The Italian school of algebraic geometry

November 23, 2020

I often wonder what on earth motivates the discussions on Peter Woit’s blog, which this week has a bizarre exchange concerning the increasing lack of rigour that plagued algebraic geometry in the first half of the 20th century. I have something of a personal or family interest in this, since my father did a PhD in algebraic geometry in the 1950s, supervised by one of the surviving members of the dying “Italian school”. The main item of decoration in the living-room when I was growing up was an illustration of the associative law of addition on a cubic curve in general position, executed in marquetry.

When I became a mathematical adult, we occasionally tried to discuss mathematics: I would try to explain what my PhD was about, and he would try to explain what his PhD was about. Neither aim was successful. But there were some points of contact: we both understood, in our different ways, the group of the 27 lines on a cubic surface, and the group of 28 bitangents of a quartic curve. And I think I gained something by seeing them in a more geometric light than the purely algebraic context I had been used to.

The context of the discussion on Peter Woit’s blog seems to be a comparison of various types of lack of rigour, and perhaps a cautionary tale that lack of mathematical rigour in theoretical physics may lead to the same kind of collapse that attended the “Italian school of algebraic geometry”, and for the same reason. But algebraic geometry was rescued by mathematicians, who were appalled by the increasing number of results that were claimed, but false, and worked hard to build a solid foundation and rigorous methods for a reliable mathematical theory. The same is not happening in theoretical physics.

Why not? Because nobody in theoretical physics listens to mathematicians. Those of us mathematicians who have made a serious study of the foundations of theoretical physics can point out in gory detail exactly where all the mistakes are. Well, maybe not all of them, but enough to be going on with. Physicists are only interested in mistakes when they are actually contradicted by experiment. Mathematicians can point out mistakes at a much earlier stage, and save billions of pounds in unnecessary experiments. Just read the last 100 or so posts on this blog – I may have repeated myself once or twice, but this seems to be necessary in order to get the message across. 

Why does Peter Woit, and all the people he allows to comment on his blog, think it is OK for theoretical physicists to try things at random and hope they get something consistent with experiment? Why do they not start by ruling out everything that is mathematically inconsistent? Oh, I know why – because *everything* they try is mathematically inconsistent. Please, people, listen to the mathematicians, and take rigour seriously. If you don’t, you are condemned to another century of nonsensical theory, and failure to obtain a consistent unified picture of the universe we live in.

Electromagnetism in an accelerating frame

November 23, 2020

As everyone knows, the theory of special relativity is a theory of electromagnetism, that shows how different observers, travelling at constant velocity with respect to each other, see different electric and magnetic fields,  but see the same underlying physics, described by Maxwell’s equations. The coordinate transformations on spacetime between such observers form the Lorentz group SO(3,1), and it is the invariance of Maxwell’s equations under this group that is the key result.

The electromagnetic field transforms under the Lorentz group like the anti-symmetric rank 2 tensors, which is to say that it is a complex 3-vector field, acted on by SO(3,C), that is isomorphic to SO(3,1). The complex structure is defined by a group U(1) of scalar multiplications, that commutes with SO(3,C). This group U(1) exists in the mathematics, converting the electric field to the magnetic field and back again. It does not act on spacetime, only on the electromagnetic field. In physics, this group is interpreted as representing the photon.

Now what happens if we have a pair of observers accelerating with respect each other? If we assume that they are observing a small piece of spacetime that can be adequately approximated as flat, then the (local) coordinate transformations now form the group SL(4,R) of all linear transformations with determinant 1. How does this group act on U(1)? Well, it doesn’t. That is, the electromagnetic field loses its complex structure under the action of SL(4,R). What does this mean? It means that mutually accelerating observers disagree about the complex structure of the electromagnetic field. As long as we keep to one observer’s definition of inertial frame, Maxwell’s equations work fine. They also work fine with the other observer’s definition. The only problem comes when we consider both observers simultaneously.

In practice, on a macroscopic scale, this rarely causes any problems. One chooses a coordinate system suitable for the problem at hand, applies Maxwell’s equations, and obtains a result that is consistent with experiment. Job done. No problem.

