“Non-empirical physics”

Peter Woit has a new post with this title, which depresses me, not because of the fact that he quite rightly condemns “non-empirical physics” as an oxymoron of the first water, but because of the utter negativity of the discussion that ensues. The way to combat the futility and absurdity of “non-empirical physics” is to do some proper, old-fashioned, empirical physics. If the empirical results contradict somebody’s theory, however eminent they are and however entrenched in the establishment it is, then the theory is wrong.

Then you need to find and fund some theorists who are prepared to say, the theory is wrong, let us find a better theory. Stop funding theorists who say, the theory is right, we just need to make it more complicated to deal with a few little problems. (We do not need any more Ptolemaic epicycles.) And above all, stop funding those theorists who say, the theory is right, and the universe is wrong.

The progress of science has always relied on empirical observations of relationships between quantities that may not previously have been considered to be related. There is nothing more fundamental in physics than this. Unless perhaps it is the discovery that quantities that were previously thought to be related are not in fact related. (The Michelson-Morley experiment is a classic example of the latter. The Wu experiment is an excellent example of the former.)

One of the fundamental problems in physics is to explain the masses of the elementary (or other) particles. In other words, how are these masses related to other things? There are some explanations for some of them, of a rather epicyclic nature: I have seen papers that claim to be able to calculate the neutron mass from other, supposedly more fundamental, concepts, for example. But mass is fundamentally a gravitational concept, and if your explanation of mass fails to involve gravity, then I fail to see how it can explain anything.

Mass must be related to gravity, in some way. How it is related, we don’t know. But why don’t we do some good old-fashioned empirical physics, and look for relationships that have not previously been considered? Why are the neutron and proton masses so close, and yet not equal? They differ by a smidgeon, just about .138 per cent. Why are the lengths of the solar day and sidereal day so close, and yet not equal? They differ by a smidgeon, just about .274 per cent. So what? I’ll tell you what: one of these smidgeons is almost exactly twice the other.

Now if you are doing non-empirical science, you will say, there is no way these numbers can be related by any physical theory, this must be crackpot nonsense. But if you are doing empirical science, you might say, OMG, you cannot be serious, can you, really?! And then you might think about it, and still you think, no, this is impossible, how could there be any such relationship? But if you are a real (empirical) scientist, you will investigate further, as I did, and you may find many more such coincidences.

Then you have to evaluate whether these coincidences can be plausibly attributed to pure coincidence. This is something that physicists have to do all the time, especially in high-energy physics, where a new “discovery” is a blip in a huge mountain of data, whose significance or otherwise is never obvious at first glance. Now my coincidences can’t be evaluated in quite the same way, and I cannot distinguish between 4.9 sigma and 5.1 sigma, but I can tell you without a shadow of a doubt, these are not all coincidences: some of them may be, but not all.

That is empirical science, as it used to be done. Yet in this day and age, apparently it is crackpottery.

What is physics?

As a mathematician, I have given a certain amount of thought to the question “What is mathematics?”, and I confess myself somewhat dissatisfied with the “silly clever” answer that mathematics is what mathematicians do. Mathematicians in general seem to like this answer, perhaps because “silly clever” answers are a speciality of mathematicians. But mathematicians, above everybody else, should appreciate that this is a circular definition and therefore is not a definition at all.

In mathematics, the purpose of a definition is to distinguish between things that are (A) and things that are (not A). But in fact this is impossible, and all one can actually hope to do is to distinguish between things that are (A and not B) and things that are (B and not A). That is what all the early 20th century work on logic established beyond all reasonable doubt.

In other words, mathematicians have proved that it is not possible to answer the question “what is mathematics?” The kinds of questions one can hope to answer are (a) “what is mathematics but not physics?” and (b) “what is physics but not mathematics?” When framed in these terms, the questions and their answers are a good deal more illuminating. Thus, we may take a particular subject which we assume is part of mathematics and/or physics, and ask ourselves which it is.

But in order to do this, we need a definition to distinguish mathematics from physics. We don’t need to define physics, and we don’t need to define mathematics, but we do need to define the distinction between the two. Now I have no doubt that you have your own definition, but I would like to propose my own: physics is the study of our universe, and mathematics is the study of all possible universes.

