Theories of Everything with Curt Jaimungal - Philip Mannheim: The Story of Conformal Gravity

Episode Date: July 6, 2026

SPONSORS: - Don’t sleep on [@ultrapouches]. New customers get 15% Off with code TOE at http://takeultra.com! #UltraPouches #ad - Shopify: Accelerate your efficiency. Free trial at http://shopify.com/theories - I personally subscribe to The Economist. TOE listeners get 35% off the annual subscription. No other podcast has this! https://economist.com/TOE This podcast is about dark matter — and Professor Philip Mannheim thinks it isn't missing at all, just hiding in plain sight as the rest of the visible universe. In his first-ever podcast, on ideas he's pursued since 1972, Mannheim lays out conformal gravity: a theory built on Weyl symmetry rather than Einstein's equations, which explains galaxy rotation curves without dark matter, tames the cosmological constant, and fits the accelerating universe without fine-tuning. With longtime collaborator Carl Bender, he also resolves gravity's infamous "ghost" problem using PT symmetry instead of Hermiticity — suggesting the graviton may not even exist as a particle. I hope you enjoy. FOLLOW: - Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e - Substack: https://curtjaimungal.substack.com/subscribe - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs - Crypto: https://nowpayments.io/donation/TOE - PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 TIMESTAMPS: - 00:00:00 - Einstein's Relativity Foundations - 00:05:47 - Riemann Tensor and Curvature - 00:11:50 - Flaws in Einstein's Equations - 00:17:23 - Quantum Gravity Renormalizability - 00:23:28 - Conformal Symmetry Principles - 00:30:09 - Dynamical Mass Generation - 00:37:08 - Inflation and Horizon Problems - 00:43:55 - Renormalization Group Fixed Points - 00:51:05 - Composite Higgs Solution - 00:57:03 - Weyl Tensor Action - 01:03:44 - Solving Dark Matter - 01:09:36 - Cosmic Coincidence Problem - 01:16:27 - Hubble Tension and Redshift - 01:24:31 - PT Symmetry and Ghosts - 01:31:21 - Probability Conservation vs. Hermiticity - 01:38:13 - CPT Theorem and Fermions - 01:45:14 - Quantum Decays and Resonances - 01:51:05 - Parity and Right-Handed Neutrinos - 01:56:14 - Gravitons and Wavefunction Collapse - 02:02:24 - Cyclic Universes and Singularities - 02:08:24 - String Theory Fine-Tuning - 02:14:45 - Unitarity and Poincaré Stresses - 02:21:02 - Gravitational Lensing and Tenure - 02:26:55 - Nature and Experimental Data LINKS MENTIONED: - Philip's Papers: https://inspirehep.net/authors/998943 - Newton's Law Of Gravity: https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation - Geodesics: https://en.wikipedia.org/wiki/Geodesics_in_general_relativity - Equivalence Principle: https://en.wikipedia.org/wiki/Equivalence_principle - Introduction To General Relativity [Book]: https://amazon.com/dp/0070004234?tag=toe08-20 - Foundation Of The General Theory Of Relativity [Paper]: https://isidore.co/misc/Physics%20papers%20and%20books/St.%20John's%20College's,%20TAC's%20curricula's,%20et%20alii%20sci.%20papers/1916-%20The%20Foundation%20of%20the%20General%20Theory%20of%20Relativity%20(Einstein).pdf - Dynamical Symmetry Breaking As A Bootstrap [Paper]: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.12.1772 - Dynamics Of The Universe And Spontaneous Symmetry Breaking [Paper]: https://ui.adsabs.harvard.edu/abs/1980ApJ...241L..59K/abstract - Mass Generation, The Cosmological Constant Problem, Conformal Symmetry, And The Higgs Boson [Paper]: https://arxiv.org/abs/1610.08907 - Living Without Supersymmetry [Paper]: https://arxiv.org/abs/1506.01399 - Exact Vacuum Solution To Conformal Weyl Gravity And Galactic Rotation Curves [Paper]: https://ui.adsabs.harvard.edu/abs/1989ApJ...342..635M/abstract - No-Ghost Theorem For The Pais-Uhlenbeck Oscillator [Paper]: https://arxiv.org/abs/0706.0207 - Fitting Galactic Rotation Curves With Conformal Gravity [Paper]: https://arxiv.org/abs/1011.3495 - Cosmic Coincidence Problem: https://en.wikipedia.org/wiki/Cosmic_coincidence_problem - Cosmological Perturbations In Conformal Gravity [Paper]: https://arxiv.org/pdf/1109.4119 - Implications Of Cosmic Repulsion For Gravitational Theory [Paper]: https://arxiv.org/abs/astro-ph/9804335 - Exact Solution To Perturbative Conformal Cosmology [Paper]: https://arxiv.org/abs/2101.02608 - CPT Symmetry Without Hermiticity [Paper]: https://arxiv.org/abs/1611.02100 - Jenny Wagner [TOE]: https://youtu.be/Bj4Ra75vvTc - Neil Turok [TOE]: https://youtu.be/_xxLW71vT4s - Chiara Marletto [TOE]: https://youtu.be/Uey_mUy1vN0 - John Donoghue [TOE]: https://youtu.be/dG_uKJx6Lpg - Cumrun Vafa [TOE]: https://youtu.be/kUHOoMX4Bqw Guests do not pay to appear. #science Learn more about your ad choices. Visit megaphone.fm/adchoices

The missing mass that we've defined as the art matter problem isn't missing. It's the rest of the visible universe, and it's been hiding in plain size. It's been there all along. Is the graviton then composite or fundamental? It's neither. It doesn't exist. Man, talk about fighting words. Nature will keep you honest. This is the legendary Professor Philip Mannheim, and today I'm excited to bring you his first ever podcast.

He's been working on a theory which solves dark matter and quantum gravity, simultaneously for literally decades. You start out with a standard theory. It doesn't work for galaxies, so you invent dark matter. It doesn't work for cosmology. You invent dark energy. It doesn't work for quantum theory. So you invent string theory.

I haven't done that. I've just taken the theory and I've solved it. On this channel, I, Kurtzai Mungle, interview researchers regarding their theories of reality with rigor and technical depth. Today, what the professor argues is that we've been wrong about dark matter. We also have no big bang in the universe, according to his theory, add further speculations on what collapses the wave function

is a particle that we don't see. It's one of the most startling discoveries in human history. I can't describe it any other way. Welcome to the story of conformal gravity. Professor, what precisely is Einstein's theory of gravity? Oh, well, I have to give you a somewhat extensive answer to that. Einstein, in the early 20th century, developed what's called the special theory of relativity.

And he was dealing with a problem. And the problem was there was Newton's laws of motion, and there were the Maxwell equations of electromagnetism, and they had different symmetries. And what Einstein realized was there had to be a universal symmetry, which we now call Lorentzian variance, which meant modifying Newton's law. And you can see this very simply.

If you take Newton's law, it just says, you apply a force, you get an acceleration. Keep on applying the force. The acceleration will get bigger and bigger and bigger, and eventually you'll be able to go faster than the speed of light. And so something had to change if you were not going to be able to go faster than the speed of light. So what he did was he came up with special relativity,

and in a sense this just generalized Galileo's object, and Newton's first law of motion, that there's no force felt for uniform velocity, and Einstein made it for uniform covariant velocity. And so he finishes up with the theory in which observers kind of move with arbitrious speeds, but uniformly up to the speed of light, and the physics must be the same. Now, missing from that were two things.

One was, well, the observer's not required to only go at uni, in a form velocity. The observer is allowed to accelerate. And there was another theory of Newton's called Newton's Law of Gravity, which did not obey the relativity principle. So he had two problems that he had to solve. And it turns out that the solution to those two problems are different, even though we usually look at Einstein gravity as a package. And the first issue was, Well, suppose you take Newton's second law of motion, the force is equal to mass time's acceleration, and you rotate the system.

Then you generate a new term, and Mach was very concerned about that new term, and he said, well, maybe it's fixed by an interaction with the distant stars. But what was really happening was that Newton's law of motion, as just written by Newton, and it starts out, force is equal to mass times acceleration, but then is generalized to special relativity, still was not invariant

under an arbitrary change in the coordinates. So if the observer chose to rotate, the physics should not change, but the equation changed. And Einstein found a way of writing down a more general form of Newton's second law of motion so that it would not change when the observer changed his speed,

or rotated. And that one is what we call the geodesic. And let me write it, state it down. M, D2, X, lambda, D, tau squared. That's the generalization to special relativity of Newton's Law of Motion, plus gamma lambda mu mu, D x-mue d tau, D x-nu d tau,

D x-nu d tau equal zero, where gamma lambda mu-mune is the first derivative of the metric. It's called the connection. And that equation is a general co-operative, and so if you make a transformation, that equation will transform and will remain zero. Now, it has two features.

It's a sum of two terms. The acceleration, the D2x land, DTau squared, and the gamma land, the mu, D x, mu, D, T, Tau, D, neither term is separately a general coordinate vector. In other words, both of them will transform, but the combination of them with the unique weight, unique weight is plus the connection, not plus twice the connection, the unique weight will remain invariant. And so that answers the question of how do you write Newton's law of motion

in an accelerating coordinate system. Now, at this stage, you've still got nothing to do with gravity. And this is still true in flat space. And we do it all the time when we change from Cartesian coordinates to polar coordinate. that suddenly doesn't make space time curved. So this is simply a property of coordinates. But Einstein then discovered something quite remarkable, two things that are absolutely remarkable. One is, if he could get gravity to come out of the connection term,

then because they had the same weight, you would get two forms of the equivalence principle, that the inertial mass is equal to the gravitational mass, namely it's n times that whole function D2x lander D tau square plus gamma lander mu mu, D tau, and so on. So that would guarantee you that the M that appears with the acceleration,

which is called the inertial mass, and the M that appears with what's going to be the gravity, the connection would be the gravitational mass, and they would have to be equal. Because if they were not equal, that quantity would not be a general coordinate vector. So it has very unique he also noticed, at least I think he did, I don't know if we said so, was that if you take

the connection at any given point you can make a coordinate transformation that would make it zero. At a point? At one, at any given point, but it would not be zero at the next point. But at any given point, you could make it zero. And if that were the gravitational field, then you would. you have shown that it's equivalent to an acceleration at that point. And that was, that's what we call the equivalence principle, the equivalence of gravity and acceleration. Now, the question is,

when is it real? If you can move it away, how do you know if it's real? And if you read the relativity textbook of Adler-Bazan and Schiefer, they say, well, you might be able to transform so that you have a rotating flywheel, but a rotating flywheel certainly does a lot of of damage. Right. So then the question is, when is it real and when is it, and when is it just an artifact of making a transformation? And the answer is, but you can't change the second derivative. The connection is the first derivative of the metric, but making the change in the first derivative doesn't change the second derivative. So you can't get rid of that as well. And the second derivative gives you the remand tensor. And so the remand tensor, by definition of a tensor,

If it's non-zero, there is no coordinate transformation that can make it zero. And therefore, what Einstein realized was that if gravity was described by the Riemann tensor, then you would have a general coordinate invariant description of gravity. So from looking at the geodesic alone, you learn, the metric is the gravitational field, the metric couples in a general coordinate invariant way. and that when the geometry is not flat, namely when the remand tensor is non-zero, then you have an effect that we call gravity.

Okay, now, all of that is intrinsic to Einstein's program, and in my opinion, any theory of gravity must contain all of those ingredients. However, what that does not tell you is what is the gravitational field. You now know that the connection depends on the metric. You know that the remand tensor depends on the metric, but you don't know what the remandensor is. So you need an equation to fix the remand tensor. And then you will be able to solve the theory, get the gravitational field, and then you'll be able to determine the response of a particle to a gravitational field.

So those are the steps you need. So how does Einstein fix an equation that's going to tell him what the metric is? The discovery that the metric is the gravitational field, in my opinion, is one of the most startling discoveries in human history. I can't describe it any other way. I mean, I was used to writing down the line element. D.S-squared is Aeta-mu-nu, D-X-nu, D-X-new, is the D-S-word. And he turned Ata-Mu-new, which,

is 1-1-1-1-minus-1, he turns it into gravity. I mean, I've known this for many, many years, and I'm still startled that he was able to do that. But it doesn't tell you what the metric is. And he then does the following. He says, well, I want to, I've now answered question A, namely, how do we describe physics in an accelerating coordinate system? But what do I do about Newton's law of gravity? Well, if you write down Newton's law, of gravity, you write down the Poisson equation, del sward phi equals row. And it's a second derivative. And the Riem and tensor is also a second derivative of the metric.