But suppose one wants a classical model of a hydrogen atom, with an electron orbiting a proton. Then the mutual acceleration of electron and proton is enormous, and cannot be ignored. It is therefore inconsistent to assume that there is a single copy of U(1) applicable to the entire interaction. This inconsistent assumption is the basic assumption of quantum electrodynamics, and of quantum field theories more generally. Problem. Job not done.

So how many independent copies of U(1) are there? The Lorentz group SO(3,1) is 6-dimensional, and SL(4,R) is 15-dimensional, so there is a 9-dimensional manifold of copies of U(1).  If we (incorrectly) take one of them as universal, then the other 8 don’t go away: they are still there, describing how electromagnetism looks to different observers, and they still have an effect on the physics one sees at a subatomic level. So they appear in the standard model somewhere. Where? I hazard a guess that they appear as the gauge group SU(3) of the strong force, and that they therefore describe how electromagnetism behaves inside the nucleus of an atom, as seen from the outside.

Now we are in a complete mess. First of all, these 9 dimensions of U(1) do not form a group. Second of all, only one of them commutes with the Lorentz group. So the gauge group of the strong interaction is not a group, and does not commute with the Lorentz group. It is really amazing that quantum chromodynamics manages to get *any* answers correct, given the egregiousness of its false assumptions.

But the problem is much worse than that. It does not just affect the theories developed in the 1970s. It affects the theories developed in the 1920s. I am not telling you anything new: these problems were well known in the 1930s, and were a major concern of Einstein’s for the rest of his life. It affects the Dirac equation itself.

To understand this, recall that the discovery of electron spin necessitated the introduction of the spin group SU(2) into the theory of quantum mechanics. Dirac went one further, and introduced the “relativistic spin group” SL(2,C). So far, so good. The group SL(2,C) is a subgroup of SL(4,R), so can quite happily be incorporated into a relativistic theory. And you can see what types of observers it describes: it describes observers who are rotating around each other. Nota bene: these observers are mutually accelerating. It is therefore absurd to identify this group with the Lorentz group, which describes mutually non-accelerating observers. Nevertheless, this absurd and inconsistent assumption underlies the whole of the standard model. In effect, the standard model acknowledges that the electron in a hydrogen atom moves very fast, but denies that the electron accelerates. No wonder physicists have difficulty relating quantum physics to classical physics.

Vacuum or vacua?

November 21, 2020

Perhaps the most embarrassing of all the contradictions in fundamental physics is the difference of 120 orders of magnitude between the vacuum energy density in particle physics and in astrophysics. This is clearly not a minor difference of opinion that can be sorted out by some tweak to one or other theory. Worse, it is not just a theoretical difference, but a difference that is supported (at least to some extent) by experiment. High energy physicists really do see a high energy vacuum, and cosmologists really do see a cold dark vacuum.

So what happens if we assume they are both right? What is the difference between the cold dark vacuum in inter-galactic space, and the high energy vacuum in a particle accelerator? I can think of two: the temperature, and the gravitational field (or acceleration). Both of these are normally thought of as properties of matter, not properties of the vacuum. And both are properties of bulk matter, not properties of elementary particles. 

I don’t think anyone seriously wants to quantise temperature. Moreover, the difference in temperature between the Earth and inter-galactic space is only two orders of magnitude. To make the kind of difference required, we’d need to use something like the temperature of the Big Bang. There may be a theory like that, but in my view it is too speculative to talk about such things.

But the whole idea of quantum gravity is to extend the theory of gravity to the elementary particle level. Moreover, the gravitational field on the surface of the Earth is at least 10 orders of magnitude greater than the gravitational pull of the centre of the Milky Way, and one gains/loses several more orders of magnitude by going far away from the galaxy.

So here’s the idea: the energy density of the vacuum depends on the detailed geometry of the motion of the particular piece of vacuum under investigation, relative to the ambient gravitational field. I am not sure exactly how many parameters this allows us to vary, but I think it is about 9. In any case, I attribute the vacuum energy that particle physicists measure to the energy of the gravitational field. Pulling the vacuum through the gravitational field releases some of that energy.