You may not agree with my definition, but I think you have to agree it is succinct, it is clear, and it can be applied in practice. Let’s try it on an example: is string theory physics or mathematics? The answer is obvious: string theory is the study of all possible universes, and therefore it is mathematics, not physics. There are two famous books, by Peter Woit and Lee Smolin, published in 2006, which argue this thesis in depth. I am prepared to admit that string theory is mathematics, and these two books argue conclusively that it is not physics, so I don’t think there is really much room for any further discussion.

Integral representation theory

In my papers on potential applications of representation theory to particle physics, I work with real representations, in order to obtain the real Lie groups used in the standard model. However, I remarked early on that this is really only an approximation to the underlying integral representations (that is, representations with integer matrices). The only problem is that integral representation theory is seriously hard.

Almost everything goes wrong already in very small cases. Even the Krull-Schmidt Theorem fails almost always: so you can write your representation as a direct sum of indecomposables, but there is no uniqueness theorem for this decomposition. I started to speculate on whether this might tell us something about particle physics. For this purpose, consider the proton (and probably also the neutron) to be indecomposable, but not irreducible. The irreducible constituents can be thought of as quarks, which I suppose might be 1-dimensional, so that the nucleons are 3-dimensional. (Larger representations may be needed to incorporate all the necessary properties.) Then perhaps the failure of the Krull-Schmidt Theorem might be interpreted as saying that although you can decompose an atomic nucleus as a direct sum of protons and neutrons, this decomposition is not unique. Maybe this tells us something important about the structure of the atomic nucleus?

Another problem is that the number of distinct indecomposables may be infinite. This happens already for the Klein fours-group. Again, this might be useful for physics, if we want to regard an atomic nucleus as indecomposable (but not irreducible, of course, except possibly in the case of hydrogen).

Another problem is that, even for cyclic groups, number theory rears its ugly head: the number of indecomposables in this case is finite, but this number depends on the class number of the cyclotomic number ring.

Number theory comes in again as soon as we have a 2-dimensional irreducible, for example for dihedral groups. Now we have to consider the action of SL(2,Z) on the representation, and if we’re not careful, the whole theory of modular functions and modular forms comes falling down on our heads. Well, some physicists do seriously consider that this part of mathematics is useful for fundamental physics, and who is to say they are wrong? I hope they are wrong, but I fear they may not be.

Now, if I am right that GL(2,3) is the group whose integral representations we need to understand, then we have some serious work to do. This group can be split into a normal subgroup Q_8, and a quotient S_3. Since this is a split extension, there is also a subgroup S_3, but it is not unique (in fact there are 8 subgroups S_3). Roughly speaking, Q_8 tells us about spin, and S_3 tells us about the weak and strong forces. I can tell you everything that is important about the integral representations of S_3, and analyse the weak doublets and colour triplets in some detail. But until I can extend to S_4 I cannot tell you about momentum or angular momentum, and therefore cannot tell you about mass, and how to distinguish the three generations.

So that’s my progress report. Now back to work.

The weak interaction

I have been trying for years to understand what particle physicists mean when they say that the gauge group of the weak interaction is SU(2). After a while, I stopped listening to what they say, and started watching what they do. Because what they say and what they do are three different things. 

First of all, they make ladder operators – these are the operators that convert neutrons to protons and vice versa – one step up and one step down the ladder. Ladder operators are not properties of groups, they are properties of Lie algebras. The Lie algebra of a compact group has no ladder operators – they are in the Lie algebra of the split group, in this case SL(2,R). But in fact both the group and the Lie algebra are needed, so what is actually used is the full algebra of 2×2 real matrices. Well, yes and no, physicists plaster complex numbers all over the place, so that what they actually use is an algebra that is isomorphic to this matrix algebra, but they do not notice this isomorphism because of all those complex numbers. And then, electroweak mixing picks out a particular copy of U(1) in this matrix algebra. Here U(1) is really Spin(2), so that the actual structure that particle physicists use is the Clifford algebra Cl(2,0). So when they say the gauge group of the weak interaction is SU(2), what they actually mean is the gauge algebra of the weak interaction is Cl(2,0).