So if the metric is the gravitational field, it's very natural to generalize Newton's, the Poisson equation to an accelerating coordinate system by using what are called the Einstein equations which are based on the Ritchie tensor and the Ritchie scalar using the Einstein tensor. So it's what you would do if you wanted to ask the question, which is what I believe Einstein asked. What does Newton's law of gravity, namely the Quasson's law, look like in an accelerating coordinate system? Einstein works that through, comes up with the Einstein equations, and they give him two things. They give him, one is Newton's law,

and they give the V squared over C squared correction. And then that is tested in the perihelion of Mercury and in the gravitational bending of light. However, it's phenomenal, I hate to say this, it's phenomenology, because he starts out with del squared phi equals row and works his way up to general relativity. But why would he choose,

where did del square phi equals row? Michael's Road come from, only from the experience that we previously had of Newton's Law of Gravity. So it's not deriving Newton's law of gravity. It's actually finding a generalization of it, which would hold in an accelerating coordinator system. And that was the steps, those were the steps he took. Now, just a moment, I have a quick question. Okay, please. So what's wrong with already knowing where you want to end up and then thinking, okay, I need to make my theory such that in some appropriate limit, I need to end there. That actually sounds like good physics, because it's like, we know this guy works approximately for the majority of cases. Yes. We should get there.

Okay. What I'm saying is it's on a different conceptual footing than the equivalent principle, general coordinate invariance, the metric is the gravitational field, the ream and tensor, when it's non-zero means there is a gravitational field, all of that is generic. The Einstein equations are specific and they're constructed to do a specific thing, which you're absolutely right. At that time, that was the specific thing you wanted to do. There's no question about that. However, we run into a problem, and the problem is that people look at the Einstein theory

as a package. Namely, they don't distinguish between accelerating observers and gravitational field. And they are different because even if there's no gravity, you can accelerate in the presence of an electromagnetic field, and in fact,

electromagnetic fields cause accelerations. So they all have to work. So he picks on the second order to pass on equation because that was the one that was there. And when it works, Everybody said the theory of gravity is solved. But what I'm trying to show you is that there was a flaw, not necessarily a super floor,

but there was a step in the reasoning that wasn't as secure as everything else. Now, what have we done when he does this? We call what he did general relativity. But that's not a third description, because special relativity means invariance transformations. It doesn't mean any specific equation of motion. So general relativity should mean invariance under general coordinate invariance with the metric being the gravitational field.

That's what it should mean. It should not mean and the gravitational field of basically Einstein equations. But because he did the two at once and because it worked, everybody takes the view that you can't touch anything. Now, let me show you. immediately where it can go wrong. And the question is, where did we get Poisson's law from?

Well, Newton gives us a one over our potential, and del squared phi is row to solutions one over r. Well, suppose I gave you del four phi equals row. A fourth order derivative equation, the solution is a one over r plus an r. If I give you del six phi equals row, I get a 1 over R, an R, and an R cubed. In other words, Newton's law of motion 1 over R is not uniquely tied to the second order Poisson equation. But all those other terms, the ones that grow like R or grow like R cubed, aren't important for the solar system,

because the solar system was still very short distance. So those terms could always have been there, and you wouldn't know from looking at the solar system. Right. It's when we go to galaxies and we jump up by a factor of a thousand, that all of a sudden the question is, does Newton's law of gravity continue to hold untouched? Or is there something else going on,

which we easily call dark matter? We meet dark matter in the regime where we did not derive the Einstein equations in the first place, namely it was geared to the solar system. But what Einstein does by generalizing it to the Einstein equations, It then becomes the Einstein equations on every scale, and not just on the solar system. Now, this objection was raised by Eddington in about 1920. He pointed out, he said, look, you can get the Einstein equations by varying what's

called the Einstein-Hilbert action, which is just the Ritchie Scalar. He says, well, you can also get them from varying the Ritchie Scalar squared. But you get these extra terms. So there was nothing unique about the second order Poisson equation. And I've been arguing for a long time now that if you want to make the claim that the universe is full to the brim with dark matter, you have to find some fundamental principle that would tell you why you should be using the second order Poisson equation. And that's the challenge. And so that's where I think there's a difficulty in the theory.

I'm not saying that the universe isn't full of dark matter. I mean, that doesn't follow from what I've said so far. But nonetheless, there is this loophole in the way that Einstein gravity is constructed. So it's a theory which tells you, I have a very specific equation of motion. and that's the one that you use, and that's the one that gives you Newton's Law. But what I'm arguing is Newton's Law was built into it. It's output if you had some other reason

to use the Einstein equations. But as long as the way Einstein built it up, as far as I can see, it's Newton's Law of Toussons Law and Newton's Law were input. Now, can we change the Einstein equations? well, of course, everybody, but everybody, including Einstein, changed the Einstein equations by adding on a cosmological constant. So from the very beginning, Einstein was saying,

well, the theory is not unique. I can add on extra terms. The one he added on was the cosmological constant, which has had one of the most checkered histories of anything in physics. But the fact that he was able to do it meant that what he was working with was not unique. Right. If it were unique, then you either would be able to add it or not, you would know. But because the Einstein equations were not unique,

that left open the possibility of adding on a constant lander, which is the cosmological constant that Einstein himself did, or adding on the Ritchie Scalar Squared. which is what Eddington did. So there's a lack of uniqueness. And if my view is that we have to resolve this lack of uniqueness, one way or the other, and that will tell us whether we should be living in a universe

with dark matter and dark energy and so on. Okay. Now, so that's Einstein gravity. Now, at the same time, people started to think, about the quantum version of Einstein gravity. And here the interesting thing is, Einstein is responsible basically for quantum theory and for gravity, and yet when you put them together,

you run into trouble. It's very ironic. So what is called quantum gravity does not mean find me a theory of, a quantum theory of gravity. It's find me a theory of gravity whose low energy limit is Einstein. I remember the doubt before I launched this podcast. What if no one shows up? Starting anything

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but not find me a theory of gravity. And the difference between those two is something like gravity has this set of features, one of them being general covariance, another one being we're going to get to this, Ritchie Flatton is at, say, the solar system level, another one being that you want it to follow geodesics and so forth. And you could also have that it follows Einstein's equations,

but Einstein's equations don't follow from this feature list. That's right. They're independent of that list. Yes, got it. Exactly. That's very good. So as far as I'm, I can see, there is a lack of uniqueness in the theory of gravity as it's formulated. When people talk about quantum gravity, they mean we already know the classical theory of gravity, and therefore we will try to find a quantum theory that will have that as its

classical limit. Now, we've done that with electromagnetism. Neutralmagnetism is a very unusual theory because in the following sense, is Maxwell writes down the Maxwell equations. Maxwell doesn't know anything about relativity, but he writes down the Maxwell equations. Einstein comes along, introduces special relativity, and the Maxwell equations don't change. Then Einstein uses general coordinate invariance, and the Maxwell equations still don't change. And then quantum field theory comes along, and the Maxwell equation still don't change.

I mean, the most amazing thing is that when people write down quantum electrodynamics, they write down the Maxwell action using quantum fields. But how can they do that? Because I know, Maxwell didn't derive it for quantum fields. The answer is, the quantum electrodynamics theory happens to be renormalizable. And therefore, the output classical fields, which are really matrix elements of the quantum operators, will obey the same equation. So if we want to discover a theory of quantum gravity, we would try to mimic what we've already

successfully done in quantum electrodynamics. And we would look for a theory whose matrix elements, gravitational fields, whose matrix elements would behave as the classical theory that we'd like it to behave as. So we're sort of not doing things the right way. And the reason I say that is historically, we started out with classical physics and then got to quantum physics. But we now know that's the wrong way to go. I mean, we always say that classical physics is the limit H bar goes to zero,

where H is plagues constant. But H bar never goes to zero. It's what it is. There's no mechanism to switch it off. What there is is a way to get into, so that you get classical results. But nothing to do with letting H-bar go to zero. So the question that I started to think about was, well, in order to get a proper quantum

theory of gravity, we need to know what classical limit we're aiming for. Yes, okay. Whereas my colleagues have done something completely different, they say, we already know the the classical limit and we're going to try to work it through. And, well, they haven't succeeded, is the best way to say it. But I certainly don't say that they've not invested an incredible amount of thought into it. Let me see if I got this straight. There's a symmetry, which we're going to get to called conformal symmetry, which you say

and your colleagues and your collaborators, I mean, will solve the issues of dark matter and dark energy twinly, as well as QG, quantum gravity. Yes. And that it's not so unnatural. So some people may look at that and say it's contrived, but you say, no, no, let's run a thought experiment. Let's imagine Einstein didn't happen, but quantum theory developed. Yes.

And you have Q of T, QED, QCD, and so forth. And then you think, okay, I don't have this classical gravity doesn't exist in this thought experiment. You have a wish list, as I mentioned, a feature list of what I want gravity to have, features I want gravity to have, general covariance, et cetera, et cetera. But you don't have Einstein's field equations. Yes. What would be natural to you, knowing quantum theory,

would be to write down a renormalizable. Yes. Field equations, set of field equations. And something that is thrown out called a vile curvature for Einstein is something that you say, maybe we don't need to throw it out. And if we included it, then we not only have this conformal symmetry, but we have renormalizability, we have all the best of scenarios,

which also happens as a consequence. We're not even trying. It also explains dark matter and dark energy. Yes. So is that roughly correct? Yes, it's roughly correct, and you had John Donahue on your program at one time. John and I have both agreed on the following,

that if quantum field theory had been developed before Einstein gravity, we would not have gone to that particular, theory, we would have looked for a renormalizable feel theory from the very beginning because SU3 cross-sue-2 cross-E1, the strong electromagnetic and weak interactions are renormalizable theories. So we have to ask, why are they renormalizable theories? And we already know that Einstein is, the Einstein theory is not renormalizable, and the huge efforts in luke quantum gravity, in string theory, and so on, have been.

mean to try to make it into a consistent quantum theory. And that's a very extensive and very committed exercise that those people have done. But what I ask is, well, can I find a renormalizable theory of gravity? Now, what makes quantum electrodynamics renormalizable? Well, the answer is it has a dimensionless coupling constant, whereas Einstein gravity is not renormalizable because it has a dimension full coupling constant. Now, we've already met that with the weak interaction. The Fermi wrote down the Forfermi interaction with G Fermi that was dimensionful,

and the theory was not renormalizable, but we discovered we could rewrite it as a gauge theory. And so we found in the end that there was a different formulation of the quantum theory that would get us back to the Forfirmie interaction at low momentum, momentum much less than the mass of the W or the Z bosons. So that's suggesting to us that we shouldn't be looking at Einstein gravity as the way we have been. We should only be looking to see what we get at a low energy limit. So the first question was, why is quantum electrodynamics renormalizable? and the answer is because it's got a dimensional as coupling constant. But I could have given you a Maxwell action,

which would be, instead of it being F mu nu nu, F, mu, N, it could have been FMU-MU-Box, F-MU, where Box is second derivative. That's just as good electrodynamics. There's nothing wrong with it. It's covariant. It bays special relativity. But it's not renormalizable.