The 9 parameters appear in particle physics as the 9 masses of 3 generations of electrons and quarks. (Neutrino masses are both speculative and irrelevant here.) These are the basic values from which particle physicists calculate the vacuum energy density. The only thing is that physicists assume these mass values are universal constants – but this assumption is what leads to the contradiction, i.e. the discrepancy of 120 orders of magnitude. Experiment proves conclusively that they cannot be universal constants, they must be parameters that depend on the gravitational field.

Our task, therefore, is to determine exactly how they depend on the gravitational field, and why. I have provided a number of ideas in this direction. If you don’t like them, then think of some better ones. It is clear to me that the neutrinos provide the mechanism for communicating information about the gravitational field to (and from) elementary particles, and therefore for determining their masses. It is therefore possible, as I have been saying for years, to use a macroscopic theory of gravity to calculate good estimates for the masses of the elementary particles. All we have to do is find the right equations.

Chromatic uncertainty

November 15, 2020

I have long maintained that there is a chromatic uncertainty principle that runs parallel to the Heisenberg Uncertainty Principle. The latter says there is a limit to the simultaneous precision of measurements of position and momentum. It is in effect a duality between the 3-vector of position, and the 3-vector of momentum. It seems to me that there is a corresponding duality between the 3-vector of colour, and the 3-vector of generations, for elementary fermions. For electrons, the three generations are very well defined, and have very precisely measured masses. That means their colour is indeterminate, or very ill-defined. 

This is not how the standard model interprets colour, however – here it is said that the electron has no colour. This is really just semantics: what is the difference between saying the electron has no colour, and saying that its colour is not defined? Or indeed, translating from mathematical language to physical language, saying that its colour cannot be measured?

Quarks have a rather less-well-defined generation: their masses cannot be measured all that accurately, and the uncertainty in their masses (generations) might indeed be attributed to a dual (un)certainty in their colours. The CKM matrix describes how this uncertainty manifests itself in certain probabilities that a quark appears in a different generation from the expected one.

Neutrinos, on the other hand, do not appear to have well-defined generations at all. The well-attested phenomenon of neutrino oscillation makes clear that a neutrino in flight does not have a measurable generation. The generation can only be measured by an interaction, and experiment tells us that this generation cannot in most instances be predicted. So colour-generation duality implies that a neutrino in flight may have a well-defined colour, since it does not have a well-defined generation.

This suggestion is of course enough to get me banned from every physics discussion group on this planet. But it seems to me a rather better explanation for neutrino oscillations than the standard model version, which postulates a non-zero mass for neutrinos, that is not supported by a single independent experiment.

So what can it possibly mean to say that a neutrino has a well-defined colour? The only property of a neutrino in flight that can be measured (or, to be more accurate, inferred) is its momentum. Taking out of account its speed and/or energy, this leaves just the *direction* of motion to correspond to the “colour”. Since directions form a 2-sphere in 3-space, this has the right number of degrees of freedom. So the colour/generation uncertainty principle implies that the colour of a quark is really the same thing as its direction of motion. Within QCD, this presumably means its direction of motion relative to the hadron it belongs to at the time.

Now we see that colour/generation duality is really the same as position/momentum duality: for we can locate the quark very precisely, within this particular proton, for example. That means we cannot determine its momentum at all, and it jiggles around with complete freedom. This is called asymptotic freedom, and won its discoverers a Nobel Prize. It is really just the same thing as Heisenberg Uncertainty, however.

Well, if we now try to take this idea to its logical conclusion, we must conclude that the generation of a fundamental fermion is a *position* of some kind. At least, we can probably do this for quarks. It is less clear for electrons. Here we measure a property called “mass” that is independent of the direction of momentum, and is independent also of its direction in space from the observer. But the duality implies that the mass is not independent of *all* directions in space. What does this mean? Experiments make clear that the mass does not depend on any directions defined by electromagnetism or the weak interaction, and electrons are not affected by the strong force. Therefore the mass can only depend on directions defined by gravity.

I hope you followed this argument: I have shown that, in the absence of any new physics, the phenomenon of neutrino oscillation, combined with the Heisenberg Uncertainty Principle, not only implies the asymptotic freedom of quarks, but also implies that the measured mass of the electron is determined by the gravitational field.

