Anomalous magnetic moments

It has been suggested to me that the way to draw attention to my results is to explain the anomalous magnetic moment of the muon. The magnetic moments of the electron and muon differ from the originally expected value by approximately 1/10 of one percent. Calculations using quantum electrodynamics correct this, and predict values that agree with experiment to at least 8 or 9 significant figures. But more accurate experiments reveal that in the case of the muon, the new predicted value is still off, by approximately one part in a billion (). This is what needs to be explained.

While I cannot at this stage produce a numerical prediction, I can show that there is at least one other contribution to the magnetic moment, in addition to the electro-dynamic contribution. Whether this contribution is big enough to account for the anomaly, I do not know. There are two ways to approach the issue, either via quantum gravity, or via electro-weak mixing.

I have shown that the electro-weak mixing angle of approximately 28.5 degrees is actually made up of two separate angles, of 23.44 and 5.14 degrees. The standard model cannot therefore describe the mixing in full detail, since the two angles do not necessarily contribute equally to the mixing. That is to say, the standard model does not completely separate electrodynamics from the weak force, although it gets very close. Hence calculations using standard model electrodynamics do not quite capture the true picture.

Looking at the issue from a gravitational point of view, I have shown that there is a mixing angle of 23.44 degrees between quantum electrodynamics and quantum gravity. Again, this mixing is not taken into account in the standard model. Hence there is a gravitational magnetic moment that has to be added to the electromagnetic moment in order to obtain the moment that is measured experimentally.

Whichever way we look at it, there is a contribution that comes from interactions with neutrinos, in addition to the interactions with photons that the standard model recognises. Once the contributions of the neutrinos are added into the calculations, I expect the corrected values to agree with experiment.

The number of elementary particles

The counting of elementary particles is very much model-dependent, and dependent in particular on what one means by “elementary” and what one means by “particle”. The standard model counts 45 elementary fermions, plus the same again for anti-fermions. This is far too many to be described by a Dirac spinor, so several extra labels are required. I have shown that by replacing the Dirac spinor with a vector/co-vector pair, one can recover mass information, but only by incorporating interactions between an intrinsic vector/co-vector for the particle and an extrinsic vector/co-vector for the observer.

In other words, observed properties of fermions may allow us to distinguish up to 64 elementary particles, even though intrinsically there can be at most 8. These 8 are directly associated with the 1+3+3+1 dimensions of time, space, momentum and energy. It would seem uncontroversial to identify neutrinos with momentum. The 3 dimensions of space must then correspond to quarks, which have a position but not a momentum.

Which three quarks should we take? It is not obvious, but earlier discussions suggest that   up, down and strange would be a reasonable choice. Their total charge is zero, which is a good sign. We then need a right-handed lepton, such as an electron. Finally we need a positively charged particle to complete the octet. The only reasonable choice, it seems to me, is the proton. Of course, in the standard model, the proton is not an elementary particle. But the experimental fact that it never decays into anything more elementary suggests that it is not unreasonable to describe the proton as an elementary particle.

At least this gives us a first stab at a list of exactly 8 fundamental fermions. It may require modification in the light of further development of the model. The papers archived on this blog give (mostly) empirical formulae that might explain how heavier particles are made out of these light particles.

Anti-particles

Dirac predicted antiparticles from complex conjugation on representations of SL(2,C). If one identifies SL(2,C)/Z_2 with SO(3,1)’ in the standard way, then complex conjugation corresponds to the chirality automorphism of SO(3,1)’. But this automorphism is a reflection in one direction in spacetime, so is not physically possible. There are various ways one might try to interpret this, but to me it says that particles with the wrong chirality do not exist in the real universe.

There is real experimental support for this interpretation, since neutrinos are always left-handed. For massive particles the interpretation is more complicated. Let us assume that Dirac’s work was essentially correct, and the only flaw was to use the quotient map onto SO(3,1)’ to obtain an interpretation of anti-particles in terms of chirality. Then we have to re-interpret anti-particles via complex conjugation on vectors, rather than on spinors.

