, this reflecting boundary condition

at the Big Bang, so the Big Bang is a mirror. But now you can ask, what about the future and past boundaries of space time? Might there be a similar reflecting boundary condition there, I mean, what if, for example, that the top boundary is related to the bottom boundary by, you know, they're not independent of each other.

It's symmetric. And so we could imagine gluing them together, just like we glued the two halves of the space time together at the bang, might that produce a reflecting boundary condition at the top? No, it turns out actually the solutions are not analytic at the top here. They have essential singularities when you follow the solutions of the field equations to the future or past De Sitter boundaries. By De Sitter boundaries I mean we seemitter, by de Sitter boundaries, I mean,

we seem to observe positive vacuum energy in the universe today. And so if you take that seriously, the future and past boundaries of the space time are called the future and past de Sitter boundaries. And they, you can ask how fields behave at those boundaries. And it's very different than at the Big Bang. They're not analytic at those boundaries. They have essential singularities.

So the upshot is that there's no reflecting boundary condition at the top and bottom. So we have this situation that we have at the middle surface here. We have a mirror boundary, but at the top and bottom we have something very different, a free boundary with no particular conditions on the fields. So that means that, sorry I'm making a mess again, I'll just clean up. You know, that means that in the middle here, the fields start out, if you will, close to the bang, the fields are pinned to a very special, the reflecting boundary condition pins them to a very special, restricted, atypical place

in their phase space. So that corresponds to a state of very low entropy. Whereas up here, there's no such pinning taking place. It's a completely different free boundary. So in other words, there's just one space in the space time where you're really pinned to a state of low entropy and then the fields are free to wriggle into a state of higher and higher entropy as they get further from the bang in either direction. So in other words, yeah, just the basic principles of symmetry and analyticity lead to a difference,

a fundamental difference between the middle of space-time and the boundary at the middle of space-time versus the two boundaries at the opposite ends of space-time that end up predicting that the arrow of time should point away. In other words, the direction of entropy increase should point away from the bang in either direction, which is what we observe. Okay. So another set of phenomena that are explained by, that have a different explanation in this context have to do with a new calculation we have of the gravitational entropy of the universe.

And so I wanna explain what I mean by that and how we calculated it and then how it gives a new explanation for the observed flatness, homogeneity, isotropy, flatness, homogeneity, and isotropy of the universe and maybe an explanation is too strong a word, but it gives a hint about why the cosmological constant is positive but so extremely tiny. The observed cosmological constant in natural units is 10 to the minus 120, so it's 0.000 with 120 zeros and then a one. And we'd like to explain why that's so absurdly small. Okay, so what's the idea with the gravitational entropy? Well, this is the,

so we do the following thing. So this is, first of all, this is the Friedmann equation. This is basically Einstein's equations for gravity. Oops, Einstein's equations for gravity are called the Einstein equations. And the special version of those equations that apply to a cosmological space-time, to the universe as a whole, are called the Friedmann equation. So this is the Friedmann equation. And it's the Friedmann equation for a universe with all of the bits that we know are there in the real universe. So there's a contribution from radiation,

there's a contribution from ordinary non-relativistic mass, and there's a contribution from the cosmological constant. And then in principle, the universe might also have spatial curvature. Spatial slices might be positively curved, like a sphere, or negatively curved, like a hyperboloid, like a saddle, or flat. And we're trying to explain why the actual universe we live in has spatial slices, which are observed to be completely flat. Okay, so the first thing that we do is we found the general solution of this equation. It's a very beautiful solution, actually. It's an elliptic function, which means that it's the special type of function that if you think about it... So we're getting a solution now for the scale factor, a of tau, that's what a solution of this equation is.

So the cosmological scale factor has a function of the conformal time tau. But now instead of restricting the conformal time tau to be a real number, like normally we think of the time on our watch as being a real number, but this solution is a special type of function which is crying out to be interpreted in the complex plane. So we're gonna let tau become a complex number. And so then I'm now plotting the real and imaginary parts of the solution in the complex tau plane.