Oh, would that it were so simple! If we now watch what they do in practical calculations with the Feynman calculus, we find that they don’t actually use Cl(2,0), they use the direct sum of two copies of the complex numbers, with generators 1, i, gamma_5 and i.gamma_5. Why? I have no idea. It seems perfectly clear that i is represented by the matrix (0,1/-1,0) and that gamma_5 is represented by the matrix (1,0/0,-1), so that i and gamma_5 must anti-commute. Yet for some reason the assumption is made that i and gamma_5 commute. I have not the slightest idea where this nonsensical assumption came from.

What was the point of going to all that trouble to construct a non-commutative gauge group and Lie algebra and Clifford algebra, if you are then going to throw it all away and use a commutative algebra instead? The commutative algebra that is used in the Feynman calculus does not capture the physical reality of what is actually going on in weak interactions. It reduces a 4-dimensional real algebra, that is capable of describing beta decay as a process involving four particles, the neutron, proton, electron and antineutrino, to a 2-dimensional complex algebra that can only describe two particles at a time, and cannot therefore explain where the mass comes from or goes to. What about muon decay? Again, we need four particles, so we need the 4-dimensional real algebra. And we need the non-commutativity in order to get some mass.

When I was employed as a Professor of Mathematics to teach mathematics to students, I found that one of the most challenging parts of teaching was trying to explain to students why their proposed solutions did not work, or did not make sense. I like to think that I got reasonably good at it after a few decades of practice. That challenge was many orders of magnitude easier than the challenge of trying to explain to theoretical physicists why their cherished “standard model of particle physics” doesn’t make sense, and doesn’t solve the problems they claim it solves. But it is not a difference in kind, it is only a difference in degree. The model uses group theory in an incoherent, nonsensical way, with arguments that consist of complete non-sequiturs, an absolute disregard for whether a matrix belongs to a group, a Lie algebra, a Clifford algebra, or a representation, or whether coefficients are real or complex, or whether groups are finite or infinite, split or compact, real or complex, or act by multiplication or by conjugation, or whether isomorphic groups are equal or not, and so on and so on and so on. Students who don’t listen to their teachers condemn themselves to a life of ignorance. But, once a teacher, always a teacher, and that is what I shall be to my dying day. The fact that the students don’t listen is demoralising, of course, but it is something I have become used to.

Lectures in Lausanne

On 7th June 2017 I gave a lecture in Lausanne in which I claimed that SO(3,4) was the group that was needed for unification of particle physics and general relativity. The context was a summer school with two courses of 7 lectures, one on representation theory of general linear groups, given by Geranios (Harry) Haralampas, and the other on subgroup structure of simple groups, originally due to be given by Colva Roney-Dougal, who had to withdraw, and in fact shared between me and Inna Capdeboscq. For my last lecture I asked the audience if they wanted to hear my prepared lecture on subgroups of Mathieu groups, or instead hear about my theories for how the representation theory Harry was talking about could be applied to the unification of fundamental physics. They opted for the latter, so I ceremoniously tore up my lecture notes and began.

Since then, this particular idea has disappeared behind a whole host of new ideas, and I wouldn’t be mentioning this here if it weren’t for the fact that the latest group algebra model has brought it back into focus. What happens is that there is a 7-dimensional representation with structure 3+4, whose tensor square is equivalent to the group algebra plus a scalar. That means that the group algebra can be represented as trace zero matrices, in such a way that the group acts the same way on rows and columns. I take this as essentially a coincidence, but it has the effect that all of fundamental physics can be effectively modelled with 7×7 matrices.

Now both the 3 and the 4 are orthogonal representations, so that the finite group embeds in both SO(3,4) and SO(7). These groups are not often used in physical theories, but their subgroups SO(3,3) and SO(6) are. These subgroups break the spinor representation 4, which means that these theories put it back in by going to the spin group Spin(3,3) = SL(4,R) or Spin(6) = SU(4). The former theory is called general relativity and the latter is the Pati-Salam model of particle physics. In both cases the symmetric matrices now split as 7+20, and the antisymmetric as 6+15.