Complex conjugation negates two directions in spacetime, which must be taken to be time and one direction in space. Such a transformation has determinant 1, so can be achieved in the real universe. This fact permits anti-particles to be observed in the laboratory. Applying this to neutrinos, we map left-handed neutrinos to right-handed anti-neutrinos, which is what we see in experiments. By contrast, true chirality takes left-handed neutrinos to either right-handed neutrinos or left-handed anti-neutrinos, which are not seen in experiments.

To me, this proves conclusively that Dirac was correct to use complex conjugation for antiparticles. It also proves conclusively that the identification of SO(3,1)’ with a quotient of SL(2,C) is a mathematical coincidence without any relevance whatsoever for physics. Isomorphism is not the same as equality.

 

New physics?

The Large Hadron Collider was predicted to do two important things: firstly, to find the Higgs boson, which it did; and secondly, to find some new physics, which it did not. Well, to be fair, it has not so far. It still might. But this seems fairly unlikely from the present perspective. So the question is, is there any new physics to find, or have we essentially already found all of it?

Of course, if you talk to physicists, they will be desperate to tell you there is a huge amount more to find. Which in one sense is undoubtedly true: physics is incredibly complicated and interesting, and the universe is full of weird and wonderful things. There is no possibility of running out of interesting things to discover any time soon.

But the question is about the fundamental building blocks of the universe: are there any more elementary particles, or any more forces of nature? Of course, this can never be ruled out completely, but the evidence is that we have looked so hard for so long, and we have not found anything unexpected in a very long time, that the reasonable conclusion is we have already found everything of significance.

Theory has not woken up to this fact. Theory is still in the mindset of trying something at random, predicting something, and asking experiment to test the prediction. This is an ante-diluvian attitude. The current situation is that experiment has discovered essentially everything there is to discover, and it is now the turn of the theorists to find a theory that  explains what has been found. Navel-gazing, and coming up with an abstract theory, and asking “does this one work?” is no longer a reasonable approach.

There are a few really iconic experiments that illustrate specific places where the theory must be wrong. But theorists are in denial. To me, as a mathematician, looking at the mathematical structure of the theory, there are two crucial places where the conflict between theory and experiment indicates there is obviously an error. One is the dark matter problem, and the other is the neutrino oscillation problem. Experts can and do construct patches to cover over these problems, but that is not the same as solving them.

My analysis, that you can read more about on this blog, is that the neutrino oscillation problem indicates an error in the foundations of quantum mechanics; and that the dark matter problem indicates an error in the foundations of general relativity. I can tell you what these errors are: all you have to do is read my blog with an open mind.

There is no new physics. Experimentally, we have discovered everything of significance. We just need a theory that describes what we actually see, instead of a fantasy universe that exists only in the minds of those who believe in it.

Quantum chromodynamics

The confusion between chirality and duality really comes into its own when it comes to the strong force. The essential point to re-iterate from the previous post is that the relationship between a particle and its antiparticle is a chirality relationship, and not a duality. The entire theory of the strong force is based on the assumption that it is a duality. As far as I can see, very little of the current theory can survive this error, though I am not an expert on the theory, so I may be wrong.

QCD is based on the group SU(3), although in practice this means not the compact real group, but the complex group SL(3,C). By un-picking the complexification carefully, it should be possible to re-write it in terms of the real form SL(3,R) instead. This is a subgroup of the group SL(4,R) of relativistic quantum mechanics. If we want first to look at a non-relativistic version, then the group becomes SO(3). Moreover, this copy of SO(3) is actually equal to the rotation group of space, inside the rotation group SO(4) of Euclidean spacetime.

In the standard model, quarks live in a 3-dimensional representation of SU(3). This is not possible, because then they would be bosons. They must instead live in a 1+3 representation. In the standard model, the anti-quarks live in the dual representation. This is also not possible, as already discussed. The antiquarks live in the same 1+3 representation, with an odd number of coordinates negated in order to reverse the chirality. Hence pseudo scalar mesons live somewhere in the tensor square of 1+3, that splits as 6+10 for SL(4,R).