So these are both the complex tau planes here. And what you see about the solution is that it has two periods. It's periodic as I follow it in the real time direction along the horizontal direction, but it's also periodic as I follow it in the imaginary time direction. So in other words,

there's a kind of fundamental domain here, the black,, which I've drawn in these diagrams, where if you just take the solution in that black square, you can think of that as like a tile, and if you just keep repeating that tile again and again, you get the whole function. So in other words, another way to say that is that you can think of the function

as just living on this tile. And you can just think of, don't think of the whole plane, just think of taking the top edge of the tile and gluing it to the bottom edge of the tile and take the right edge of the, take the left edge of the tile and glue it to the right edge of the tile. And when you do that, the tile becomes a torus like this,

at the surface of a donut. And you can think of the function as really living on the surface of the donut. The interesting thing about that is that, well, it's exactly the same thing that Gibbons and Hawking noticed in the 70s about black hole solutions. In particular, they noticed, I should say, in particular, they know that the crucial thing for this argument

is that the solution for a general cosmological solution turns out to always be periodic in the imaginary time direction. So as I go around the donut in this direction. Now there's an interesting connection as I go around the donut in this direction. Now there's an interesting connection between periodic imaginary time and inverse temperature. Exactly, yes. Is that going to be explored here

or am I just reading too much into this? No, yes, yes. So very good question. Gibbons and Hawking noticed that all of the different black hole solutions all had this property that they were periodic in the imaginary time direction. And then as you say, there is this interesting, there is this kind of slick argument that shows that if you have a solution like that, which is periodic in the imaginary time direction, that it has an associated temperature, which is basically the period, or one over the period in the imaginary time direction, and an associated

entropy, which is basically, at least the semi-classical approximation to that entropy is just the integral of the so-called action around one period in the imaginary time direction. And for people who are skipping around to this point in the video, you are not saying that our universe is a Taurus, you are saying that the scale factor looks like this? Yes, that's right, Yes, that's right. This is a plot of the scale factor of the universe. And instead of thinking of it as living on the infinite complex tau plane, you can think of it as living on a Riemann surface, so called Riemann surface, which has the shape of a torus. But the spatial topology of the universe

is something different. That's either an infinite three-dimensional Euclidean space or a three-dimensional sphere, or it could be a three-dimensional torus for all we know, as long as it's big enough that we can't see around it. But anyway, that's a different story. All right. Okay, yes. So they use that observation to calculate

these famous formulas for the temperature and entropy of black holes. And we just do exactly the same thing with these new cosmological solutions. And in particular, we get the entropy these new cosmological solutions. In particular, we get the entropy of the cosmological solutions by just calculating the action around one period in the imaginary time direction. So that's what I say here. We obtain a general formula. It's a nice analytic formula in terms of these so-called complete elliptic integrals for the gravitational entropy of an FRW universe, so for a cosmological

space-time in other words. And the first thing that we find is that if you just look at that formula and ask which universes have the highest entropy, according to that formula, there are universes where the spatial curvature is always negligible, flat universes just like the ones that we see, and universes where the cosmological constant is positive but as small as it can be. So we don't explain why the cosmological constant

has a particular value. I wish we did have an explanation for why it has the particular value that it does. But we think it's an interesting hint that the entropy is biggest for universes that are flat just like ours and that have ultra small positive cosmological constants where ours is certainly ultra small.

It's this 10 to the minus 120 value. Briefly, are you able to be more precise with the word tiny here? Like is there a bound? No, that's the thing. So we don't, unfortunately we don't have an argument for why, we don't have a lower bound on it. According to our formula, the smaller the vacuum energy is, the bigger the entropy gets. So, yeah, I mean, I think I suspect, you know,

as I said, our formula is a semi-classical formula. So it's just if you want the kind of leading approximation to the formula for the entropy. So if this explanation is right, presumably there is, you know, after the semi-classical approximation, you calculate the quantum corrections to that. And presumably, those would, you might see a quantization

effect where maybe there's, yeah. So it must mean that our formula is only approximately, if this is on the right track and the properties of our universe are explained as a consequence of a maximum entropy argument that our universe is the one with the highest entropy, which by the way, the idea would be that the highest entropy space, you know, the, well, maybe I'll get to that in a moment. I'll get to that in a moment. So, so, so to just continue the story about calculating the entropy.