And the reason it's not renormalizable, but by adding in the DELS squared, you have to add in something else a coupling constant which has a dimension, because the whole thing has to have, the action still has to have dimension four. So why would we not use F, mu, NU, DL Squirt, F, Munu, DL, F, Munu, for electromagnetism? And you say, it's not renormalizable. And I say, well, you have to decide whether Einstein's lack of renormalizability would also exclude it. If you want to exclude F, mu, mu, Nude, DL squared, F, mu, mu,

because it's not renormalizable, then you should exclude Einstein gravity as well. So you have a conundrum. So instead, what I realized was that anything other than F-Mu-New squared would not be conformal invariant. And conformal invariance, and it simpler says,

it says you take a triangle and you stretch it up, then all the angles remain the same. But if you have a little weight hanging on one of the arms, then as you stretch it up, that arm will sag. And so if you have masses,

you lose the conformal symmetry. So when Vyle introduced this, and it was very soon after Einstein, about 1918 or so, the problem with it was, well, things do have mass. But we learned much, much later in the 1960s with the development of goldstone bosons

that mass can come in the back door. It can come in through the vacuum. It's not a property of the operators or the equations of motion, it's a property of the states that you calculate the operators in. So it's possible, now that we know about spontaneous breakdown,

it's possible to go back and say, well, now we can have a universe which has a conformal limit. You see? And so if you do that, you then explain why quantum electrodynamics is renormalizable after all. Now, when you do it for gravity,

as you noticed, you're led to fourth order equations of motion. The reason being is each, the remand tensor is two derivatives of the metric. Remen tensor squared is four derivatives of the metric. The action is integral D4X and therefore the coefficient is dimensionless. Now you can have global scale invariance, which just means you have richie squalor squared, richie tensor squared, and remand tensor squared, or you can have local conformal invariance, namely that you do this locally

at every given point of the space time. Then you'll led to a unique action, which is the square of the vial tensor. And the vial tensor is approximately the traceless piece of the remand tensor. And so you could have this theory which would be based on the vial tensor, and it leads up to fourth-order equations,

and I started to think about that theory because conformal symmetry would be pointing me in the same direction that it's pointing me in SU3 cross SU2 cross U1. So that would make gravity as close as possible to the standard theory. People that I spoke to all told me the same thing and that was that this theory had states of negative norm,

namely ghost states, It was not unitary, it's not physical, and it can't be allowed. I knew that. I didn't quite know what to do about it, but about I'd been working on something else. I've been working in particle physics. I'm not a relativist, not an astrophysicist. I'm a particle physicist. And I did my first postdoc in Brussels with Robert Brought and Francoiseins-en-Glau.

And they had, they were many body theorists. And their ideas on what we now call the Anglo-Brow-Brout-Higgs mechanism came about because they knew about Anderson's mechanism for making photons massive in a superconductor. And at the same time, renormalization group ideas were just coming into condensed matter through the work of Wilson and Kadenoff and so on. and it was gradually being transcribed into particle physics. And so I had this training in renormalization group, and from Brussels, I came to the Institute for Advanced Study in Princeton,

and at that point, I asked the following question. At the phase transition, the whole system is correlated into long-range order, and that's a scale symmetry. You have a correlation length that extend the full length of the critical, crystal, and that's the scale invariance. But the order parameter is zero. There's no magnetization. The magnetization only comes when you go below the critical point. But at the critical point, which is what the condensed matter people were doing, you had exactly scale symmetry. So I started to ask the question, well, could I get scale symmetry as a way of generating masses?

Namely, not looking at a system at the critical point, but looking at a system, a T-E- zero. And I found a dynamical way to do that in which I had a conformal symmetry that was restored because I was at what's called a fixed point, namely that the coupling constant obeyed very special conditions. And I discovered at the same time that I had to look at the expectation value of si bar side, which is what I was looking at. That's the mass operator. And when its dimension, it's typically it mentions three, because it's three halves for each spinner, when its dimension drops to two, then the theory becomes so infrared divergent that it spontaneously breaks and you get long range order. And so I had, by that point,

I had understood that there was a way to get mass generation in a theory with no mass scales. Hmm. So I then pressed on and I worked on Grand Unified theories and so on. And in the late 1980s, I got a National Academy of Sciences, National Research Council Fellowship to the Goddard Space Flight Center in Maryland, a branch of NASA in Maryland for a year. I went there and I started to collaborate with the person, called Demos Casanist.

Now, Demos was an astrophysicist, which I was not, and he also knew relativity, which was very helpful. He had originally been a student of Dave Schramm at Chicago, Enrico Fermi Institute, got a position at NASA,

and he had actually discovered inflation before Goof. He had written a paper that if the universe went through a decidophase, then there would be no horizon problem. Now, it's very interesting that I mentioned on Gler and Brout, they, at the same time working with Edgar Gunzig,

had also shown that if the universe went through a decidipase, there wouldn't be a horizon problem. Alan Gooth's work came a little bit later, but what he showed was that, yes, you also solve the flatness problem, which, by the way, dark energy has unsolved, but we'll get back to that a bit later, perhaps. Sorry, just a moment. In order to make this more comprehensible to people,

they may be confused at what is renormalization exactly. So can we give a quick three-minute crash course on renormalization group, which is not precisely the same thing, and then also conformal symmetry as distinct from just mere scale invariance? Okay. All right.

Renormalization is, you're faced with the following issue. you write down a theory, you take the free theory, you quantize it, you get an infinite number of modes, because a field is described at every single point of the space time. It's not just fried on a line, it's all over, and therefore in momentum space, that means an infinite number of modes. And then when you start to interact, switch on an interaction, and you interact, then you can excite every one of these modes. and that's an infinite effect because you're swimming over an infinite number of modes

and that's bad news. However, in a renormalizable theory, you can redefine the parameters that are already in the Lagrangian to make the renormalized charge, the renormalized mass, in such a way that you can cancel those infinities. Now, in Einstein gravity, that doesn't work.

you can cancel the infinity in first order, but then it comes back in second order and you can never catch up because it's not a renormalizable theory. But renormalizable means that the theory can handle the infinite modes that you put into it by quantizing the theory.

That you can do summations over modes in such a way that you can still get finite answers. I subscribe to the economist. Their science and their AI coverage is among the best I've found anywhere. And I say that as someone who reads plenty of it. I'll give you some examples. They just ran an analysis on how attitudes towards science are changing in American politics and what this means for research and funding in scientific institutions moving forward. This sort of high quality reporting is fantastic. They even covered

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And they're also, of course, known for global affairs, both political and economic reporting. They are top tier, and interestingly, and flatteringly, Toa is one of the only podcasts that the economist partners with. So as a listener,

you get an exclusive 35% off. That's not a deal that they have just anywhere. Head to Economist.com slash T-O-E to subscribe. That's Economist.com slash T-O-E for 35% off. Now, Feynman said it was a dopey procedure, and there's some ancient views, not ancient, but 100-year-old views from Dirac, and he said that he didn't know. He saw it as sweeping infinities under the rug, but it's much more rigorous than that. It's become much more rigorous since the 80s. So do you also view it as a trick? Okay.

I don't view it as a trick, but if we could get that trick to work for gravity, I think everyone would be happy. I certainly don't think it was a trick. Dirac actually tried to deal with this infinity problem by introducing an indefinite metric. And I'll come back to that when we talk about PT symmetry. Perfect. So there actually is a solution to that problem. Powley picked up on it as well. And in fact, that's how Pauli got to the Powley Villers regulator,

which is 1 over K squared minus M squared minus M prime squared, so it's a difference of two propagators. And Powley Villars is able to do this renormalization while preserving gauge invariance. Right. But then you've got that minus sign, which we'll have to deal with, and there's a lot of story, a whole story in that.

It's like whack-em-all. Like, you solve one problem, and then some others creep up, and you just have to whack them down and something else pops up. Well, with the best world in Newheld, I would say that dark matter, dark energy, string theory, supersymmetry, extra dimensions, anthropic principle, is doing exactly what you just said. Okay, so now conformal symmetry means...

Ah, okay. Conformal symmetry goes like this. You can make... The Lorentz Group has six generators, three boosts and three rotations. and the Poincaray group adds in four translations. So that six plus four is ten. You could also make a dilatation,

which is just a one-time stretching, which gets you to 11, but then there's another four transformations you can make, which are called special conformal transformations, which involve reciprocals of the coordinates, and those 15 objects close on a group. And that group is the conformations, formal group.

And it's the full symmetry of the light cone. Right. You can see why, if the light cone says DS squared is equal to zero, then 25 DS squared is also equal to zero. So what you finish up with is the light cone has more symmetry than Lorenz and Poincoré.

However, standard gravity is based on making local laurenton French transformations and local Poincorri translations, but not on making the conformal transformations local. That's what Vial does. We'll get back to that. So renormalization is a procedure which will work, and it's fully legitimate,

and people have proven rigorous results about whether a theory is renormalizable or not, What I've described for you a moment ago is exactly how SU2-Cross-U-1 got off the ground. It was, you replaced the GFirmie by the exchange of a gauge-boson. You made the gauge-bos on massive by the Angler-Broud-Higgs mechanism, and then you had to show that the radiative corrections, the renormalization loops, are under control. And they were. And that's why we think the theory is correct.

And it's been tested, so it's hard to see why it would be wrong. So that's renormalization. Now, the next question was, well, when you do renormalization, you have to introduce a scale, which is a cutoff, because you've got this infinite number of modes, and then you introduce what's called a counter term, which depends on the cutoff in the opposite way, so it cancels it.

and a renormalizable theory says you can get away with the finite number of counterterms and always get rid of this lander. But the lander intrinsically breaks the scale because it has scale. And therefore, you had a different question, which was, well, I've got a lander in first order, a lander in second order, a lander all the way out to every order. Can they organize themselves in such a way that they all cancel each other? And you can see the following. Let's suppose I have a term that goes like E to the minus lambda.

It's one minus plus lambda squared over two, minus lambda cubed over three, where all those terms have unique coefficients. And if everything works out and all the coefficients come out the way I just described, then that entire sum, which looks like it's violently divergent, would go like E to the minus lambda, and therefore vanish when Lander goes to infinity. that happens when you have what's called a renormalization group fixed point.

Namely, it turns out that it controls all of the radiative corrections. Each one of those terms, that landa, land a square, they're all quantum loops and they're called radiative corrections. They're all controlled in such a way that the sum of them is well behaved. And in a crystal, at the critical point, you ask the following question. you have a high-temperature ferromagnet and you lower the temperature. How does it know it's supposed to go into a ferromagnetic phase? How does it know it's supposed to produce an order parameter?

And that is you need a cooperation of all of the modes of the theory into what's called long-range order. Yes. And what made people like Wilson really excited was that you could get long-range order from short-range forces, that you did not need long-range forces to get long-range order. And the best example of that was the icing model. And so you have this long-range symmetry, which is a conformal symmetry, which occurs if you're at a fixed point of the renormalization group. Now, there are two options for this fixed point.

One is that it's the physical coupling constant, and that that's a nidepressant. and that's an idea that goes back to Gelman and Lowe in quantum electrodynamics and the other is that you go that you asymptote to that value and that's Wilson's point Wilson's point was

the zero of what's called the beta function doesn't have to be it doesn't have to be the physical coupling constant it can be but it didn't have to be and then people discovered that if this funny beta function which usually goes like this, instead went turned down, then the theory becomes what's called asymptotically free.

And that was a bombshell. That was the work of Dave Pollitzer, Frank Wilczek, and David Gross. And when they discovered that non-abillion gauge theories were asymptotically free, and at the same time, Jehoft and Feldman had shown that non-abillion-gauge theories were renormalizable, which was not known at the time. and then Weinberg, Salam, and Glashow had been promoting weak interactions based on these non-ebellion gauge theories. And then everything came in. The neutral currents came in.

The W and the Z bosons came in, and a few years ago the Higgs boson came in. So everything comes in. And it then looks very much as though the fundamental theory of nature, at least for three-cross-two-cross-one, in strong, electromagnetic and weak, is renormal. normalizable gauge theories. So at this point, and they're all renormalizable because they're conformal invariant. Except. There's one snack in the standard theory, and that is that we write down this funny double-well

potential. And the double-well potential, you can write it as land of phi to the four minus mu-squirt phi-squared, and the minus-mew-squared has a scale. otherwise the potential wouldn't go up and come down again. So in the standard theory of introducing the double well potential, you break the scaling variance in the Lagrangian, and there's no scaling variance. However, what I had been doing in dynamical symmetry breaking

with si-bar-si replacing the scalar field, then there is scale invariance and it's broken in the vacuum. So I had a different way of approaching things, and as far as I can see, conformal symmetry can only be relevant in the real world under two circumstances. One, that we are at a renormalization group fixed point, so that all those lander lander squirts are all organized themselves, and two, that there is no fundamental scale of field. There's no elementary scale of field. that then it must be a dynamical bound state. And I've written papers saying the diagnostic that we need to identify

is can we tell if the Higgs boson, the 125 GV that we've discovered, is it in the elementary Lagrangian or is it a dynamical bound state? In other words, is the Higgs composite? Yes, is it composite? So if it's elementary, if it's elementary, and it goes into the fundamental theory,

then you no longer have scaling variance because of the minus mu squared, 5 squared. So you need mass to be dynamically generated. Yes. And if the Higgs was composite, it would have this dynamical generated mass which would have conformal invariance? They follow together.

Yes. Now, if the Higgs is elementary and it's in the Lagrangian, then it's the god particle because it gives mass. to everything. If on the other hand, it's dynamical, then the vacuum is the God vacuum, because the vacuum gives math to everything. And so, and what's the benefit of having a dynamical scalar field? The benefit is that you no longer have to deal with the hierarchy problem.