The antisymmetric matrices form the adjoint representation, and in the Pati-Salam model the 15 dimensions of SU(4) are supplemented by the 6 dimensions of SU(2) x SU(2) to make up the 21 dimensions. Of course, Spin(7) is not the same as SU(2) x SU(2) x SU(4), so that ultimately this model does not work. General relativity does not include the extra 6 dimensions, but instead identifies the 6 dimensions of SO(3,1) with the 6 dimensions of SL(2,C) to make up the difference. It is this egregious error which is responsible for the whole sorry mess of curved spacetime. The 20-dimensional (6×6) part of the symmetric matrices holds the Riemann curvature tensor, and the other 7 dimensions hold the matter, so that by adding the two together the curvature is revealed as an illusion. I am not sure which three of the 10 components of the stress-energy tensor one can do without here – or perhaps two of the three triplets are combined in some way.

So maybe I was right all along. The great advantage of SO(3,4) over SO(3,3) is that it already contains spin groups SU(2) inside SO(4), so you do not need any extra spin groups or Clifford algebras or anything of that kind. In effect, the vector representation of SO(3,4) behaves like a spinor, and it is the square of this “spinor” that contains all the physics.

Mathematics and music

As a mathematician who is also a musician, I have been asked countless times why mathematics and music go together. My answer for many years has been one that I know people do not want to hear: both require many thousands of hours of relentless and often soul-destroying practice to do well. Anyone who has the personality to deal with that has the potential to do well at both. Of course, what most people want to hear is that it is a special gift, because that lets them off the hook of examining their own incompetence in both fields. While that may be part of it, the real point is that to do well in any human endeavour requires dedication and hard work. And it is the particular nature of the hard work that seems to me to connect mathematics and playing a musical instrument: both are absolutely unforgiving, and the clear distinction between right and wrong means that most of the practice consists in repeatedly getting things wrong, which many people find hard to deal with.

And in both cases, a faultless technique is both essential and irrelevant at the same time. In mathematics, just as much as in music, a faultless technique without imagination is sterile: I have come across many mathematicians in my career whose technique is far better than mine, but who have no imagination, and cannot play the mathematics. Just as I have come across many musicians in my career whose technique is far better than mine, but whose music-making has no soul. The technique is not an end in itself, it is a means to an end. In music, technique is there to make something difficult and delicate sound easy and assured, so the listener can be transported to a place of beauty and imagination. We do not want music to sound difficult – we already know it is. The same is true of mathematics. A real piece of mathematics sounds like a piece of music – you may not understand a word of it, but it transports you out of this world into a place you can hardly imagine. And a true virtuoso makes it all sound so easy, as years and years of practice appear to slip effortlessly off the chalk onto the blackboard. The goal of the real mathematician is for the listener to say at the end of it: that’s easy, why didn’t I think of that? Too many mathematicians these days revel in a formidable technique, and play a kind of atonal post-modernist music that no-one wants to listen to. The mark of a true master is playing something really new in C major.

The latest model

My write-up of the latest model, based on the group algebra of one of the two double covers of the rotation group of the cube, has appeared today as arxiv:2104.10165. It has many advantages over all my previous attempts at unification, and in a sense seems to “contain” them all. I have high hopes that this may be the last model, and that it contains exactly enough algebra on which to base a complete unified theory “of everything”. I will update it shortly with a summary of the implications for quantum gravity, and then start work on the real issue, which is how to explain the values of the 24 or 25 unexplained parameters of the standard model. 

At the same time, my paper arxiv:2011.05171 on “Subgroups of Clifford algebras” has received three genuine referees’ reports, and although they ask for “major revisions” I am reasonably confident that I can revise the paper to their satisfaction. This is the first time that any of my papers on physics has been sent to a referee who has taken it seriously, which I think is a cause for celebration in itself.

Brains

Brains have evolved over billions of years, mainly for the purpose of endowing the owner of the brain with a superior power to outwit, evade and overpower their enemies, in order to survive in a hostile environment and pass on an inheritance to their children. The main purpose of a brain, therefore, is to negotiate the difficulties of everyday life. The brains of theoretical physicists, like the brains of most other people, are largely devoted to this task. Anyone who wants to employ clever people, whether they be physicists or anyone else, to think about problems other than the problems of day-to-day life, needs to ensure that they do not need to spend too much of their brainpower on dealing with the problems of everyday life.