So far, I've just talked about calculating the entropy for completely homogeneous isotropic universes with no perturbations in them. But as I said, the actual universe started off with tiny little perturbations. So now you can redo the calculation adding those cosmological perturbations. And if you just add general cosmological perturbations, tiny little density perturbations, tiny little gravitational waves, you know, you find that some of those increase the entropy, some of those decrease the entropy. In general, they can go either way. But if you restrict, if you say the Big Bang is a mirror, so you restrict yourself to perturbations

satisfying the mirror boundary conditions, the reflecting boundary conditions, then we show that those all cost entropy. So those are all entropically disfavored. So in other words, then we find that the entropy also favors universes that are homogeneous and isotropic, because these inhomogeneities we were adding

were exactly the small, yeah, are disfavored. So homogeneous and isotropic exactly is the state that you get if the perturbations are as small as possible. OK, so in other words, in summary, the gravitational entropy is largest for universes like ours, those that are homogeneous, isotropic, and flat, and with a tiny positive cosmological constant

lambda. Although, as I say, we don't have any explanation for why the cosmological constant isn't even smaller. Okay. So now I'm getting close to the end. So another point I want to mention is that this picture, the CPT symmetric picture seems to suggest a very nice formula for the wave function of the universe.

So maybe I'll just say what the formula would be. So the idea is that, so this is the idea of the wave function of the universe was invented by DeWitt, but there was this very beautiful suggestion for what it might be by Hawking who had the, he felt dissatisfied by the fact that there seemed to be two different types of laws of physics. On the one hand, if I give you a system at an initial time, if I specify the state of the system at an initial time t1, and then I want to calculate,

you know, evolve it to a later time t2 and ask what is the probability that it's in such and such a new state, there's a rule for doing that evolution, which is the so-called path integral. It calculates the amplitude, the quantum mechanical amplitude for going from state one to state two, and then you square that to get the probability for going from state one to state two. But then what determines state one in the first place?

Well, in the lab, maybe the experimentalists just went in and said it to be whatever they chose, but in cosmology, you seem to need a second independent rule for determining what determined the initial state. And he said, you know, that's sort of dissatisfying. Wouldn't it be nice if somehow the path integral was the answer to both questions? And so he proposed, along with Hartle, Jim Hartle, that the so-called no boundary wave function as the wave function of the universe, where the idea is that the amplitude of any three-dimensional spatial configuration of the universe at any moment in time, the amplitude that it has a particular configuration

is the path integral over all four-dimensional configurations that have no boundary except for the three dimensional configuration that you're looking at. So I won't go into the arguments for why that's a, what motivated them to take that proposal, but it's a very elegant minimal proposal. It seems to just involve the path integral and no other information beyond it. But it had various problems. It didn't define a convergent integral, and it didn't seem to agree with observations, basically, in short. So here we have an alternative option suggested to us by

this CPT symmetric picture, which is that we can instead say that the amplitude associated with any three-dimensional spatial configuration, like this top configuration pictured here, is not the path integral over all four-dimensional... So instead of taking the path integral over all four-dimensional configurations that have no other boundary, we're going to take the path integral over all four-dimensional configurations whose only other boundary is the CPT conjugate, the CPT mirror image of the first configuration. So that has, anyway, maybe, that has a number of nice features associated with it,

but one thing I wanna highlight is that it then forces on topological grounds every path that goes from the top configuration to its CPT image on the bottom has to go through a bang just by virtue of reversing itself to get from one to the other. So it's as opposed to the Hardell-Hawking proposal, which was designed to focus on configurations that avoided any singularity. Here we instead have these configurations

that are forced to pass through a Big Bang like singularity, like the one that we see in our past. So it gives a kind of a nice explanation for why our universe began with a Big Bang because it has kind of topological character. Okay, so the final point I wanted to say was about scale invariance and a proposed fix that we make to the standard model vacuum in order to achieve it.

So for starters here, I just have two pictures of two famous, beautiful scale invariant configurations. So scale invariance means a kind of a situation where as you zoom in on something, it looks the same. And so here's the Mandelbrot set, which famously has that property. And then this on the right-hand side is the Penrose tiling, which has a kind of discrete scale invariance,

where if you zoom in by certain discrete steps, it looks the same. So now in physics, local scale invariance, meaning that a kind of scale invariance where you can do a different zooming in at different points in space and time. That kind of local scale symmetry is called Weyl symmetry.