You see, the hierarchy problem says the following, that when you calculate the self-energy of a scalar field, it can emit well, we say it couples to the fermions, so emits a fermion and absorbs a fermion, and that gives a quadratic divergent energy, and we have to renormalize it down. Now, you could have said, oh, but that also happens for photons.

But photons has gauge invariance, and that protects you. And the scalar field doesn't have any invariance, and so you have no real way of making its mass come all the way down to 125GV. There was a mechanism that was specified to do that, which is called supersymmetry, but we now know from the LHC not having found any super particles,

we know that even if supersymmetry exists, there are no supersymmetry particles in the same mass region as the Higgs boson that could solve this hierarchy problem. So the failure of the failure of, the LHC to find super symmetry is actually a serious challenge for the standard SU3 cross SU2 cross U1 theory. Now, scale invariance could be global or local.

You could change the scale globally, just over the whole system, or you could point by point. When it's local, it's called conformal symmetry. So that's the background that I was working in on considering conformal. symmetry as playing a role in mass generation in elementary particle physics, namely in SU3 cross-sue-2 cross-U-1. And I showed that if you do that, then you get dynamical goldstone boson, massless particle, you get a dynamical Higgs boson with the mass. And that's, I've suggested that that should be the true Higgs. Now, when I got to NASA, and I started to collaborate with Demos Casanas,

this was in the 1980s. We started to think about the question of the cosmological constant. Because if you think of this double well potential, the difference between the upper maximum and the lower minimum, that's the energy of the vacuum, and that is the cosmological constant. So when you have spontaneous breakdown, you produce a cosmological constant,

even if you didn't have one to begin with. In phase transitions, that just says you release free energy when you make the phase transition. And so there's this energy that's suddenly appearing. And no matter what theory of gravity you have, we do know that as the universe cools down, it does go through the electroweak phase transition. And that's at least at a T.E.V. And a T.E.V is 10 to the 15 degrees. and so a black body would be T to the 4 would be 10 to the 60.

And so we know that we release a cosmological constant, which is at least 60 orders of magnitude bigger than the data could possibly support. So that's the cosmological constant problem. And supersymmetry had this wonderful idea. You have a contribution from a fermion and a contribution from a boson. Because of Fermi statistics,

the Fermion loop has an extra minus sign and they cancel. And so supersymmetry became extremely fashionable because it could do that. It could control the vacuum energy. It could control the Higgs energy. It became an ingredient for getting string theory down from 26 dimensions to 10 dimensions, so it becomes the super string theory. And of course, super particles would be an ideal candidate for dark matter. So it covers an enormous set of issues in particle physics.

However, it still needs to be found, and that's not happened. So supersymmetry has the problem that it only controls the cosmological constant if the supersymmetry is exact, namely the mass here and the mass there cancel. They can't cancel because we know the super particles, which we haven't found, must be at least a T.E.V. And therefore, they don't cancel, and therefore the broken supersymmetry, gives you a very big cosmological constant. And so supersymmetry, which was a good idea in order to control the energy of the vacuum,

really doesn't do it. Okay. So Demos Kazanas and I was thinking about this in the 1980s. This is long before the accelerating universe. That didn't occur until 1998. And we were looking for a mechanism to control the cosmological constants. and I realized that there was another symmetry

beyond super symmetry, which is conformal symmetry. Because if it's exact, remember, the angles remain the same when you scale everything up. And so we started to think about the idea that gravity might have a conformal symmetry. And we realized immediately

that the action would have to be the square of the vile tensor. it would lead us to fourth order equations of motion. And so we started to work on that. And of course, I was fully aware, and my colleagues were constantly telling me, don't work on this theory because it has a ghost. It's not a unitary theory.

I didn't know enough to be able to refute them. But I said, never mind. Let's see if nature likes the theory, because if it does, nature will know how to solve the ghost problem, even if I don't know how to solve it. So that was my strategy. So we write down this C-squared theory. This is 1987, 1988. The equations of motion are fourth-order derivative equations.

They're absolutely grotesque. In those days, there was no software package that you could just open up and say, give me the vial tensor. So we had to write our own software package from scratch. We used a program called Maxima, which I don't think exists anymore, and it took us four months, four months, just to get the equations correct. Now, you can see that Demos and I really believe this. We're going to invest four months. And we didn't even know it was going to be four, we didn't know it'd be four months.

It could have been more, and we didn't even know we'd succeed. But we did succeed, and we got the equations exactly, and it only took us one afternoon to solve them. And when we solve for the geometry outside of a static, spherically symmetric source, we got a one-over-our-up potential, which we were very, very excited about, because our problem had been from the beginning. if we write down a theory which doesn't contain the Einstein-Hilbert action, because G Newton has a scale, how do we get one over R? And so we said, well, at least let's try to see if we can get one over R. And we did, and we got the one-over-R. But because it was a high-derivative theory, there were other solutions.

And the second solution that we got was a linear potential. And we just looked at it, and we almost fell through the floor. Because we realized if the one over R is falling and the linear potential is rising, the average is flat. And we had the opportunity to explain flat rotation curves. Now, you might think linear potentials are bad news, but they're good news in the following sense. If the total velocity is flat, is constant, and the Newton piece is falling, which is Kepler, then the piece that's missing must be rising. The piece that's missing isn't flat.

The piece that's missing is rising because it's the rising plus the falling which makes the flat. And so we had a chance to get the linear potential. We then realized, and when we said this in the paper we wrote in the astrophysical journal when we got the solution,

we said, well, if linear potentials are important, then the objects on the other side of the universe are putting out their own linear potentials, and they must be affecting local dynamics. And that's Max principle. Right. And so we finished up, reincorporating Max Principle into gravity.

It's not there in Einstein because Einstein's geometry is asymptotically flat. So in the end, it doesn't satisfy Max principle. But here the geometry is not asymptotically flat, and it does. And so the Max principle, in our view, is, there's an interaction between the local and the global physics. Now, let me ask you a quick question. Sure.

Dirac also had a large number hypothesis, which also connected the global to the local. Yes. And I'm curious if your work validates his or if it's orthogonal to it. I don't know enough to be able to answer that. I'm sorry. It's really a question of why G. Newton

should be so much smaller than let's say the electric charge. If you have two objects together, a meter apart, the electric force between them is 40 orders of magnitude bigger than the gravitational force. And Dirac was trying to explain that. I have not been able to do that any more than Einstein does it,

because Einstein just says you stick G Newton into the equations of motion. It doesn't tell you what G Newton to use. He just used the one that Cavendish had. Yes. But you've assumed, the answer in a sense. I recognize that question. I haven't been, I haven't thought about it. Ah, okay, so what was the next thing that you did? The next thing we did, and it's also in this

paper we wrote back in the late 1980s, we asked, well, we live in an expanding universe, but when we do galactic rotation curves, we're looking at the behavior of a particle moving inside a galaxy. So it's in the rest frame of the galaxy. So we asked, what would the expanding universe look like to someone who's in the rest frame of a galaxy? And we worked it through, and the answer was a universal linear potential. And so we came up with two linear potentials, one that was local, that we've recovered from the local source, and the other one of which was global. which was coming from the rest of the universe. It took me seven years to untangle those two objects,

because they were both linear potentials. But let's go back first of all to us, our solution, that for a static spherically symmetric source, you get a 1 over R and an R. Well, I said to you right at the beginning, that's what you get with dels four phi equals row. Uh-huh. You see.

So the natural Poisson equation in this theory is the fourth order Poisson equation, not the second order Poisson equation. And we still get the same geometry as Einstein for the solar system. So let's just go back to the solar system. What do we need of a physical theory? Well, we say we want to recover Einstein. But we don't, what we'd like to, what we need to discover is the Einstein's solutions in the region where they've been tested.

And what's been tested in the solar system is that the geometry outside the sun is Ritchie Flat. So we don't actually test Einstein's equations. We test the solution to Einstein's equation. Yes, exactly. Because Ritchie Flat is the immediate exterior solution to Einstein. We do test it in the inside when we do cosmology because we're inside the universe. But as far as the solar system, we're just testing outside of Einstein. So what we're actually doing is not Poisson, it's a little Plas equation.

It's D'L square of five equals zero. But that equation doesn't make any sense unless you have a way to get the coefficient of the one over R related to properties of the interior. So we do indirectly test Einstein. We know that it's Ritchie Flat, which said it's the metric is a constant over R, but that constant you can only get by solving the Einstein equations in the interior of the source

and then balancing them at the surface. But that only gives you that the coefficient is the integral of row, the density over the source. It's one particular moment of the source. And you can't from measuring the moment determine the integrand. If you wanted to get the integral,

you'd have to know all the moments. But we only get one. So that's another floor in the standard discussion, is that the geometry outside of the sun is Ritchie Flat, and Ritchie Flat is a solution to conformal gravity. Now, conformal gravity is based on derivatives of the Ritchie Tents of Vanishing. So if the Ritchie Tense of vanishes, then its derivative does as well. And therefore, you'll always get back to Einstein's solution.

But the derivative can vanish without the richy tense of vanishing, and so you get extra solutions, like the linear one. So physics that's demanded of what we know of the solar system is only that we have to recover the solutions to Einstein gravity in the region where they've been tested. If we get the equations, then of course we'll get the solutions for free. But we don't need the equations. Can we get those same solutions with a different set of equations?

And the answer is yes. you could go to fourth order theory, but then you will get different solutions when you go to larger distance. And so where Deimos and I finished, we realized that we had found a solution to the cosmological constant by a symmetry which gave us a chance

to get a solution to the dark matter problem. And then we have to do fits to rotation curves and that took me a long time, and gradually it emerged. I did most of the work with my one-time student, James O'Brien. It emerged in the end that the two linear potentials were both playing a role, the interior one and the exterior one,

and there was a competition between them because they were pushing in opposite directions. And what we found was that we could fit 138 galaxies, and that was how many were known at the time, rotation curves, using just that universal linear potential coming from cosmology and the universal linear potential coming from the local source. And we need a few more things, but we needed one more universal parameter coming from clusters of galaxies. And we could fit 138 galaxies with those fixed parameters.

The same parameters for every galaxy. Dark matter needs two parameters per galactic halo, and so it needs 276 more parameters. And so we could so easily have failed, but it turns out that it works. Now, at the same time, I was making a slightly different track, and that was, well, I showed you that when we change the vacuum, which we have to in order to generate masses,

then we get a mass scale. But then I'm going to get a cosmological constant. And that's true. I will get a cosmological constant. I'll induce one by dynamics. Now, how is it control? Well, the answer is, in the conformal theory,

the trace of the energy momentum tensor is zero. And therefore, the trace is controlled in such a way that the cosmological constant that's induced cannot be any bigger or any smaller than everything else, anything else in the energy momentum tensor. In the standard theory, that's called the cosmic coincidence, namely omega landers 0.7, omega matters, 0.3.

So they're the same today. But the point here, right, right, right. It's a natural occurrence. Now, there was one piece of the story that I left out, and that was that in order to get the cosmology to look, what it looks like in the rest frame of the galaxy, I needed the cosmology to have negative curvature. And then the linear potential would be the square root of minus the curvature.

But the curvature of cosmology is going to be something like the Hubble scale. And therefore, this extra linear potential was naturally a cosmological scale. I didn't have to force it. Now, in the cosmology, you have the background and you have fluctuation. fluctuations. The fluctuations I showed produced a quadratic potential. But again, universal because it got nothing to do with the local source. And so when I finish up, I have a local one over R, that didn't go away, a local linear coming from inside the source, a global linear and a global quadratic coming from the cosmology and the fluctuations in the cosmology.

Mm-hmm. Four-note. and I do 3,000 data points. Wow. Now, why the heck does this work? Why does it work? If you look at the galaxies and you look at

V squared over C squared R, that's an inverse length, and you just look what that value is for the whole 138 galaxies and all of them are coming out, to be of the order of 10 to the minus 30 inverse centimeters, which is approximately the inverse of the Hubble radius. Written on the data, nothing to do with me.

It's there. Now, you may have heard of a theory called Mond. That's Mon's A-0. That's why it works, because A-0 does set the scale of all these galaxies. You've had John Moffat on this program. His theory, is blog. Same story.

He has that universal acceleration. So there's a universal acceleration in the data. And it has nothing to do with me, has nothing to do with Milgram, has nothing to do with Moffert. It's just so if it's Lambda CDM, Land the CDM has to explain why there's that universal acceleration.