Degrees of freedom

November 13, 2020

There are four groups in the standard model of particle physics. Three gauge groups U(1), SU(2) and SU(3) and the spin group SU(2) or SL(2,C). In practice all of them are complex forms of the relevant Lie group, and have dimensions 1, 3, 8 and 3, totalling 15. These 15 complex degrees of freedom are sufficient to describe the whole standard model, subject to adding in a finite number of constants. The actual number of constants required varies slightly between sources, but 25 seems to be an upper bound. These constants, however, do not and cannot increase the number of degrees of freedom. The number of degrees of freedom in the model is 15. No more. No less. Exactly. 15.

The standard model is capable of explaining (almost) everything in particle physics, and its predictions over half a century have (almost) all turned out to be correct. There is nothing wrong with the standard model apart from a few minor niggles. Experiment has vindicated the model in (almost) all its particulars. The number of degrees of freedom is confirmed as 15. No more. No less. Exactly. 15.

Yet for half a century, every attempt to go beyond the standard model has increased the number of degrees of freedom. In the 1970s, the Pati-Salam model added 9 d.o.f., and the Georgi-Glashow model added 12. Georgi’s SO(10) model has 45, Lisi’s E8 model has 248, and string theory has 496. To the extent that these models make predictions, these predictions have been falsified time and time again. The extra degrees of freedom give rise to new physics of one kind or another, either proton decay, or extra dimensions of space, or super-symmetric particles, or something else which has never been observed, despite extensive and intensive searches. Experiment has shown conclusively that these extra degrees of freedom do not exist in our universe. The actual number is 15. No more. No less. Exactly. 15.

There is only one (complex) simple Lie group with 15 degrees of freedom, and that is SL(4,C). If you want a unified theory of particle physics, this is the only group that is available. There can be no argument about it, no discussion, no objections that such a theory “contradicts experiment”, no ifs, no buts. This is the only available group. Take it, or leave it. You can discuss which real form describes the universe best, and you can discuss whether you want to add a scale factor to extend the group to GL(4,C). But that is all.

In fact, SL(4,C) is already used in the standard model, but it is usually described as a Clifford Algebra rather than a group. This is the Dirac algebra of all 4×4 complex matrices. The scale factor is not used, so that there are 15 degrees of freedom in the algebra. No more. No less. Exactly. 15. Therefore the Dirac algebra already contains ALL of the degrees of freedom that actually exist in the physical universe. But that means there is no room for any equations, and no room for any physics.

To get equations one needs two different physical interpretations of the Dirac algebra, that can then be related to each other. The best way to do this is to take the Dirac spinor and its dual together, to form an 8-dimensional real object, then take the tensor square, to get a 64-dimensional real algebra, and finally put the complex structure back, to get a 32-dimensional complex algebra. As a representation of SL(4,C), this algebra splits as 1+15+6+10. The complex 6-space supports the Maxwell equations. The complex 10-space supports the Einstein field equations. The complex 15-space supports the standard model of particle physics. The complex 1-space supports a concept of mass and charge. Together, this algebra is exactly the right size to contain all of fundamental physics. No more. No less. Exactly. The right size.

One degree of freedom for charge. Six degrees of freedom for the Lorentz group. Ten degrees of freedom for the ten mixing angles and coupling constants. Fifteen degrees of freedom for the fifteen fundamental masses. The number of undetermined parameters in the standard model is 10+15=25. No more. No less. Exactly. 25. Of these 25 parameters, 10 are determined by the Einstein field equations, and therefore depend on the gravitational field and/or acceleration of the experiment. The other 15 are determined by pairing two interpretations of the symmetry group SL(4,C).

Mass or energy?

November 6, 2020

Einstein’s 1905 theory of (special) relativity shows that (total) energy E, (rest) mass m and momentum p=(p_1,p_2,p_3) are related by an equation, which Einstein originally expressed as m =\sqrt{E^2/c^4-(p_1^2+p_2^2+p_3^2)/c^2}. The mass is therefore invariant under the Lorentz group SO(1,3) acting on the energy-momentum (usually called 4-momentum) (p_0,p_1,p_2,p_3), where p_0:=E/c. That is, mass is a scalar under the Lorentz action.