Mathematically speaking, it's saying that it's like imagining that your theory has the property that for whatever metric you have, if you rescale it by any scale factor, by any conformal factor, that that's a kind of symmetry of the theory. And this kind of symmetry was advocated by a number of people over the years,

Weill and Dirac and Dickey and more recently, Tuft, as a kind of natural physical completion or extension of diffeomorphism invariance, which is the symmetry of Einstein's theory of general relativity. It's sometimes called general covariance. And so somehow the idea at a physical level is that,

so for Einstein, the idea of general covariance was that you could, you know, if I make a choice of a coordinate system, a reference frame here on Earth, then an observer in the Andromeda galaxy should be able to make an independent choice of coordinates there. My choice of coordinates here shouldn't force a choice of coordinates on them. It's a local choice. We can all choose coordinates differently, our coordinate reference frames differently at different points in space.

And the laws of physics should look the same because these are ultimately just statements about how we're describing the physics, not affecting the physics itself. And so, but you would think that there should be something similar about rescaling, choosing a different unit of length. So this transformation here, where we do a conformal transformation on the metric,

that's like saying, okay, if I have three different sticks in front of me of different lengths, and I pick up the one in the middle, and I say, you know, behold, this is the meter, you know, and I'm gonna measure all lengths on earth relative to this stick. That an observer in Andromeda, similarly should, if they have three sticks, they should be

able to make still an independent choice from the one that I made here on Earth. Again, for a similar reason, that it's really just about a local choice about how physics is being described. Yeah, yeah, that things should be formulated in a way that allow that freedom. So now what about, okay, so fine. So that's all well and good as far as motivation is concerned.

Now in the real world around us, it seems like that symmetry is broken. We don't seem to have scale and variance. It seems like, you know, my desk is one size and not another and that that matters. Exactly. So seemingly it would have to be, if this symmetry were present in nature,

it would have to either be a gauge symmetry that we fix in our normal description of things to a particular gauge. We're used to working in a particular gauge if you want. Or another possibility is that it's spontaneously broken, that it really is a symmetry of nature. In the standard model, there's a symmetry called electroweak symmetry, but because of a phenomenon called the Higgs mechanism, it's not obvious. It's not exactly spontaneously broken,

but it's hidden from us by the Higgs mechanism so that it looks like the symmetry of the standard model is just the symmetry of electromagnetism. And it's only on closer inspection that we see that the theory actually has this larger electroweak symmetry. So it could be something like that, that the universe is really a violin variant,

but there's some analog of the Higgs mechanism, which is spontaneously breaking it or Higgs-ing it to make it look like there's naively less symmetry than there is. But in any case, it's interesting that if we look at the standard model, well, if we ignore the Higgs, the Higgs is weird, the Higgs field is the weirdest part of the standard model, famously it's a very puzzling part of the standard model. Famously, it's a very puzzling part of the standard model. If we ignore it for a second, we see that the rest of the standard model is classically a violin variant. So if we just focus on the sort of beautiful part of the standard model involving the gauge fields and the fermion

fields, that part is completely a violin variant, even when we couple it to gravity. But that vial symmetry is anomalous, meaning that when we now take quantum effects into account, the usual story is that that classical vial symmetry is broken quantum mechanically. Okay, so as a summary here for people who are tuning in, some would say the Dirac equation is scale invariant, but what they actually mean is that the massless version of Dirac is scale invariant.

And when you have a mass, whether it's inherent or it's given by the Yukawa coupling breaking, then you don't have scale invariance any longer. And when I say scale invariance or when you say it here, you mean local scale invariance, which is a more general type. You can scale differently at each point. Now, I'm sure I disarranged something there, so please tell me what the correct way of stating that summary would be. So it is true, first of all, what you say,

that when you add a mass to the Dirac equation, that that does break scale invari to the Dirac equation, that that does break scale invariance. Now actually, in the standard model, you're not allowed to add any masses like that. Well, actually, for the right-handed neutrinos, you can, but for the rest of the standard model, you cannot add explicit masses because they would break the one of the symmetries of the theory, it's gauge and the gauge invariance of the theory. So in fact, the way particles effectively get their masses is by so-called Yukawa couplings to the Higgs field,

which actually do not violate violin variance. Although when the Higgs gets a vev, it's spontaneously. I see, right. Yeah, yeah. So I was also thinking of the fact that even if you ignore, even if you don't think about those... Well, yeah, no, that's actually the,

you're right, you're right. That's the easiest part to say actually. So yes, it's spontaneous, but because the Higgs has a VEV, that violates Vial invariance in the standard model. Got it. Now let's talk about this anomaly though. So here's the usual, here's a famous formula