There are 9 pseudoscalar mesons made out of 3 quarks in the standard model, so probably they live in the 10-dimensional representation. This breaks up as 1+3+6 for SL(3,R), and as 1+3+1+5 for SO(3). There is a question as to how to allocate the 10th dimension, compared to the 3 pions, 4 kaons, eta and eta-prime mesons of the standard model. Notice that the allocation of mesons to a 9-dimensional representation is based on a fundamental error – using duality instead of chirality. Hence it is neither necessary nor sufficient to replicate the standard model precisely. We should instead look back at the experiments the led to the quark model of mesons.

Then we find that in fact five kaons are observed experimentally, not four. There are two charged kaons, and three neutral kaons. The standard model has a hard time explaining the extra kaon, and while it may be generally accepted that the explanation succeeds, it looks like a fudge to me. My explanation of the extra kaon comes out immediately from correcting the duality into a chirality, which I have justified extensively on both mathematical and physical grounds.

 

Relativistic quantum mechanics

Chirality is also a feature of relativistic quantum mechanics, where any transformations with negative determinant reverse the chirality. Such transformations are not physically possible, as they correspond either to reversing the direction of time, or reversing one direction of space (as in a mirror) or reversing all three directions of space.

There is a temptation to confuse chirality with duality, but they are mathematically quite different concepts. Duality applies to representations, while chirality applies to groups. In the standard model, chirality is often applied to representations, which is a mathematical category error, and leads to a great deal of confusion. No doubt the calculations still work, but the mathematical consistency of the model is destroyed.

The confusion is not easy to clear up, because in the case of SL(4,R), though not for all groups, there is both a chirality automorphism of the group, and a duality automorphism of the group. (In the case of SO(4), there is a chirality automorphism, but no duality automorphism.) Each of these automorphisms of the group takes each representation to another representation, which is sometimes the same representation and sometimes a different one. Representations can in principle be self-dual with or without being self-chiral, and vice versa. For SO(4), every representation is self-dual, while some are self-chiral and some are not. For SL(4,R), I think it is true, though don’t quote me on this, that every representation is self-chiral, while some are self-dual and some are not.

To give some examples, for SO(4) the 4-vector representation with spin (1/2,1/2) is self-chiral, but the 3-dimensional representations with spins (1,0) and (0,1) form a chiral pair. For SL(4,R), the vector and co-vector representations are self-chiral, but form a dual pair. The anti-symmetric square of either of these is a 6-dimensional self-dual, self-chiral representation, that restricts to SO(4) as the chiral pair (1,0)+(0,1).

The upshot of all this is that it is not possible to build a relativistic quantum mechanics on the basis of a chiral pair of spinors. It is only possible to build such a model on the basis of a dual pair of vectors. This is a mathematical theorem that may appear subtle and difficult to understand, but it is important and fundamental.

The discussion of left- and right-handed spins in the previous post only applies to non-relativistic quantum mechanics, where the group is SO(4). In this case it is possible to massage the spins into the form used in the standard model. But it is not possible to extend this to the group SL(4,R), where we have to devise a new formalism. Both the vector representation and the co-vector representation restrict to SO(4) as spin (1/2,1/2). To obtain the spin (1/2,3/2) representation we need to look in the tensor cube of spin (1/2,1/2), and therefore in the tensor cube of either or both of the vector and co-vector representations.

This tensor cube splits as the anti-symmetric cube (dimension 4), the symmetric cube (dimension 20) and two copies of the “middle” cube (also dimension 20). Restricting to SO(4) we have spin (1/2,1/2) for the antisymmetric cube, spin (1/2,1/2)+(3/2,3/2) for the symmetric cube, and spin (1/2,1/2)+(1/2,3/2)+(3/2,1/2) for the middle cube. To understand leptons and baryons, we have to understand these representations. To understand mesons and intermediate vector bosons, we have to understand the symmetric and anti-symmetric square representations. Details can be found in my papers.

 

Chirality

In the standard model, the Dirac spinor is divided into left-handed and right-handed Weyl spinors, that express a chirality of 4-dimensional spacetime. As such, chirality only appears in relativistic quantum mechanics, not in non-relativistic quantum mechanics. In the latter, the left-handed and right-handed spinors are equivalent representations of the spin group SU(2). However, I have shown that the spin group SU(2) is only isomorphic to the double cover of the rotation group SO(3), not equal to it. Therefore, SU(2) and SO(3) combine to make SO(4), the rotation group of Euclidean spacetime.