You see. So where do I finish? Where I finish is, I've got 138 galaxies. I can fit all of them using the luminous matter alone, no dark matter, and I've replaced dark matter by the rest of the universe, the cosmology and the fluctuations in the cosmology. So I say the missing mass that we've defined as the original dark matter problem isn't missing. It's the rest of the visible universe, and it's been hiding in plain sight. It's been there all along.

You just didn't want to do it because you were hung up on one over R. You see, the problem with one over R is, if it's really true, and it's one over R on all scales, if you have a problem here, you can only solve the problem by putting matter where the problem is, because it's still local, whereas if the linear potential, you could put matter outside, and it can do the job. And so as far as I can see, that solves the dark matter problem in galaxies. Kurt here, note that if you'd rather listen to Toe, we're on Spotify, iTunes, everywhere with a podcast catcher. You can just search my name or theories of everything.

And also remember to hit subscribe. Now, your colleagues who are telling you, hey, you shouldn't pursue this. There's a fatal error, namely negative norms and ghosts and so forth. Yes. Did they prior to them telling you this, did they already know about all the issues that it could potentially solve? So suppose to solve six of the major issues, but it has this other problem. Did they know about that, or did they just say it has this other problem? I would say that it's possible that people realize that conformal symmetry would control the cosmological constant,

but they never pursued it because of the ghost issue. So all of these things I found after the fact, just because I knew enough about conformal symmetry that I wanted. to apply conformal symmetry to gravity. Now, I did something else, and that was, if I have a space of negative curvature, it acts like a concave lens. A convex lens will pull the light in. A concave lens causes light to diverge.

And so if the universe has negative curvature, photons are given a push by the geometry, and that's the accelerating universe. And so as soon as I heard about the accelerating universe, I realized that I'd written a paper in 1992, which had all this in it, but I had never pursued it. And in 1998, as soon as it came out, I immediately wrote a paper to show that I can fit the accelerating universe data

without any fine-tuning. That the cosmology, I told you the cosmological constant was under control, but it's more. if the curvature is negative, because a megallander plus a mega-m-m-plus-a-k is one, if a mega-k is positive, then a megallander has to be less than one. Because it had to be one minus something. Therefore, what I had already realized was the deceleration parameter had to lie between zero and minus one. No choice.

It couldn't be anywhere else. If it's anywhere else, then the theory is wrong. So I worked it through. I got an expression for the luminosity distance versus redshift. I applied it to the accelerating universe data, and it lies right on top of the data with the deceleration parameter emerging the one-free parameter in the theory as minus 0.37, right between 0 and minus 1.

Uh-huh. And then, of course, in this theory, there's no fine-tuning, so the universe always accelerates. it doesn't just accelerate very late. And so at higher and higher ed shift, this theory is going to depart from standard theory. Interesting. And what I've been doing so far is,

well, I've got the background. I've got the accelerating universe, but what about the fluctuations? And is it really true that I can accelerate all the way back to last scattering? And I've had a set of very good students. Matthew Phelps, Sanker Amara Singh, Tianania Lee, and currently,

my latest student, Daniel Norman and I, have been working on the fluctuations. And you are fully justified in holding up on saying this is great until we can fit the fluctuations. I don't know we will, but we're working very hard on it. Nature may be kind, nature may be not kind. All I can tell you is Einstein once said, and we always quote him,

Rafinit is de God, subtle is the Lord, but he went on to say, oh, but the chef is de nicht, but he's not vicious. Uh-huh. You see. So I think maybe,

maybe the, I've gone a long way, so maybe all the other pieces will fall in. But nonetheless, that's, so I've handled two separate problems, the cosmological constant problem,

and the dog matter problem. Now, let me, just state, there's a lot of question now about landa CDM itself. And my view is, if landa CDM is wrong, at some point it's going to start failing to fit data. And we've seen a little bit of this at the moment, but not enough. We've seen the Hubble tension, namely using the landless CDM for the plank fluctuation data gives us a value of the Hubble parameter, which is about 68 kilometers per second per megaparsec, whereas the usual sephiid cosmic distance ladder

gives us 73. And that's quite a serious difference. It's not much as I'm not in the standard camp. I don't believe it's big enough to say it doesn't work. However, the other issue that's emerging, and this is coming from Desi, Desi, is that it looks like dark energy might depend on redshift, that it changes with cosmological epoch. Now, if that were to be the case, that would be an absolute catastrophe for the standard theory, because the fact that you would then not make dark energy a cosmological constant does not mean that the cosmological constant problem has gone away. That was there from, that's always been there from the very beginning. Why isn't it 10 to the 60? Why is it plank scale 10 to the 120? That hasn't gone away.

It's made worse? Yeah, it's made worse. In other words, a red shift dependent dark energy is really bad news for the cosmological constant problem. But we'll have to see how that plays out. So there's something else your theory solves, or I wouldn't call it a problem per se, but it's more like a question. If I recall correctly, it's the size of galaxies. Yes. Okay. So it turns out that I've got a linear potential and a quadratic potential. They have opposite side. So if the quadratic potential gets too big, the distance gets too big, the quadratic potential becomes negative. But that's velocity squared and velocity can't be complex. Therefore, galaxies have to have a finite size. Now, to understand this, one over our potential, whatever else you want to say about it, it never cuts off.

It doesn't matter how far away you are from the source. There's always going to be a bound state possibility. But this is giving an actual reason for why galaxies have a finite scale. Now, since you asked me about this, let me go back and discuss what happened with James O'Brien. Oh, and you should know that I have the ability at any point if you wanted to show something on screen. You don't have to pull it up on your computer now, but I can show it for the audience,

so I can show finite-sized length in the galaxies. I can show curves that you said that your model was able to fit, the 130 or what have you. So any time, you just let me know, and it's on screen. Okay. Well, you can show some rotation curves of ours, if you want. What had happened with this, and this was James O'Brien's PhD thesis,

with me. When I had first applied this linear potential to galaxies, there were only 11 galaxies that were known, whose rotation curves were well enough study that you could actually do modeling with them. And while I said that the linear and the falling and the rising would balance, as you go out to larger distance, the linear tension, linear must win. And therefore, eventually the rotation curves that are flat have to turn up. Now, you should also know that for dwarf galaxies, they already are rising. It's the bright spirals that are delayed from rising because the Newton term is so big, because it's a big luminous galaxy. So I expected to see this rise. And James, I said, James, go ahead and find some galaxies for me, and we'll see what they look

quite, and he found the 138 galaxies. I mean, the data for them. I didn't even know they existed, but he found them all. And we'd ply the theory, and it doesn't work. It doesn't work, because the rise is not there in the data. At the point where I expected that everything would rise, it doesn't rise. And I realized we went through the data, and we found there were, 200 points where my theory said it should rise and it didn't rise. That sounds bad. Yes. And then now you're going to hear something quite remarkable.

I still had one extra term in the theory. That's all the quadratic term. Right. But it's one term with one number and it's universal. Uh-huh. So I realized that one term can cut down one rise, but it turns out it cut down the entire 200. Yes.

In one number, we've got 200 points that were rising and they all curved down. And if they keep on curving, then that's what you said before, that eventually the galaxy would have to end. That is quite remarkable. So what occurred to you both when you were able to fit how many? 138 you said or 200 with one parameter? 178 galaxies, 200 data points were in the awkward region. Well, we just realized it was correct.

we realized that there was something going on in galaxies that was being missed in land of CDM. Now, it may not be that conformal gravity is the answer, but what I've been saying all along is, here's this formula with the linear term, the two linear terms, and the quadratic term. If you think that landa CDM is the answer, then derive it in lander CDM. because it fits the data points. That was what I would ask my colleagues to do. But that's not happened. Now, Milgram and John Moffitt, all of us have been saying,

look, we have these theories, the universal acceleration parameters. If you don't like them, derive them in your theory. It's not, you shouldn't say what we're doing is wrong. You have to say that's up to data. And data is saying that there's, a structure that's currently being missed. Okay, now, by this point, I began to think this theory is on the right track.

It's renormalizable. It resolves dark matter, resolves dark energy, and standing in the way, there's just one question, the ghost problem. Right. Remind me the year at this point. This was, the year, the year was 2008. and I'll tell you what happened. Bender's coming.

Yes. I had always felt that nature knew how to handle this problem, even if I didn't. And I went to a conference. I've known Carl for many, many years, and he talked about the Lee model. The Lee model was introduced by T.D. Lee in the 1950s, and what it did, this was the early days of renormalization, it allowed you to calculate the renormalized coupling constant in a close form analytically, which was very, very powerful. But as you move the parameters, all of a sudden you discovered that there was a region where you suddenly got negative norms. And so this was a problem

that Pauley and Chilene and Lee and Heisenberg had all worried about, but had never come up with a solution. I had actually looked at this problem when I was in graduate school. There was a theory called the Heisenberg nonlinear spinner theory, which preceded everything, because this was still in the 50s, and it had an indefinite metric. And Heisenberg sort of said, well, you know, this solves the scaling variance problem of the theory, but it really was still a problem. And Carl had realized that when you move the way.

you move the renormalized coupling constant to a certain value and you got this negative norm at the same time the burr coupling constant became complex and the theory was no longer hermission and so everything that you've done was presupposing

that you had a hermission theory and Carl had realized have to give you a little bit of background in 1998 or So, Carl and his former student, Stefan Betcher, had been looking at the Hamiltonian p-squared plus I-X-cubed with an I, with a complex eye. So the potential is complex. You would swear that it must have complex icon values. You would absolutely bet any money that it has complex icon values, except it doesn't. All of its eigenvalues are real. and Stefan and Carl showed that using WKB approximation, and then a few years later, Dory, Dunning, and Takao proved it as a rigorous theorem

that all the eigenvalues of p-square plus IX-cubed are real. Now Carl had realized that that's a theory where you have a PT symmetry. Parity sends X-cube to minus X-cubed and time reversal sends i to minus i. So ix cubed is pt symmetric. Now, Wigner in 1960, when he introduced time reversal, had said the following.

The shreddinger equation is i-dipsy-d-t. So if I let t go to minus t, I won't keep the Schroeninger equation unless I let at the same time I go to minus i. So I need an operator that will complex conjugate numbers, not complex conjugate functions, because we do that all the time when you take the emission conjugate.

But you have to conjugate, you have to conjugate numbers, and that's called antilinear. And what Wthickner had shown that if you had an anti-linear symmetry, then the anti-linear operator will conjugate E into E-star, because it's the complex conjugate,

and that means that for every E, there should be an eigenvalue E-star. And so you had two possibilities, E equals E-star, so the energies are real, or the energy is in complex conjugate pairs. And so once you have PT symmetry, then you have the possibility of real eigenvalues. And the p squared plus Ix cubed has that symmetry.

And that's why its eigenvalues are real. So when I got interested in this program, when Carl gave this talk, he said that when he uses the PT theory, which means the following, How do you do you construct a norm, a probability. Usually you say you take the Ket and Dirac tells you, take the Hermission conjugate of the Ket,

the bra, and the bra kett is the probability. That's not the most general. What Carl had realized is if the Hamiltonia is not hermission, you take the PT conjugate of the Ket. And that norm is positive. And so when people said that they had a negative norm theory. What they meant was the Dirac formula for the scalar product, for the inner product. The direct formula was giving you a negative answer. Not the theory. You were in the wrong Hilbert space.

The Hilbert space is dictated if the Hamiltonia is not permission, then you have to use the PT conjugus. I'll come back to this in a minute. For relativistic, you have to use CPT. Right. We'll get to that in a moment. So when Carl did that for the Lee model, All of a sudden the ghost disappeared. And it was a perfectly consistent theory. So I had known Carl since forever, but we were at a conference together. And he talked about this. And he gave a car.

It was in Florida at Fort Lauderdale, run by the University of Miami. And he talked about this. And I said to him, Carl, you have a theory with a, you have a way to get rid of a ghost. And I have a theory with a ghost. Let's get together. So Carl was, at the time, was in Los Alamos for the year. So I went to Los Alamos and we sat down and we solved that problem in three days. In three days, we'd figure out the whole thing.

Now, of course, it may be three days, but we both came with 20 years of prior experience. Of course. Yes. But in three days, we'd realized that conformal gravity was a PT theory and not a permission theory. And then there was no ghost. Now, the thing that made this work was that, going back to what I said about quantum electrodynamics, everyone had always taken it as a given that whatever the quantum theory of gravity was, the Hamiltonian would be Hermission. But it never needed to be her mission.