for the anomaly in a, suppose you have a theory with classical violin variance, but now you wanna ask what is the, is there a quantum anomaly in that symmetry? Well, the answer is yes, if either of these two coefficients A and C are non-zero, and there's this famous formula that if you have ordinary spin-zero scalar fields, spin-one-half fermion fields, and if you have n-zero of these ordinary scalar fields, n-one-half of the spin-one-half fermion fields,

and n-one of the spin-one gauge fields, so those are the three ordinary types of fields that appear in the standard model. What the resulting anomaly is, and what you see from this formula is that, well, the anomaly is always non-zero. Both a and c are always non-zero because the coefficients are all positive. I mean, the coefficients are plus 1, plus 1 half, plus 62. So there just can't be any cancellation here. And for future reference I've also

included the corresponding formula for the vacuum energy per mode in a theory that has N0 ordinary scalars, N1 half ordinary fermions, and n1 ordinary gauge fields. So now what we got interested in was that what we noticed is that, okay, if you talk about scalar fields in four dimensions, there's actually, in addition to the ordinary scalar fields we we usually talk about, these are dimension, these have mass dimension one. These are basically the so-called Klein-Gordon scalar fields. There's a second type of scalar field

you can have in four dimensions, which also can be coupled conformally, can have classical conformal symmetry, which has dimension zero. Okay, so let's add some of those dimensions. Let's imagine we add some of those dimensions zero, in other words, mass dimension zero, dimensionless scalar fields to the theory.

Okay, well then you can calculate what their contribution is to each of these three formulae. So let's imagine we have N zero prime of these new dimensionless scalar fields. Then now you can ask, well, now you see some, there are some more minus signs in the game. You can now ask is cancellation possible? And the answer is yes.

Remarkably, it works. You find that for the standard model, actually, if you include 36 of these guys, And the answer is yes, remarkably it works. You find that for the standard model, actually if you include 36 of these guys, the first equation cancels to zero. But then you're surprised to find, wait a minute, actually the second equation turns out to also cancel to zero then.

Oh, and wait a minute, the third equation also cancels to zero. So, stated another way, what's interesting is that if you look at the general solution of these equations, it's only possible to satisfy them if this is the general solution. And so, okay, the first thing you notice actually, which I forgot to mention is that N zero, the number of ordinary scalar fields has to be zero. So in other words, that is saying that that ordinary Higgs fields like the Higgs field

in the standard model, that in order for this cancellation to take place, those Higgs fields have to not be fundamental fields in this story. They have to be emergent or composite fields. So anyway, that's part of the story here. And of course, many people have speculated over the years that the Higgs is a composite field because of its strange properties if you think about it as a fundamental field. There's this so-called hierarchy problem

if you think about it as a fundamental field. Okay, but so the other thing is that you notice that cancellation is only possible if the number of spin one half fermions is four times the number of gauge fields. So in other words, for most theories like the standard model, it's just not possible, even by adding these dimension zero scalar fields, it's not possible to cancel the vacuum energy and the two Weyl anomalies. But in the standard model, it is, because actually, well,

the number of gauge fields in the standard model, there's eight gluons plus three weak vector fields plus one plus one U1 gauge field, so that the these are the electroweak gauge bosons, so there's 12 gauge bosons in total. But then the number of fermions is three generations times 16 fermions per generation is 48. So it is indeed exactly four times the number of gauge fields only because there's three generations of fermions. So this is actually the anomaly cancellation here

takes place only in the presence of three generations which is another very interesting thing here that most anomaly cancellations in particle physics in the standard model in particular, there's a lot of anomalies that cancel, but none require a particular number of generations. They all cancel generation by generation. So they would cancel

regardless of how many generations you have. Here, this one is requiring us to have three generations, which is an unexplained fact about the standard model. So then it's a prediction of your theory to have three generations? If there's, yeah. Do you say it's a prediction or it's a compatibility or it's a condition or what?

What is the word you would use? It depends what you take as a given because there's all sorts of things we don't know about the standard model. And so it depends what you, depending on what you take as a given, you can infer other things. So for example, yeah, if you take as a given, the fact that the symmetry of the standard model is what it is,

okay, so that forces you to have 12 gauge bosons, and you take as a given that the fields in the standard model transform as they do under the standard model gauge fields, under the symmetries of the standard model. So these are the basic properties of all the different quarks and leptons in the standard model are determined by how they transform under symmetries. That's how the structure of the standard model works. So it's forced upon you by specifying the representation of the symmetry in which they transform. So if the fermions all transform in the way that we know they do, then that's forcing this 16 upon you too. So if the symmetry of the standard model is what it is and the fermions transform in the way that they do, in other words, if they all feel the forces that they do, then the three is explained.