This invariance of mass therefore carries through to the theory of (classical) electromagnetism, in which special relativity provides the theory to unify electricity and magnetism. It carries through to quantum electrodynamics, where mass is a scalar in the Schroedinger equation, and in the Dirac equation. In this context, a scalar is both invariant (the same for all observers) and conserved (the same for all time). Experimentally, however, we know that mass is not conserved in the weak interaction, and it is not conserved in astronomical events such as the collision and merger of black holes.

Mathematically, indeed, Einstein’s equation has more symmetry than just Lorentz symmetries. If we define p_4=mc, then the equation becomes p_0^2-p_1^2-p_2^2-p_3^2-p_4^2=0, so that the symmetry group is obviously SO(1,4). Then we see that the energy is invariant under the group SO(4) acting on mass-momentum. Now SO(4) contains two normal subgroups isomorphic to SU(2), one “left-handed” and one “right-handed”, such that the left-handed one can describe how the weak interaction converts mass into momentum.

Conservation of energy is the most fundamental principle in the whole of physics. Invariance of energy is not usually assumed, because it is incompatible with the assumption of invariance of mass. But in any theory of physics that is independent of the observer, a quantity cannot be conserved (for all observers!) unless it is also invariant. Therefore, either we have to abandon the principle of conservation of energy (unthinkable!), or we have to abandon the principle of invariance of mass (unthinkable!).

A huge amount of effort has gone into theoretical physics for 90 years to try to avoid this inevitable conclusion. Physicists are determined to have their cake (energy is conserved) and eat it (mass is invariant), but the universe just isn’t like that. It is mathematically impossible in a relativistic theory to have both conserved energy and invariant mass.

So I ask: mass or energy? Which do you want? Particle physicists insist on mass, while relativists insist on energy. Fight it out amongst yourselves. I don’t want to get involved. But I know what the correct answer is, and I have spent most of this year trying to explain it on this blog.

A poem

November 3, 2020

Mass or energy.
One or the other.
Not both.
Einstein told us that.
In 1905.

A quaternionic Dirac algebra

October 29, 2020

There are many ways to construct the Clifford Algebra Cl(3,3), but the most relevant for physics is to take the algebra Cl(1,3) generated by Dirac matrices \gamma_0,\gamma_1,\gamma_2,\gamma_3, and adjoin quaternionic scalars i,j,k that commute with all the gamma matrices. This cannot actually be done consistently with 4×4 quaternionic matrices, as one ends up with the wrong signature for the quadratic form. But it can be done with 8×8 real matrices. There is then a wide choice of Clifford Algebra structures, from which to choose a suitable one for physics.

To exhibit the 3+3 symmetry I choose i\gamma_0\gamma_1\gamma_2\gamma_3, j\gamma_0\gamma_1\gamma_2\gamma_3 and k\gamma_0\gamma_1\gamma_2\gamma_3 for the first three, and \gamma_1,\gamma_2,\gamma_3 for the last three. Note that the standard model prefers to break the symmetry to 1+2+3 by replacing the first generator (defined to be \gamma_5) by \gamma_0. The product of all 6 generators is known as the pseudoscalar element, and is \gamma_0 (total energy) in my choice of generators, or \gamma_5 (which I shall show can be interpreted as rest mass) in the standard choice. 

Now there is a canonical choice of subalgebra Cl(0,3) generated by \gamma_1,\gamma_2,\gamma_3. This algebra is a direct sum of two copies of the quaternion algebra, obtained by projecting the generators with one of the two orthogonal idempotents (1\pm \gamma_1\gamma_2\gamma_3)/2. Hence one of them contains elements like \gamma_1+\gamma_2\gamma_3, while the other contains elements like \gamma_1+\gamma_3\gamma_2. Therefore they have obvious physical interpretations as left-handed and right-handed spins for particles at rest. The spin group itself is the Lie group generated by the products of pairs of the gamma matrices, and is isomorphic to Spin(3) = SU(2). The Clifford Algebra also contains the larger group Spin(4) = SU(2) x SU(2). The `relativistic’ extension to Cl(1,3) extends the algebra to the 2×2 quaternion matrix algebra, and therefore mixes the left-handed and right-handed spins together. The spin group extends to Spin(3,1) = SL(2,C). In other words, all the standard structure of spin is available, but with a slightly simpler mathematical structure than in the standard model.