Her mission is a too stronger requirement. Now, when I started to work with Carl, I realized the following. I knew a little bit about this, very little, but I knew from my days as a student that if a Hamiltonian is her mission, the eigenve values are real.

But I also knew that that's only one way. Right, right. You can have something that's real, that's not her mission. Yes. When you say Hermitian, do you mean self-adjoint in its more general form? Yes, self-reduits. Her mission in the 19th century understanding of second-order differential equations,

that you can get rid of surface terms. I see. And that's exactly, that's how Dirac arrived at Hermitian quantum mechanics in the first place. And because he copied what people already knew from the 19th century. So you can, it's true. and you said it, if the Hamiltonian is not a mission, it does not follow that the eigenvalues are not real. So I started to think about this,

and I said, well, if hermiticity is sufficient to give a real eigenvalue, what's the necessary condition? So you open a linear algebra book, and you won't find it. And the reason is it's not a linear condition. It's an anti-linear condition. The theory has to have an anti-linear symmetry, and that's the necessary condition for real eigenvalues.

And so what this meant was, and why I think people have never been able to make Einstein gravity into a quantum theory, is beyond the issue of renormalizability. They're insisting that the Hamiltonian be hermission, because it always has been emission. And in the fourth order theory, it isn't. but it has an anti-linear symmetry. And that's the second floor. It doesn't mean that, therefore, the Hamiltonian is not omission, but what I'm saying is by restricting to omission Hamiltonians,

you were looking at too narrow a problem. Because it was obvious that the metric, the classical metric, is real, and therefore the Hamiltonian must be permission. But it doesn't have to be omission for the metric to be real. It only has to have an anti-linear symmetry. Okay. So we worked that through, and we published a paper in physical review letters. And what we basically had shown was that the 1 over k squared minus 1 over k squared minus M squared

propagator was renormalizable and unitary because you were in, it looked like, you see, when I write to you a propagator, 1 over k squared minus 1 over k squared minus k squared minus m squared, two terms canceling each other. You see the big advantage is they both go like one over k squared, but with the minus sign, the net effect is one over k to the four. And so for very high momentum, the one that we needed the cutoff, because we've got all these modes, now you don't have to worry because now you've got renormalizeability.

And when you had John Donahue on your program, that's what he was talking about, was the second plus fourth, the one over k squared comes from the second, and the other one comes from the fourth order. And from second plus fourth, you have a renormalizable theory of gravity. But if you look at the propagator, you've got the one over k squared, and the other one has a minus one over K squared minus M squared.

So if you just take the normal vacuum T55 vacuum for the propagator and put in intermediate states, you'd put the sum N-KET, N-Brah, minus the sum M-KET-M-Brah. you put it in and you get that propagator. And you've got states of negative no. And the theory is not unitary. However, you can't do that.

You can't look at a C number propagator and tell me the Hilbert space. If you know the Hilbert space, you can get the matrix elements, but you can't from the matrix elements get the Hilbert space. And what Carl and I showed in that theory was that when you thought you were writing those sum on n-n minus sum on n-n, M equals one, you had presupposed that those states were normalizable. If they were, then that would be correct. And what we had showed, and this grew out of work that I'd done a little bit earlier with

Aaron Davidson, who was another longtime collaborator of mine, he was at Ben-Gurian University, we had shown that the states were not normalizable. And therefore, the whole discussion of ghosts, you were in the wrong Hilbert space. Now, does this also work in QFT? So I imagine this works in QM, but is this Lawrence Cullery? Yes, of course. What Carl and I did initially was we did it for quantum mechanics. I long since generalized it to quantum field theory, yes.

But it's exactly the same story. Now, we discovered that the solution to the ghost problem was not that we got rid of the ghost. We never got rid of the ghost. We just showed that the reasoning that caused you to think there was a ghost was not valid. Right, right, right. Everybody that was trying to get rid of ghosts was adding on terms that would cancel something, but you never needed to do that because it was never, the reasoning was wrong. And what, of course, how does it work is instead of having vacuum T-5-5 vacuum,

the left vacuum has to be the PT conjugate of the right vacuum and not the emission conjugate. And then when you put in intermediate states, it's no longer modular. and therefore you can get a minus sign. I mean, those are the details, but the basic point was, and the thing that saves the theory, is that the states were not normalizable. Now, Carl had developed a whole apparatus

for dealing with non-normal states, namely, there was a way to continue them into the complex plane. So if you had something that went like E to the plus X squared on the real axis, it went like E to the minus X word on the imaginary axis. And so this theory lives on the imaginary axis. And of course, people were trying to do quantum gravity with everything living on the real axis.

And that's, in my opinion, why they couldn't succeed. So that was the first side of the story, was the PT. Now, the second side of the story, and I realized this as soon as I got to know about PT, was that there was a connection to CPT. And the CPT goes the following. The Lorentz Group breaks up into four units, four regions. There's the one with the real coordinates.

Then there's the one which is related by parity, one which is related by time reversal, and one which is related by parity times time reversal, which is PT. Now, parity reverses the spatial piece, time reversal reverses the time piece, so PT reverses all four components of XMU, and that's Lorentz, that's consistent with Lorentz. And so what you discover is that there's an extension of the Lorentz group, which involves a PT transformation into this other region where XMU is replaced by minus Xx.

mu. And that turns out to be the covering group of the Lorentz group. And so if you have that as a symmetry, then Carl said, look, PT is natural. It's related to the Lorentz group. Hermeticity is an artificial requirement on a function. It's a mathematical requirement. Symmetry of the Lorentz group is a physical requirement. So you get vial spinners out of this? Not yes, almost. So that's PT. Now you do it for coordinates. Now you do it for fermions, so I did it for fermions. It didn't work. The reason is, for fermions, P is gamma-0, and T is gamma-1, gamma-3.

But gamma-0, gamma-1, gamma-3 is not a Lorentzian variant. You have to multiply by gamma-2. Gamma-0, gamma-1, gamma-3, gamma-2, is commonly called gamma-5, is a Lorentzian-V variant. but gamma 2 is charge conjugation. So for fermions, that transformation is C-PT, not P-T. And so what I'd realized was was that C-PT symmetry was intricately connected to the complex Lorentz group. And so if the universe is complex-Lorentz invariant, then C-PT can follow. follow. And then if C is conserved, then you're back down to PT, which is what you get for the

non-relativistic case. However, that's not the whole story. And the reason is, if I write down something like Cy Barci, which is a Lorentz scalar, it's a CPT invariant, but it's now got a coefficient. How do I make the coefficient is going to be conjugated by a, you know, the, the coefficient, is going to be conjugated by time reversal. So it may not be real. So can I make the coefficient real? And the answer is yes, if I impose probability conservation. And so I replace the entire discussion of CPT theorem by complex Lorentzimvaryance

plus probability conservation gives me CPT. Now, any non-relativistic theory that descends from relativity would automatically have probability conservation. And if you have a non-Hermission Hamiltonian, you need to have the CPT conjugate be the bra, not the Hermission conjugate, and then probabilities conserved. So the key feature, and I think this is what people have missed, the key feature of quantum mechanics is not hermiticity is probability conservation.

Hermiticity implies probability conservation, but probability conservation does not imply hermiticity any more than hermiticity implies reality, but reality of eigenvalues doesn't imply hermiticity. So probability conservation is, in my opinion, the most important feature of the theory. And that's how the PT works. Okay, now, as soon as I started to work with Carl, we solve the problem of the ghost, and now I can tell you why it's guaranteed that this is the solution. You take the fermion and you couple it to geometry through the spin connection, which is how we describe the propagation of spin a half particles in a gravitational field. You take the action, sidebar i gamma mu, dmue plus capital gamma mu psi. That's the coupling

of a Dirac fermion to the spin connection, and you do the path integral over Si and Saibar. You get the C-squared action. You've integrated out the fermions. The only thing that's left is the metric, and the metric appears as C-square. Why is it C-squared? Because the Dirac action was conformal invariant

with the spin connection. So it was already conformal, and therefore you automatically get the C-squared action. So now you've only got two on. There are only two things you can say. Either, we know the standard theory doesn't have a ghost. Therefore, the gravity theory could not have one if it's induced by integrating out the fermions.

But if it did, then the entire SU3 cross-SU-2-cross-U-1 would collapse when it's coupled to gravity. So you've only got two choices. Either there's no ghost or SU-3 cross-SU-2-1 is wrong. And there's no reason for a ghost because you're starting out in a theory which is ghost-free, at the standard model, and you just integrate out the fermions.

So in the end, there could never have been a ghost in the first place. But how do you show that there's no ghost? You've got to construct the Hilbert space and get the bras. The bras are called the dual space, and the whole point of these theories is that the dual space does not have to be

the omission conjugate of the kets. Physics is richer than quantum mechanics is richer than that. And that I think is what Carl has really established is quantum mechanics does not require the dual space to be the Hermission conjugate the same way that Dirac originally introduced it. It's a richer theory.

And it's that richer theory which works in the fourth order case. Okay, now, when I did this, I realized that Wigner has said there's an E and as an E star. Therefore, I now have a way to describe decays. Because if the energy is complex, then it's E plus I gamma, and then E to the I times E plus I gamma gives you E to the gamma T, which can decay. So for the first time in my career, I had a way of describing decays. But you paid a price. You needed both of them.

You needed one that grows, because it's a complex conjugate pair, one grows like E to the gamma T, and the other grows like E to the minus gamma T, and when you combine them, the gamma's cancel. Because you've got to get probability conservation because that went into the CPD theorem. and therefore each mode, the growing mode, cancels the decaying mode and vice versa. So let's see how that works. We go back to Rutherford. Rutherford discussed radioactive decay.

And you ask, well, it's E to the minus gamma T, you measure the gamma and you measure the half-life. And that's how we do half-lives of materials. And that looks like it's not unitary because emission Hamiltonians don't have complex item values. So that doesn't look like it could possibly be unitary. So what's going on? Well, it's decaying into something.

And the decay products are building up. As the population of the state that's decaying declines, the population of the states that it's going into increases. So one decay is E to the minus gamma T, and the other one is E to the plus gamma T, and the total population remains fixed. So what you really have to do when you do decays is you've got to include the decay products,

and then you have CPT symmetry. Okay, now, I wrote a paper in 2018 in J-Fiz-A, I was discussing all of this, and I said that if I have an E to the minus gamma T, then Vigner tells us, that there's a time delay H bar over gamma, which is just the uncertainty principle. So when a particle travels by a square well,

it's delayed for a while before it's released. And that's called the Bright Vigna formulation of resonance formation. So the physics is the wave is held for a while, and then it escapes the well, and then it continues on. And it's held for a time, H bar over gamma. time delay. But I said that there's another one

with a minus gamma. And that gives a time advance. A negative time delay. And they're both operative. And then I saw an article in a strange place,

the BBC, of work by someone else that was on your program, Freim Steinberg, who had said that when you excite an atom into the first excited state, it's supposed to decay with a rate H-bar over gamma, where gamma is the radiative width

of the upper level. But he said it doesn't. It decays right away. And I wrote a paper which said that the reason it's decaying right away is because the time delay is cancelled by a time advance. And so the net effect is you have a line width, but you'll have a time, you'll have a time for the transition, which is not given by H-bar over the line width. It's given by H-bar over the energy difference between the first and the first and the excited state, the grounds state in the excited state. And so when I wrote that paper, I began to have some confidence that the correct way of describing decays is by having both modes at once, because that's what PT symmetry requires. If I may, let me tell you, everything that I've told you in this interview has solved problems that occurred to me when I was a graduate student.

Yes. Yes. And when I was a graduate student, I was being taught the grand laws of physics. We used a textbook by Messier, which was an excellent textbook. I've used it when I teach quantum mechanics. I had an excellent set of teachers Carl Levinson, Harry Lipkin, Egal Talmi, Amherstah Shalit,

and I learned a lot of quantum mechanics from them. But I also, as I was listening to them, some things didn't sound right. And I identified four different things, well, four and a half different things that didn't sound right. I'll tell you the half in the moment. The half was a discussion

of the Heisenberg non-linearous theory with its indefinite metric, which I've told you how to, to resolve that through the work with Bender. But the four things, the first one was, I said the following, the Hamiltonian generates time translations. Parity generates space reflections.