You need three generations of such fermions. So it's a little bit of a more... Because there's a number of things about the standard model we don't know. It's hard to say what to take as inputs and what to take as outputs. But the point is it relates properties of the standard model that previously were not related to one another and that happened to turn out to be related in just such a way as they need to be

in order for this thing to cancel. So anyway, it's certainly an interesting numerological hint that this cancellation is nontrivial. OK, so now if you take these 36 dimension 0 scalar fields, an interesting fact about them is that in ordinary dimension one scalar field, like the Higgs field, if you look at its spectrum in flat space,

it's very blue. It looks nothing like the spectrum of perturbations we see in the early universe. And so if you wanted to turn a field like that, if you wanted to convert its spectrum into a spectrum like the one we see in the early universe, you have to put it in an accelerating, inflating background spacetime, and then it basically bends the spectrum and makes it a, turns it more scale invariant. So that's how inflation works.

The interesting thing about these dimension zero fields is that they just automatically have scale invariant spectra by virtue of their different dimensionality. And so, you know, they already have scale invariant spectra in any FRW background or just even in flat space. And so then you can look at what the imprint of that spectrum is on the primordial curvature perturbations that we actually observe in cosmology and the cosmic microwave background. And so that's what we did in this. We made a first attempt. It's

really just a first attempt, I emphasize. There's a number of approximations and assumptions we had to make to do this kind of first quick and dirty calculation. What we end up calculating, the details are not important here, but what I want to emphasize is that we get this formula for the shape of the primordial power spectrum. The key thing I want to emphasize is that this is an observable quantity that's measured by the cosmic microwave background experiments. And in particular, it involves a coefficient in front. So we get a certain formula for that coefficient

which is determined entirely in terms of measured standard model parameters. So we get a predicted formula, or in other words, a predicted number here for this amplitude. And then the coefficient, the exponent here, ns minus one, that's called the spectral tilt. We also get a formula for that, again, determined in terms of an already measured standard

model coefficient, namely the fine structure constant of the strong force. So that's not the usual fine structure constant, which is the electromagnetic fine structure constant of the strong force. Hmm. So that's not the usual fine structure constant, which is the electromagnetic fine structure constant. This is the analogous constant for the strong force. And so if you just compare, so here's, so if you plug in the numbers, for details, I refer you to the paper.

Because you have to plug in the numbers and then extrapolate in wavelength over a very large length scale in order to compare with observations, there's this big uncertainty in this first number. So we get this value 13 plus or minus 4.5 times 10 to the minus 10. And that's maybe, you can think of that as a once, plus or minus 4.5 times 10 to the minus 10. And that's maybe you can think of that as a once

our estimated one sigma uncertainty in the extrapolation. So that's within, that ends up surprisingly being within a factor of two and within two sigma of the observed Planck value for that amplitude. And then for the spectral tilt, our value when we plug in the numbers, we get this value 0.958, which agrees very well with the observed spectral tilt in the Planck satellite. So we're very excited about that.

As I say, it's just a first pass at the calculation. So it needs to be done more carefully, but we're very- I'm sure you're extremely happy. I mean, there's a great deal of consonance here unexpectedly. I mean, as I say, I'm excited about the whole picture because I think it gives a bunch of, it just explains a bunch of, it gives proposed explanations for a bunch of things

that are not, that may not, many of these things have alternative explanations from inflation, but then some of them do not. So for example, this relationship determining the two measured parameters of the primordial power spectrum, the amplitude and the spectral tilt in terms of the already measured, completely in terms of the already measured parameters of the primordial power spectrum, the amplitude and the spectral tilt in terms of the already measured, you know, completely in terms of the already measured parameters of the standard model, that is, you know, different from, you know, you don't get that from inflation. So yeah, so anyway, so here I've just given a summary because it's been a long story I'll just just try to briefly briefly say it so so taking seriously the analytic extension of the cosmological

solution of the Einstein equations led us to sort of two basic hypotheses one is the well led us to this hypothesis that the Big Bang is a mirror, and then to a new result, that this was sort of, this first result was from extending in the real-time direction to negative time, and this second one was by taking seriously the extension in the imaginary time direction, a new formula for the gravitational entropy. And then together, those two things, the fact that the Big Bang is a mirror and the new formula for the gravitational entropy gave new explanations for the dark matter being a how it, the dark matter is right handed neutrino and how its abundance can be explained.