There is also a canonical choice of subalgebra Cl(3,0), which is completely different, as it is isomorphic to the 2×2 complex matrix algebra. The `spin’ group in this case is generated by i,j,k, and can be identified with the gauge group SU(2) of the weak interaction. It commutes with Cl(0,3), so with spin SU(2), but not with relativistic spin SL(2,C). This is as it should be, in order for the Weinberg angle to run with the energy scale, and is the same as in the standard model, in which \gamma_5 anti-commutes with \gamma_0 (energy). The pseudoscalar element is now i\gamma_5 = \gamma_0\gamma_1\gamma_2\gamma_3, which squares to -1 instead of +1, so there are no projections associated with this pseudoscalar. Instead, the pseudoscalar generates a Lie group U(1), which commutes with i,j,k, so with the weak gauge group SU(2). Together they generate the electroweak gauge group U(2), such that electroweak mixing is all about the mixing between i and i\gamma_5, just as in the standard model.

This mixing breaks the symmetry between i,j,k, and this symmetry-breaking is also reflected in the structure of the Clifford Algebra, which is thereby converted into Cl(1,2), generated by \gamma_5,j,k. The spin group similarly is converted into SL(2,R) generated by i, j\gamma_5, k\gamma_5, but the pseudoscalar remains the same, i\gamma_5. Now it is important to realise that this symmetry-breaking involves a choice. This is a choice of one of the three generators of Cl(3,0) as being special, or equivalently a choice of one of i,j,k as being special. I interpret this as a choice of one of the three generations of fermions as being special, i.e. physically stable. The fermions, I believe, are represented by the odd part of the algebra, and the bosons by the even part. There is a variety of different interpretations available, but essentially there is one charge or hypercharge defined by the pseudoscalar, and three generations or masses defined by the three generators.

In particular, \gamma_5, as one of the generators, represents the mass of the (first-generation) electron. In the real world, mass and charge are mixed together in a way that defies explanation, at least in the standard model. So in reality there are four mass-charge terms for fermions, being the four dimensions of the odd part of the algebra. I take these as the masses of the three generations of electron, and the proton. Similarly, there are three mass-charge terms for bosons, which I take as the masses of the Z, W and Higgs bosons.

To measure the actual values of the masses, we have to project onto the energy term, which lies outside the algebra Cl(3,0). Whatever happens, we have a projection from a 7-dimensional space of particles onto a 1-dimensional space of mass. All we really need to know for the mass of bulk matter now is the mass of the neutron, since we must know that the neutron mass is very close to the proton mass. As I have explained on numerous occasions, there are two formulae that relate the neutron mass to the 4+3 fundamental masses. One is the fermionic equation e+mu+tau+3p=5n, the other is the bosonic equation Z + W^+ + W^- = 2H + 2n.

If you want to read more about this, you can read my preprint posted at since the arXiv are playing their usual silly games with this submission.

Cl(2,3) or Cl(3,2)?

October 13, 2020

I have shown how the Dirac algebra of 4×4 complex matrices can be profitably regarded as the Clifford Algebra Cl(2,3), generated by gammas 0,1,2,3,5, or Cl(4,1), generated by i0,i1,i2,i3,5, that is 123,023,013,012,5. I have never understood why Dirac felt it necessary to introduce the scalar 01235, when Cl(1,3) and Cl(3,1) both contain the imaginary scalar 0123, which is already perfectly adequate for expressing the mass term in the Dirac equation.

Nevertheless, the extra gamma matrix proved useful later in modelling the weak interaction. There is still the question as to whether spacetime has signature (1,3), as particle physicists believe, or (3,1), as everybody else believes. In other words, is it better to use Cl(2,3), as Dirac did, or Cl(3,2)? Now Cl(3,2) is not isomorphic to Cl(2,3), but the two algebras contain the same physical information, so it is perfectly possible to use Cl(3,2) instead of Cl(2,3). Then the same physical concepts appear in different mathematical clothes, which may (or may not) be easier to work with.