In earlier times, such people were either from wealthy backgrounds and did not need to worry too much about everyday problems, or simply didn’t care about having enough to eat or a warm place to live. Such people could often be found quite cheaply, by providing them with a warm enough place to live and enough to eat, and allowing them to spend their brainpower on thinking about other things. Hence the success of the mediaeval universities. Nowadays, we have largely reverted to the older style, that if you want to think for yourself you have to be either wealthy, or prepared to live in a tent.

It is hard to understand how we came to this pass. The premise is irrefutable. In my time in academia, over the best part of half a century, the fiscal powers have done their utmost to ensure that many of the cleverest brains waste most of their time in pointless tasks aimed at ensuring their own survival. The whole point of these powers employing these brains is precisely to avoid this problem. After a while I refused to play the game any more, because life is finite and I wanted to spend what remaining time I might have thinking about the real problems, not about the problems of how to write a grant proposal to pay a meagre salary to someone I would then have to spend a lot of time working with on problems I was no longer interested in. The result was predictable: I took early retirement by mutual agreement with my employer.

Too many theoretical physicists are having to spend too much of their valuable brainpower on ensuring their own survival, instead of thinking about the fundamental problems. No-one these days can afford to spend ten years thinking about a problem in the way that Einstein did. That is why there is no new Einstein. And never will be, while this crazy situation continues in universities the world over.

Failure of unification?

I think I am going to have to give up reading Peter Woit’s blog. The relentless negativity of it is just so depressing. In today’s discussion he again goes over the top to over-hype the undoubted failure of one or two attempts at unification as a failure of the whole idea of unification. If that is your attitude, you might as well just give up and go home, and stop infecting everybody else with your negativity.

My PhD supervisor taught me that it was important to work on four problems at the same time: one easy, one hard, one trivial, and one impossible. That is what I have always tried to do, and continue to do – except that, now I have retired, I’ve more or less abandoned the easy and hard problems, and restrict myself to the trivial and the impossible. But it is essential to understand the word impossible in the sense that the US Armed Forces use: “the difficult we do immediately; the impossible takes a little longer”.

This is the sense in which we need to understand the “impossibility” of unification. You have to believe in the possibility of unification, or you should not be in the business at all. The current situation is that two or three promising ideas for unification have failed. How many ideas are there for unification? Hundreds and hundreds. How many ideas do people like Peter Woit listen to? None.

The problem is obvious. It is not the lack of ideas. It is the lack of listening.

Muon g-2

There has been a lot of excitement about the latest measurement of the gyromagnetic ratio of the muon, and the confirmation that the measured value of g-2 differs significantly from the value calculated using the standard model. Both experiments that measured this value, 20 years apart, were conducted in the same 15-meter diameter storage ring, so that there is no control for the size of the ring, and therefore no control for the variation in the direction of the gravitational field from one side of the ring to the other.

It is completely clear that experiments of this kind *must* control for effects of this kind, and if they don’t, then the conclusions cannot be considered robust in any way. The rush to predict new forces and new particles is premature, when the experiment has not ruled out the possibility that the “new force” is nothing more nor less than our old friend gravity. If you have a candidate quantum theory of gravity, now is the time to use it.

Well, my quantum theory of gravity already explains the CP-violation of neutral kaon decay as an effect due to the change in direction of the gravitational field. The actual value of the effect in that case seemed to be about 3/4 of the sine of the angle between the directions of the gravitational field at the two ends of the experiment, but the apparent factor of 3/4 is not statistically significant, and a factor of 1 is completely consistent with the experimental results.

So let us attempt the same thing in this case. The angle between the two directions of the gravitational field is approximately .015/6000 radians, so that the sine of this angle is around 2.5 parts per million. The reported discrepancies between theory and experiment are around 2.4ppm for the original Brookhaven experiment, and 2.0ppm for the new Fermilab experiment, with a weighted average of 2.15ppm, described as 4.2 sigma.

The discrepancies between experiment and my conjecture are therefore .1ppm for Brookhaven, .5ppm for Fermilab, and .35ppm for the weighted average – in all cases less than 1/4 of the discrepancies from the standard model. Time for people to start listening to me?

There is no evidence at all for any new physics here. Instead there is clear evidence for a mixing of gravity with the standard model.