The arrow of time, why it points away from the Big Bang, the absence of the observed absence of primordial gravitation waves and primordial vorticity. This is remarkable. Oh, thank you. That's, that's well, I'm excited. I agree. I mean, I kind of, I kind of agree.

It seems to, that's why, that's why we're excited. Yeah. I agree with you. But the phase of the primordial density perturbation. So the, the, the ringing of the CMB, the homogeneity, isotropy and flatness of the universe and the smallness of Lambda. Although again, with this very important caveat

that actually we predict Lambda is even smaller than what we see. So that's a shortcoming. The fact that the Big Bang singularity itself would emerge as a kind of topological requirement if the proposed wave function of the universe I described at the end is correct. That's a very big if still, but we just think it's a kind of elegant proposal at the moment, but remains to be tested in many different ways. And yeah, that then to kind of adjust the standard model vacuum to avoid the vacuum

energy and the vial anomalies, we were led to introduce 36 dimension zero scalar fields. And yeah, we found that that led to this non-trivial cancellation of the two, the A and C vial anomalies, so both vial anomalies and the vacuum energy. The cancellation required three times 16 neutrinos, or in other words, required three generations with the caveats we discussed earlier. And although the calculation is preliminary, we find it very striking that that preliminary calculation gives a value for the amplitude and tilt of the spectrum that seems to agree remarkably well with observations.

And I forgot to mention this, but you might think that adding 36 new fields isn't that adding a huge number of new particles and wasn't I trying to avoid new particles? Something I forgot to mention was that these dimension zero fields are very funny and the striking thing about them is that they, when you consistently quantize them,

they're kind of trivial. They only have a vacuum state. They don't have other states that are excitations above the vacuum. So they don't actually introduce any new particles. You can't excite any particles. You can't see them in detectors. They don't mediate forces.

If you'd like to read about this, I recommend, I forgot to mention there's a very good book on axiomatic quantum field theory by Bogolubov and Todorov and, oh shoot, I forgot to put the reference, but two other authors who I'm forgetting, from 1990 I believe, and they treat this example, precisely these fields in great detail in that book and explain this result that they're kind of, you can consistently quantize it, but then the theory is only has a vacuum state. So I think of it as like a kind of,

almost like it's kind of fixing, it's some accounting. It sounds like fine tuning. I don't know, I'm not sure how so. Normally I think a fine tuning is being something about when you have to change a parameter to a very special value to get a certain result. Whereas here the issue is that these fields, because they have a lot of symmetry, the kind of symmetric sector of the theory is very constrained, very small. It only has one state in it.

So it's something more like that. So I think it's more a consequence of a symmetry. But anyway, it's quite a strange story actually because the naive quantization Anyway, it's quite a strange story actually because the naive quantization of these fields would give a different answer, which would be that they would just be an inconsistent non-unitary theory.

And so the Bogoliubov book, this story has been hashed. These fields were initially introduced by Heisenberg back in the 50s and there's a huge literature of different people quantizing them and calculating with them in different ways. And- Now, most people think of scalar fields as that of the Higgs boson and it has a particle

associated with it. So how is one who is trained in quantum field theory, the standard quantum field theory is supposed to understand what a dimension zero scalar is without it having a corresponding particle to it. Now I know we can read the references, which will be in the description, but if you could outline it. Yeah, no, good question. Good question. So what it's very closely analogous to

is that when you introduce a gauge field, an ordinary like the electromagnetic gauge field in field theory, it's a field that looks like it has four components, A mu, where mu runs over zero, one, two, three, It's a vector field. But because of its symmetry, it turns out that two of those components go away and don't carry physical particles, and only two of them, the two physical polarizations are left. So this field is very analogous to those first two components that go away.

There's very close mathematical analogy with those. Basically, there's one, you can take this one dimension zero scalar field and rewrite it as having, it has a four derivative Lagrangian, and you can take a theory with a four derivative Lagrangian and rewrite it as two fields which have two derivative Lagrangians.