As we know, Cl(2,3) is the algebra of 4×4 complex matrices. On the other hand, Cl(3,2) is the direct sum of two algebras, each isomorphic to 4×4 real matrices. Not much difference, you might think. But enough difference to make a difference, if you see what I mean. We now replace Cl(2,0), that is 2×2 real matrices, by Cl(0,2), that is the quaternions. Hence we replace the split real form SL(2,R) by the compact real form SU(2) of the weak gauge group. Does this sound like an improvement? I think it does. We also replace Cl(0,3), that is the sum of two copies of the quaternions, by Cl(3,0), that is the algebra of 2×2 complex matrices. Does this sound like an improvement? Physicists will surely think it does.

To see this more clearly, let me introduce some generators x,y,z,u,v for Cl(3,2), with x,y,z spacelike and u,v timelike. Then Cl(0,2) is spanned by 1, u, v, uv, where u,v,uv behave like quaternions i,j,k. Also Cl(3,0) is spanned by 1, x, y, z, xy, xz, yx, xyz, where x,y,z behave like the Pauli matrices. In particular, xyz generates a scalar U(1), and xy, xz, yz generate SU(2), extended to SL(2,C) by adjoining x,y,z. This copy of SL(2,C) is not identical to Spin(3,1), which would have xt,yt,zt in place of x,y,z. Nevertheless, it contains the same SU(2), which is the spin group Spin(3), not to be confused with the weak SU(2), which is generated by u,v,uv.

The symmetry-breaking of the weak force is still evident, since uv (the Z boson) is in the even part of the algebra, while u and v (the W+ and W- bosons) are in the odd part. In particular, uv commutes with x,y,z, so the Z boson is its own antiparticle, while u and v anti commute with x,y,z, so the W^+ and W^- are not their own antiparticles.

Now what have we done by converting Spin(3,1) with xt,yt,zt in SL(2,C) with x,y,z? We have introduced three new concepts, that are associated to directions in space, but which are not momentum, or the dual of momentum (position). So what are they? There are quite a number of triplet symmetries that spring to mind as possibilities: generations, colours, and direction of spin being three of them. Do we think these concepts should be observable or not? Well, x, y, z lie in the Clifford Algebra, not in the spinor. Observables are operators on the spinors, so observables lie in the Clifford Algebra. I am therefore inclined to assume that the converse should also be true, that elements of the Clifford Algebra are observable. If so, then x,y,z probably represent the three generations.

Well, now, the true picture is obviously a lot more complicated than this brief sketch can do justice to. Bosonic spins are described by the xt,yt,zt copy of SL(2,C), so that one can distinguish zt+xy and zt+yx as the two helicities of photon travelling in the z direction. All terms in this algebra are even. Fermionic spins, I believe, are described by the x,y,z copy of SL(2,C), so that one distinguishes left-handed and right-handed electrons via z+xy and z-xy. Half the terms in this algebra are odd, which is what gives the algebra its fermionic behaviour. If we multiply the left-handed and right-handed terms together, we get (z+xy)(z-xy)=z^2+xyz-zxy-xyxy=z^2+x^2y^2 = 1+1. In other words we get a positive real scalar (mass) coming out. Or, as I said before, three positive real scalars for the x,y,z cases.

So, everything we could possibly need is in this algebra. The only thing we haven’t got is the gauge group SU(3) of the strong force. But do we really need it? All it describes is fictitious unobservable concepts of “colour”. Yes, we can interpret x+yz, y+zx, z+xy as three colours and x+zy, y+xz, z+yx as three anti-colours, and 1+xyz, 1-xyz as the fourth (lepton) colour, and invent a fictitious group SU(3) to act on this space, by pretending that x and yz are the real and imaginary parts of a complex number, but what has any of that got to do with the real world?

The only game in town

October 11, 2020

It is quite unbelievable how often one hears the defence of string theory, that it is “the only game in town”. If this were true, it would be a shocking indictment of the state of theoretical fundamental physics. But it is not true. There are many other games in town. It is just that no-one pays any attention to them. There are large numbers of people with hugely interesting ideas about how to break the logjam, and work towards solutions of the fundamental problems. It is just that no-one listens to these ideas. The authoritative figures only talk, and don’t listen. They are bereft of ideas themselves, but instead of listening to other people’s ideas, they shut their eyes and their ears, and open their mouths, and intone the mantra that their own long-since failed idea is the only one in town. Nonsense. There are huge numbers of ideas out there. Several of them might even be crazy enough to be true.