Time translations and space reflections commute. Doesn't matter which sequence you do with them, because they're acting on different things. And therefore, the Hamiltonian must commute with parity. you see. And of course we knew that the weak interaction violated parity. And I was told that there was an answer to that question, which was originally suggested.

T.D. Lee had thought about this. And he said, yes, but when parity acts on a left-handed neutrino, it doesn't go anywhere because there's no right-handed neutrino. And therefore the HP commutator isn't defined by the states. And I had two objections to that. one was, who said there's no right-handed neutrino? We just hadn't found one as of the time that parity violation was discovered. And the second was, there was, there's a landabaryon which decays into a proton plus pi minus,

and it's parity violating, and it doesn't involve neutrinos. So why should parity be violated in processes that don't involve neutrinos? So I was puzzled by that, but I didn't know the answer. What did he say when you said that to him? Well, that was, and I wasn't talking to TD Lee. I was just talking to my professors at the Weissman, Inso. I see. And, well, they agreed that there was an issue.

But I didn't know what to do about that. When I started to work on dynamical symmetry breaking, it occurred to me that parity could be spontaneously broken, namely in the vacuum, not in the Lagrangium. But to do that, I'd have to change SU2 left. cross U1 to SU2, left cross SU2 right cross U1, make it chiral. And the only way I can make it chiral would be if there were right-handed neutrinos and right-handed currents.

So now I have to spontaneously break it down to SU2-left-Cross U1. But it turns out that there was a way to do it. And that is, when you do the Dirac mass, it's si-bar-si, which links the right-handed fermion with the left-handed fermion. When you look at a myeronomas, it's si-transposed C-Sci, where C is charge conjugation, and that mixes right with right, or left with left. So if the right-handed neutrinos got a myeronomas, you would break the right-handed sector. The right-handed neutrinos would become very heavy. The right-handed currents in the S-O-2 right would become heavy, and you'd be down to S-U-2.

two left cross you one. And then you could do the standard thing. So I realized that parity is probably a good symmetry in the real world. And there's another reason for doing this. And that is, if we take the strong interaction, it's QCD, QCD has a global chiral symmetry. And some of it is gauged and becomes the weak interaction. But that made no sense to me. If you're going to gauge any of it, gauge all of it. And therefore, for me, the weak interaction is the gauge of the strong interaction chiral symmetry. But that requires right-handed neutrinos exist. So my original concern was, well, parity would take left-handed to right-handed. I said, yes, but parity gets broken, even though the right-handed neutrinos are the mechanism for breaking the parity. And so everything

ties together. Now, it turns out that there's some evidence for this now because we have, we found that the neutrinos that we see neutrino oscillations are massive. Now, the reason that they're massive is if the two, any states that are degenerate, then any linear combination of them is still degenerate. Therefore, the only way you can get neutrino oscillations is if they're non-degenerate. And therefore, at least one of them has to be massive if there's an oscillation. How would you make them be such small masses? Well, there's something called the seesaw mechanism, which is a feed-down from grand unified theories through the right-handed neutrinos. So these very light particles get the square of the dirac mass

divided by the myeronomas of the right-handed neutrino. So right-handed neutrinos now are very much in vogue. Okay, so that was the first issue. The second issue was something very strange that I could not understand. The Lorentz group has Fermi... The branch group is the fundamental group of nature. Its fermions are reducible into a right-handed vial fermion and a left-handed vile fermion. So the Dirac fermion, which contains both of them, is reducible under the Lorentz group. And I could not understand how could the Lorentz group, which is the fundamental group, of nature, be reducible under the spin when the spin a half is the fundamental building block of nature.

It doesn't make any sense to me. Something was wrong. But as a grad student, I didn't know what else to do. So I left it there. Just a moment. I'm not understanding the problem. Tell me the problem as you saw it as a grad student. Okay, the problem was the left and the right-handed fermions are independent representations of the Lawrence group.

they're not connected. No Lorentz transformation can take you from one to the other. So if you try to build a theory on left-handed fermions or a theory on right-handed fermions, those are disconnected theories as far as Lorenz is concerned. But Lawrence is the fundamental group of the universe, at least was thought to be at that time. And therefore, I would have expected that the fermions,

if they're the building blocks, would have been connected by some transformation. except I didn't know anything about what it might be. When I started to work on conformal symmetry, I realized, I told you, the conformal group has these five extra operators beyond the Rensum-Poincaure, those five transformations mix the right and the left-handed fermions. In other words, the Dirac fermion,

four component and four times four, 15 plus one, and we've got six Lorentz, four translations, one dilatation, and four conformal. It's 15. The Dirac spinner is the fundamental representation of the conformal group. And therefore, the conformal group, which I've been using for gravity already,

I discussed that, the conformal group must be the fundamental theory of the universe, and it's the symmetry of the light, Why I don't like supersymmetry, super symmetry is super symmetry is not the symmetry of the light cone. And that's why I didn't like it. But conformal symmetry is the symmetry of a light cone, and therefore fermions have to be four components. And then there have to be right-handed neutrinos.

The neutrino could suddenly not be two-component anymore. My whole question about right-handed neutrino is going way back to when I was in grad school, gets resolved by the conformal group because it says right-handed neutrinos must exist. Now, if we're permitting right-handed neutrinos, then how is that not a candidate for dark matter? Because you're already excluding dark matter or salt dark matter with a parameter. So where are these right-handed neutrinos then? Well, okay, for the moment, they're out of reach. I'm sorry to have to say this, and I'm really very disappointed.

And that is, I haven't yet come up with a theory that would give me an upper bound on the right-handed neutrino mass. I've only got lower bounds, and those lower bounds are not achievable at a large Hadron Collider. And so that problem is still open, I don't know. Right-handed neutrinos as a candidate for dark matter would have to have a much, much smaller mass. It'd have to have maybe a KV mass

or so on. Can I ask you a silly question? Sure. Of course. If the Higgs is composite, do you believe the graviton then is also composite or is fundamental? It's neither. It doesn't exist.

Man, talk about fighting words. Yes. Going back to this conformal theory, when you quantize it, you find something very, very strange. And that is, it's a fourth order theory. So you would say, I'm going to have to have two gravitons

because I've doubled a number of degrees of freedom. But there's a theorem due to Weinberg, which says that if you have a spin two particle, it must couple through the Einstein tensor. And therefore, I can't have that occur. But there's an escape. clause. And the escape clause is it requires that the metric of that Hilbert space be positive definite. Now, we said a moment ago, oh, but I just showed you it wasn't negative definite,

but what I didn't tell you because it was unrelated at the time, it turns out that I get zero norm. And so what happens is the graviton becomes an zero norm state, not a positive norm state. and so you have gravitational radiation because it's a covariate metric theory but when you quantize the gravitational radiation you don't get a particle but even though we always did that with photons because that presupposed that you had a Hilbert space

of positive definite norm and Carl and I wrote a paper to show that in these fourth order theories you can actually get zero norm gravitons and so the result of that is that they're not observable Now, I'm going to say something which you should never quote me on.

Well, I've written it, so I'll say it. It's possible that the collapse of the wave function is due to the emission of the zero norm gravitons that we don't connect, that we don't observe. But that's about as far as I've gone on that question. This is interesting. I'm going to get you to go a tad further. Yes. Because there are two other theorizers that I see you be extremely close to. One is Turok slash Latham Boyle, which I'm going to just place as one theorizer.

And then the other is Roger Penrose. So taking the symmetries of a light cone extremely seriously, also reminds me of Twister theory. And then the love for conformal symmetry and his conformal cyclic cosmology is, it also has a rhyme there. Yes. Now you saying that gravity may be connected to collapse also has a rhyme. with Roger. Yes, of course. And there's so many other rhymes with Turok Boyle and that whole research program as well. So I'm wondering, what is the relationship between these two

broader research programs? I have visited Neil and Latham in Edinburgh a few summers ago, and what they're doing is they have a CPT-invariant universe, which has the universe that we have, and then there's a mirror reflection of it, and that can get rid of the Big Bang singularity. And that's exactly what Roger Penrose is doing as well, because he has a cyclic universe. And when I first started working on conformal gravity, I realized that it had a cyclic solution, you see.

And that would have been the case if the curvature were positive. But when I finally showed that the curvature was negative, it's a theory which never has a singularity. And the way the theory solves the fatness problem is by never being singular. And so, I mean, roughly the following. You write down a dot squared plus k is equal to row, is the Friedman equation.

Row goes like 1 over 8 of the 4. 1 over 8 of the 4 blows up when A is 0, a dot blows up when A is 0, and K is irrelevant. And that's the flatness problem. So to get around that, Alan Gooth puts, has an exponential growth before then, but what I do is I say, ah, yes, but you've got the wrong equation. It's a dot square plus k is equal to minus row. And then there's no singularity.

And so then you've got one universe that lives forever with no big bang, and that means that in the early universe, the radius is finite. So that's a little bit different from what Penrose and Turok and Boehler are doing. So there's no period of inflation? No, because you never needed it. Because the universe, if you look at the integral of DT over A of T, and you take A of T is T to the N, if N is less than one,

then the integral of DT over T to the half is T to the half is finite. And that's the horizon. If you do DT over T, you get log T, which blows up at T equals zero. If you do DT over E to the HT, it also blows up. It also has no horizon. So you will get a horizon for any theory for which the expansion radius is T to a power greater or equal to one, and exponential is certainly in that category. So you don't need to be exponential to solve the horizon problem, and you don't need inflation

to solve the flatness problem. Now, I said before that inflation in the end doesn't solve the flatness problem, because what it shows is that omega-M plus omega-K plus omega-Landa is one, that omega-K is suppressed by this exponential expansion in the early universe, so you get that omega-M plus omega-M- a mega lander is one. Why? However, before you had Lander,

you had omega M equals one, and therefore you had a prediction that the matter density of the universe was critical. Critical density, meaning it equal to one. But when you have omega M plus omega Lander equals one, you've now got a real problem,

because a mega rm is redshifting, omega Lander is not, and they have to add up to one today. So that's the point three and the point seven. So somebody had to adjust the very early universe 10 billion years ago so that land, such that the matter density would fall in such a way

that it would just be catching up with, that land would just catch up with it today. So you're still got a fine-tuning problem, which is called the cosmic coincidence problem. But inflation was introduced to get rid of fine-tuning. and yet you still have a new, you've solved the five, you've got a new one. So what was thought when the accelerating universe was first observed,

which was 28 years ago, it was thought that something would happen before inflation, which would be related to quantum gravity, that would lock you into those values so that it would take 10 billion years before a mega-lander could build up to one. and so people look to string theory to see if they could solve that problem.

Can you show that there's a string theory very early universe pre-inflation which will lock you into the right values of omega-M and omega-Lumber? Now, of course, that hasn't happened. People always say that string theory can't be falsified at ordinary energies,

you have to go to Planck-scale energies. But that's not true. we know that amelander's 0.7. We know it's not 10 to the 120. Therefore, string theory has to explain why it's 0.7, otherwise it's falsified. If string theory could come up with a number for a megalander, and it turned out not to be 0.7,

then string theory is falsified. And that's ordinary energies. So that's the challenge for string theory is to explain why it's not 10 to the 120. And for the moment, it looks like the more natural thing to occur in string theory is anti-descita, not decider, which would make a Megalander negative.

Now, that's not the only thing that can happen, but overwhelmingly it looks like you get a Megalander's negative. And so that problem is still there. There's still a fine-tuning problem in inflation, even without string theory. And despite everything, Omega M equals 0.3, omega lander equals 0.7 really works remarkably well for the fluctuations. And so if I'm going to be able to get anywhere, I've got to recover those results.

Otherwise, I have a theory of nothing, not a theory of everything. Even with everything else I've done, with the dark energy, with the dark matter, with the quantum gravity, with the PT theory, with all of that, I still have this problem. of the fluctuate. I still have to solve the fluctuation. It's not that I can't, it's not that I've calculated them and it didn't work. I just have this problem that I have to do the calculation. And that, that's where I am at the moment. So, I've told you about two things that bothered me in grad school, the commutation of the Hamiltonian with parity, and the issue of the Diracfermion being reducible under the Lorentz group, getting me to the conformal group and right-handed neutrinos,

But there were two other things that bothered me. What are they? Okay. We were told about the CPT theorem, and we were told that it was a great success because it predicted that the K-plus lifetime and the K-minus lifetime were equal, which they are. So particle and antiparticle have the same lifetime, which is a consequence of CPT. The only problem that I had with that was that the proof of C-PT theorem required that the Hamiltonian be her mission,

and therefore neither the K-plus nor the K-minus would be. decay in the first place. So it just did not make sense to me that you could apply a theorem that you only established for omission Hamiltonians that you could apply it to decays. Now there was a related problem, which is the fourth one, which is we learned very early on that when you look at scattering, in the scattering amplitude, you find a cross-section which peaks when delta is pi by two, when the phase shift is pi by two, and you fit it with a bright vignor, you continue the bright vignor into the complex plane and you get a pole and you identify that pole with a particle.