And those two fields then end up canceling each other basically, very similarly to how in like Gupta-Bloyler quantization of the electromagnetic field, the two unphysical polarizations get rid of each other basically. That's the gist of it. Well, professor, this is quite remarkable.

Well, thank you. It was certainly remarkably long. I think, thanks so much for sticking with me through it. I know that you're staying up late for this, so I appreciate it. Oh, no, no. Yeah, I appreciate you having me on, and I've enjoyed your podcast a lot. Thank you. Okay, well, the only sticking point of, it seems like many of the major issues in physics

are written here in a litany. The only part that's not is the measurement problem. So before we go, do you have any ideas, either how your theory interprets or solves the measurement problem, or just some other ideas that you currently don't see the connection between your work with Turok and this? Yeah, good question. Well, let me first mention that, yeah, I definitely, unfortunately I don't mean to disappoint,

because I know I'm on the theories of everything podcast, but I don't think of this as a theory of everything. I think of it as leaving many questions unanswered. I mean, for example, what are the values of all the different parameters in the standard model? Why do they have the values they do? And yeah, lots of questions are unanswered. So, I think in terms of this picture of explanations

kind of competing with one another, I'm excited that this seems to be, according to my judgment, seeming to be more likely, it's, yeah, I've begun to think that it's basically to be, according to my judgment, seeming to be more likely, it's, yeah, I've begun to think that it's basically has more explanatory power than the current standard theory of the early universe. That's of course a very minority view at the moment

and we'll have to see whether other people, as they get used to these ideas and vet them, come to agree with that broadly or not. But anyway, that's how it looks to me. But yeah, but it's not, yeah, definitely not, definitely not a theory of everything. Okay, so, but yeah, now that you mentioned it, the question, the relation to the measurement problem

in quantum mechanics, it it, the question, the relation to the measurement problem in quantum mechanics, it's a good question, interesting question. Do you want to save that for next time? Sure. I mean, I was going to say that the, you know, two half thoughts associated with that are that, you know, I think if this idea for the wave function of the universe that I mentioned earlier is correct, then you can learn a lot about what quantum mechanics in subregions of the

universe is about, because ultimately it all follows from the wave function of the entire object, which is the claim is that that is a specified wave function. So a lot of those investigations were very interesting investigations into that were done by Hartle and Hawking in the context of their wave function of the universe. And anyway, it's a long and interesting story. So yeah, maybe there's something to, maybe it has something to say about that.

I don't think I really know what it has to say about that. And yeah, another thing is just that, I mean, something that's always seemed fascinating to me, basically every, every time there's a factor of. Two in physics, like, um, like a, not, not every time there's a factor of two, but every, every, all of these precise factors of two, like, like, like the, like the, the fact that you have to spin a, you know, spin a, spin a fermion around twice in order for it to come back to itself or the fact that you have to spin a fermion around twice in order for it to come back to itself

or the fact that you have to exchange two fermions twice in order to get back to the original wave function or, I always find those intriguing. And so I think this, and I definitely think the two sheetedness of the universe in our picture is related to that. I mean, one way to say it is that you only see the two sheetedness when you look at the tetrad, which is this kind of square root of the metric.

If you looked at this metric, which involves the square of the tetrad, the square of the scale factor, if you want, you know, you wouldn't see that there were two sheets because you know, the two sheets are differentiated by the fact that A, the scale factor is positive on one sheet and negative on one sheet. If you squared it, you wouldn't see that. And so, yeah, so I, in relation to all of that, I mean, there's another famous square root is that quantum mechanics is somehow the square root of a probability theory. You take the wave function that you calculate in quantum mechanics and you have to square

it to get probabilities. So I think an interesting possibility is something like that the wave function lives on one sheet, basically, and its complex's sort of the, yeah, it's the combination of the two that gives rise to a full real probability. Are you aware of David Hestene's work in geometric algebra? I know a little bit about geometric algebra, but I don't know, I don't think I am familiar with his work. All right, well next time we'll talk about David Hestain's, I'll talk about him in geometric algebra, as I think it's something that's relevant to your work. Thank you so much for coming on, Professor.

All right, well it is great, it is great, great, great, great being on. Anyway, yeah, thanks, thanks so much for having me on. Okay, take care. Okay, you too. Bye. Also, thank you to our partner, The Economist. Firstly, thank you for watching. Thank you for listening. There's now a website, curtjymungle.org, and that has a mailing list. The reason being that large platforms like YouTube,

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