In this situation, SO(4) is made out of two copies of SU(2), which are swapped by any reflection, and are therefore chiral. Hence we can make left- and right-handed spinors by acting on vectors with the left- and right-handed copies of SU(2). One can then translate all the necessary calculations from the standard model to the new model, or vice versa. But notice that in the new model, chirality exists also in non-relativistic quantum mechanics.

Whereas the standard model only recognises one spin for each representation, here we have three: a left-handed spin, a right-handed spin, and a middle spin. The chiral spins can take half-integer values, but the middle spin, being a label for a representation of SO(3) rather than SU(2), can only take integer values. The middle spin is obtained by multiplying (tensoring) together the left and right spin representations.

The defining representation of SO(4) on vectors, or co-vectors, has spin (1/2,1/2), which means the chiral spins are each , while the middle spin is . The middle spin therefore contains things that correspond to observables in the standard model. There are several different copies of this representation, that have different interpretations. The spin 0 representation is a scalar, which might be interpreted as mass, or charge. The spin 1 representation is a 3-vector, and can represent position, momentum or angular momentum according to context.

The square of the Dirac spinor contains four copies of spin 0+1, which can represent four fermions, such as one proton and three generations of electron. Each has a scalar mass and a 3-vector angular momentum, related by a sign that incorporates the charge. This representation can also be interpreted as the electro-weak interactions between fermions, in which case one copy of spin 0+1 splits apart into the Higgs boson and the photon, while the other three represent the Z, W+ and W- bosons. (Actually, there is probably a bit more mixing than this, to give the photon a helicity, but these are minor details.) In both cases, we get an approximate mass relation, that the sum of the three “spin 1” pieces is almost equal to twice the “spin 0” piece. These relations are not exact, because they require a “spin 2” correction, which I shall describe next.

The spin (1/2,3/2) representation has a similar pair of interpretations. Here the right-handed spin is made up of a symmetric triple of 1/2 spins, and describes baryons, made of three quarks. The full left-handed spin consists of 4 or 8 copies of the spin 1/2 representation, so contains enough spinors for two types of leptons and three colours of up-type  and down-type quarks. The middle spin now consists of one complex copy, or two real copies, of spin 1+2, that is dimensions 3+5. In terms of bosons, this representation describes the 8 gluons. In terms of fermions, it describes the baryon octet. The spin 1 part contains the proton, and two xi baryons. The spin 2 part contains the neutron, the three sigma baryons, and the lambda. For more details, see my papers.

Interpretation of quantum mechanics

Although I have proved mathematically that the spin group of non-relativistic quantum mechanics cannot be equal to the double cover of the rotation group of space, physicists will be reluctant to accept this without an analogous physical argument. Quantum mechanics arose out of the discovery that elementary particles do not seem to behave in the way that spinning spheres in classical physics behave. Hence any analogy with spinning spheres must be taken with a large pinch of salt.

The standard identification of the spin group SU(2) with a double cover of the rotation group SO(3) of space implies that the only information about time that can be contained in the spin group is a single sign. But this sign does not even contain the information as to whether time is running forwards or backwards – it is a sign that is called “up” or “down” and is an abstract sign that is difficult to interpret, although its physical behaviour is well understood.

One can alternatively use Hamiltonian duality to identify SO(3) with the symmetry group of momentum, so that it is then the energy coordinate that is missing. Since it is the combination of energy and momentum that determines the (rest) mass, non-relativistic quantum mechanics contains no information about mass. That is the essential reason why the standard model of particle physics needs to input all mass parameters by hand, from the results of experiments.

Dirac extended SU(2) to SL(2,C) and therefore SO(3) to SO(3,1), thereby including the time coordinate, and the mass. Hence in particular the Dirac equation incorporates the mass, and allows one to consider reversing the direction of time, hence predicting anti-particles. The discovery of anti-particles was then interpreted as vindication of Dirac’s method. Nevertheless, Dirac’s work does not solve the problem of mass, which remains as intractable as ever.