And how do we, the particle data tables contain lots of resonances that have been observed by seeing peaks in the cross-section. Now, the problem is you can't have an isolated complex eigenvalue. You either have non or you have them in complex conjugate pairs. because of CPP? Because of CPP symmetry. First of all, I proved CPT theorem without hermiticity, only using probability conservation,

and then I saw that I needed a complex conjugate pair of solutions. And then they would be, they could be eigenstates of the Scaffering Hamiltonian, and now you will immediately ask me, well, don't I then get two resonances? Well, the answer is no, and it's going back to what we said before, it's 1 over E minus E1 minus I gamma 1, minus 1 plus I gamma 1. That's the propagator, and there's a minus sign in the middle.

And when you add those two together, you just get one resonance when E is equal to E1. But if you added them together, you'll get something totally different. It's the relative minus sign that gives you unitarity, except everybody tells you the minus sign means it's a ghost. But Carl and I had shown it wasn't a ghost in the first place. So now we can put in the relative minus sign. And that's the Hamilton, that's exactly the propagator that Dirac was using to try to get physics without dead wood.

I think it was, he was trying to get better behavior asymptotically. And Leon Wick did the same thing. and so what Carl and I were able to show was that even with the relative minus sign you still have unitarity and you'll have to have it because I'm getting the complex conjugate pair because of probability conservation

and so if I have probability conservation I must get unitarity because that's what unitarity is. It says you have time independent evolution. So all of those things hang together. So then I came up with something very surprising. I only released this very, very recently.

I now go, I went to the square well, and I plotted that propagator that I just gave you 1 over E minus E1 minus I gamma minus 1 over E1 plus Igamma. I took the modular squared to get the cross section. I plotted the cross section, and it's a bell shape. Okay. which is a resonance and it's an exact resonance

and you can fit it with a bright Vigna but you have to fit it with a bright Vigna with a different gamma with a different width and that's not the right width and that's what people have been doing and they've been fitting it with this width using the bright Vigna

and I'm saying is that that's not correct because the bright Vigna was never an eigenstate of the Hamiltonian and you can see it in the following way You take the square well, it's got a real potential. So you write down the eigenvalue equation, determinant of H minus lambda i equals zero. H is real, if the potential's real.

So that's a real equation. Real equations only have two kinds of solutions. Real? Or complex conjugate pairs? Therefore, there must be complex conjugate pairs, and they must combine, the way I just described with the relative sign

in a way that still preserves probability conservation and give you a normal bell-shaped cross-section and that's then a real-life elementary particle and so what you have

is you send in a wave and you get scattering of two particles and then the the cross-section grows like either the plus gamma-t gets to the peak

and then it falls like E to the minus gamma T. So you need both effects to produce the one peak. And that's basically the same thing that I was describing about Ephraim Steinberg's experiment. It's the same idea.

The time delay and the time advance cancel each other. Have you spoken to Steinberg? No. No. Not yet. Not as of yet. So you can see that things that I was worried about in graduate school,

I discovered that there were actual answers to them, which were not the answers that could be given at the time, and that they've really tied in with everything that I've been doing. And I've really followed my nose in a way. I wanted to get control of the cosmological councilman. I went to conformal symmetry. I solved the theory. I found I could get rid of dark matter.

I applied it to the accelerating universe. I found that I could fit it without fine-tuning. I applied it to the early universe and found that I had no horizontal flatness problem. And then when I quantized it, I found that it was a PT theory, which, again, I didn't know at the time I started working on it. And all of this is just, it's sort of flowed automatically

from one step to the next. Do you have any ideas to how the Higgs acquires its dynamical mass? Oh, yes. Okay. When you look at electrodynamics with a fixed point,

then you get scale invariance with anomalous dimensions. And so the Fermion propagator scales as p-squared to a power. That's what anomalous dimensions mean. You get scale invariance with a power. If that power is negative, then you can show that the bare mass is zero.

That's the same condition. But as you make the power more negative, you make the theory more convergent in the ultraviolet, but you at the same time make it more divergent in the infrared. And when it comes down by one whole unit, the infrared divergences become so severe that they force you into a different Hilbert space. and that's the mechanism

and then the electron mass is dynamical so if you like when I first started thinking about this you know there's this idea that goes back to Poincorre of how do you stabilize an electron if it's energy

if MC squared is E squared over R namely if it's self-energy is the entire mass and Poincoree introduced the Poinceree stresses those stresses Those stresses are coming from the vacuum. So in a sense, this dynamical theory that I described to you is really a theory of point-pera-ray stresses.

And then the massive electron is stabilized. Now, what about when it comes to the dark matter, the evidence for it isn't just rotation curves. Of course, of course. So what about, say, collisions, galaxy collisions or other structure formations or cluster dynamics, etc.? Or lensing. Okay, the fluctuation theory that I'm working on

will address the anisotropy in the CMB and will address a large-scale structure. Whether it'll do it correctly, it remains to be seen. However, the issue of gravitational lensing is much more subtle. You see, you just take the lensing formula in a standard textbook, like Weinberg's book, that presupposes that the light is coming in

from an asymptotically flat geometry because that's the standard, Schwartz-Till metric. Now, if I have a linear potential, light is not coming in from an asymptotically flat geometry, and the whole calculation has to be revised, and there's a spirited discussion in the literature as to what is the correct gravitational lensing formula when you have a non-assentotic geometry. Now, you might say, oh, but what about the sun? How do we get gravitational bending by the sun? And the answer is we never measure gravitational bending by the sun.

What we do is we measure the light without the sun, and then six months later, the light passes the sun. We're measuring the difference. And in the difference, the background drops out. So we're not measuring the trajectory of a photon as it goes by the sun. We're measuring the change in the trajectory as it goes by the sun. Those sounds are the same. Can you tease out the difference for me, please?

Okay. The calculation of the bending of light by the sun is the bending in the local geometry. And you're doing two measurements, one with the sun in the way and the other with the sun not in the way. And you're measuring the difference. The difference, it didn't matter what was going on away from the sun. That's the same in both cases, so it drops out in the difference. So the bending by the sun is a difference experiment. Lensing by a galaxy due to a distant quasar and the light coming by, you're seeing the entire trajectory.

In other words, you can't do the experiment with the cluster of galaxies out of the way. So it's not a difference. And that's why it's different, and that's why it's become so very complicated. And I'm fully aware of that. Yes, there's many more things I still have to do. Let me ask you to wrap up a personal question, if you don't mind. Sure. I'm sure as you're developing this theory, as you have developed this along with your collaborators over the past couple decades,

that you've encountered criticisms of various sorts, like you mentioned a ghost, and what about this, and what about that, and what about some that I've thrown at you today. But I'm also curious if you've received more than just criticism, more like derision that what are you doing? You're a fool for working on this. This is obviously down the wrong path or something like that. Well, maybe. Well, I don't know if I'll even put this part in, but I'd notice from running this channel that people who have their own theory, they tend to look down at other theories. It's not just mere critique at some sort of scientific level. Yes, I get a lot of resistance.

Let me put it this way. Everything I've written eventually gets published, but it puts a demand on my nervous system to get it published. You see, it was bad enough that I'm working on an alternate gravity theory, but I've also given up hermiticity. So I'm getting flak from every direction. but the one thing that saves me is tenure. Tenure allows you to work on any problem you want.

But I've followed this theory because I've tried to describe it to you in this interview. I've just followed the steps one after the other, and that's where it's led me. It's not that I've never invented anything. I've just taken the theory and solved it to see what it looks. looks like. And I mean, I can tell you now, but some of these things take years to figure through. But once you figure it out, it's very simple to describe it. But it's not gone off in different directions. And look, I mean, you start out with the standard theory. It doesn't work for galaxies, so you invent dark matter. It doesn't work for cosmology. You invent dark energy. It doesn't

work for quantum theory. So you invent string theory. but you keep on patching it up. I haven't done that. I've just taken the theory and I've solved it. And it'll take me wherever it'll take me, and I have no control over that. And then I fit data. I mean, I shouldn't be fitting all this data

if I'm so completely on the wrong track. The reason I laughed is just because I thought you were going to say, what saved me, Kurt, and you were going to say something ethereal, and you're going to say, the Dirac's confidence saved me or the Laplacean

or the Laplacean or my wife saved me or something like that and he says something extremely practical. Tenure. That's also true. Tenure is the thing. It gives you the freedom

to work on things that your colleagues don't approve of. Whereas if you're not tenured and you're seeking tenure, you have to work on mainstream physics. You will never get hired if you don't work on mainstream physics. But once you get hired, once you get tenure, you're free to work on non-mainstream physics. Now, I would not choose to work on non-mainstream physics if I didn't think the mainstream physics was,

I wouldn't choose to work on non-mainstream physics. I didn't think the mainstream physics were getting nowhere. I mean, you know, you have the dark matter problem, the dark energy problem, the quantum gravity problem. and you just proceed as though you can ignore them. And I don't understand. I don't understand why people would do that. But I'll tell you the difficulty that I have, and it's this. I gave a talk once, I gave talks to community groups.

I gave a talk to a middle school, as a middle school children, and I asked them, who knows the name of a scientist? Every hand shot up, and they all said, Einstein. And then I said, who knows the name of another scientist? Total silence. And that's the danger of challenging Einstein. But I'm not challenging him. I'm taking what he's done, and I'm looking at where there are issues that could still be addressed. I don't mind if you don't like what I've done, but you've still got to address the issues I've raised. Professor, what advice do you have for students who are watching?

Well, don't trust experts. You know, Vigner was once asked, what's the definition of an expert? And he said, it's the person who's made the most mistakes in the field. And I think that's true. I mean, physics is, we make mistakes. And don't be worried if you make mistakes. Trust your own judgment. I also tell everybody, don't work on conformal gravity until you get tenure.

I think that's the more practical advice there again. Practical advice. By the way, I say that a bit cheekily. I do hope people look into your theory, and I'm going to place many of your papers on screen and in the description so the researchers who are watching can inform themselves. Yes, okay. what I wanted to say was while we have this grand dark matter head of this

there's an intercurrent of dissatisfaction with it I mean it's been going on now for 40 years and they keep on not finding anything and nobody's given up yet I mean there are no time limits in physics it may take 100 years to find it that's the way the physics works

But nonetheless, I think that people are dissatisfied, but not ready to jump. And if this goes on, I mean, John Moffert and myself, Monty Milbram are, you know, we're really out in left field. And whether people would ever join us remains to be seen. fairly, it's not just that we're against dark matter, we have developed theories to deal with the absence of dark one. And that's the, that's, it's not just words. I mean, and, you know, the standard people have, I've really got to come up with, with, with, with, with, with, with, one of these formulas, either the mog or myself or the mon formula and derive it from landa CDM. You can't just say that, you know, what we're doing is nonsense. I'm saying fine, but the data don't

so. And, you know, physics, maybe I can give another answer to you. In the end, physics is an experimental science. And I've always said that the number of wonderful mathematical theories exceeds the number of physically relevant theories by n-1. So don't be too seduce. by the mathematics. You've got to keep data in mind. When I did my PhD, I did it with a phenomenologist named Uri Maor. It was a Weizmann Institute in Israel. And that's what I learned from him. Always keep an eye on data. Nature will tell you what to do. You should not, you can't proceed without nature. Let me explain. You take the rotation group and it's designed for integer bin. Well, it has half integer spin representations. But that doesn't mean they exist. They may exist,

they may not exist. The rotation group alone will not force them to exist. Uh-huh. You see. What does force them to exist, by the way, is the conformal group. Uh-huh. Because then four-component fermions becomes a fundamental representation. So mathematics will take you so far. And that's the language of physics. That's what we do. But you've always, we should never lose sight of data. And that's what I tell my students, is never lose sight of data. Nature will keep you honest. Professor, thank you for spending so much time with me.

Right. And I hope you enjoyed it. And I know the audience enjoyed it. It was and is an honor. Thank you. Thank you very much for inviting me. and so giving me a chance to explain what I'm doing, it's basically my entire career.

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