Indeed, the extra complications of relativistic quantum mechanics give the impression that the problem has been solved, when it hasn’t. It is best to consider non-relativistic quantum mechanics in order to see the problem most clearly. I shall pursue the analogy with rotating spheres, but repeat the warning that this is an analogy only.

A rotating sphere has a fixed axis in space, say the z axis, and is moving in the x, y and t directions. In the macroscopic world, the symmetries of x, y, and t are expressed by the group SO(2,1). But in the quantum world, all groups are compact, so the symmetry group must be SO(3). But it is SO(3) acting on x, y, t, not on x, y, z. A model rather like this underlies common interpretations of the photon, travelling in the z direction and spinning in the x,y plane. A photon has no mass, which is again a consequence of the fact that we are effectively using only three of the four coordinates of spacetime (or of 4-momentum). To put it another way, the constant speed of light implies that the z coordinate and the t coordinate contain the same information, so one is redundant.

Massive particles must use all four coordinates of spacetime. Their spins therefore cannot be described by SO(3). Their spins can be described by SU(2). But as already explained, if SU(2) is simply a double cover of SO(3), then we are only using three spacetime coordinates. So we cannot describe a massive particle this way. We must instead use a copy of SU(2) acting on four spacetime coordinates.

In the language of quantum mechanics, there are no spinors, only vectors. But the “spin” of a massive particle is completely different from the spin of a photon, because it is described by a completely different group.

 

What does it take to change a light-bulb?

Theoretical fundamental physics has been in the dark for too long. I am trying to change the light-bulb. The job specification is clear, but what about the person specification? Essential characteristics are: ignorance (most important), stupidity (almost as important), stubbornness (vital), laziness (required). Highly desirable: selfishness, arrogance. Contrariness (desirable). A highly developed contempt for authority (essential). Clearly I am the best qualified candidate for the job!

Group theory and astronomy

You’ve probably noticed that the tools I have used to analyse and correct the standard model of particle physics are group theory and astronomy. It is probably not a coincidence that the School of Mathematical Sciences at QMUL was at one time almost a “School of Group Theory and Astronomy”. It still had some of that ethos when I joined in 2004, though by the time I took early retirement in 2016 there was not much group theory left, and the Astronomy Unit had moved into the School of Physics.

There was a weekly “Pure Mathematics Seminar” which was dominated by group theory, but which from time to time hosted talks on astronomy. I was exposed to regular updates from colleagues on the state of astronomy, and the latest exciting news. So I could not miss the announcement in 2007 of Garrett Lisi’s “Exceptionally simple theory of everything”, which claimed to show how the Lie group E_8 could explain everything about the universe. This claim unfortunately has not lived up to expectations, but I was hooked, because E_8 is without a doubt the most interesting Lie group of all.

There is a “proof” in the literature that E_8 cannot work because it is too small. It took me several years to realise that in fact E_8 cannot work because it is too big. It seemed to me that the correct dimension must be somewhere in the region of 15, while E_8 has dimension 248. So I tried SU(4), and I tried SL(2,H), and I tried G_2, but they didn’t work.

In the meantime, I realised that if the ultimate goal is to unify the theories of the very big and the very small, then some of the numbers that are used in particle physics must also appear in astronomy. So I looked for suspicious coincidences. When I’d found three, on 5th January 2015, I went round the corner to the pub and bought 2 or 3 bottles of their best champagne. I knew I was onto something. I had located the problem, or one of the problems: the standard model does not take account of the acceleration of the laboratory.

I looked for a physicist to work with, but no-one seemed to be interested. Even now, when I have found around 18 suspicious coincidences, and explained many of them, no-one seems to be interested. One reason for this may be that since Einstein’s general theory of relativity was launched in 1915, the foundations of physics have been dominated by geometry. Everyone thinks that the solution to the problems can be found by piling on more and more abstract geometry. But this approach has been tried for 100 years and has failed. As I have shown, the problem is in the group theory, and the solution is therefore also in the group theory. And the correct dimension is 15: the group is SL(4,R).