The answer is 5. What is the question?

January 29, 2026

If you watch “Mock the Week” you will be familiar with the format. In this case the discussion is about whether the question is “What is 2+3?” or “What is 4+1?” This discussion has been going on for at least a hundred years, with no resolution in sight. String theorists swear by 2+3. Woit says that is “not even wrong” and claims it is obviously 4+1. Normally in a conflict between Woit and string theory I find myself on the side of Woit. But not in this case. I am now convinced that in this instance Woit is not even wrong, and “not even wrong” is right.

You see, the main question about particle physics is “Why 2+3?”, or in technical terms, why is the gauge group of the nuclear forces SU(2) x SU(3)? There is a related question about the charge on the up quark: “Why 2/3?” There is no mention of 4+1 anywhere in particle physics. But it was only once I started to consider the question “How 2/3?” that I started to make progress in understanding what is really going on behind the scenes.

That is because I found the answer in the equation tan(2W) = 3/2, where W is the “weak mixing angle”, which is without a doubt the most important angle in the whole of particle physics. So after that I could turn my attention to the questions “Why 2/3?” and “Why 2+3?”. Or, to be more precise, to the question “What is 2+3”?

And I found that the answer is 5 Maxwell equations. The 3 Maxwell equations in here are the ones that relate the magnetic field to the current. These are the most important equations in the whole of pre-nuclear energy. They are the most important equations in the whole of pre-quantum technology. They are still the most important equations today, because you can’t run a wind farm or an electric car without these three equations.

If we write the equations in “natural” units, so that the vacuum permeability and the vacuum permittivity are both 1, then the speed of light is also 1, and the equations are written in vector form as curl(B) = J+dE/dt, where B is the magnetic field, J is the current, and E is the electric field. The rest of the equation is the mathematics that describes how the magnetic field varies in space (curl), and how the electric field varies in time (d/dt).

There is another companion equation that describes how the electric field varies in space, that is div(E)=Q, where Q is the electric charge (or technically the charge density). That gives you 1+3 Maxwell equations. Normally you would add another 1+3 equations div(B)=0 and curl(E)+dB/dt=0, but these are equivalent to Lorentz symmetry of the electromagnetic field, so if we assume the symmetry, then we don’t need the equations.

But we do need one more equation, that is conservation of charge. Electric charge cannot be created or destroyed, it can only move around, or cancel out. The equation that enforces this physical property is div(J)=-dQ/dt. Or is it dQ/dt=-div(J)? We know what the answer is, but what is the question? It shouldn’t matter, really, should it? Multiplying the equation through by -1, doesn’t change the equation, does it? Well, yes, actually it does. It doesn’t change the answer, but it does change the question. And therefore it is important. It is the same as the difference between the questions 4+1 and 2+3.

We know that the ultimate question of particle physics, the universe and everything is 2+3, so this is the question we must ask. And it tells us to write the equations in 5×5 matrix form with the differential operator (1, d/dt, del) acting on the columns (0, Q, J), (-Q, 0, E) and (J, E, -B) to produce the answer (0, 0, 0). That means that in the matrix form, Q and B are antisymmetric, while J and E are symmetric.

Now when I was taught Maxwell’s equations, back in the dark ages (1970s), I was taught that E is antisymmetric, not symmetric. This is wrong because if E is antisymmetric, then spacetime is Euclidean, not Lorentzian. Of course, I didn’t know that at the time, but I do now. The difference between 4+1 and 2+3 has nothing to do with this, it is the difference between Q being symmetric (and therefore J antisymmetric) or antisymmetric (and J symmetric). And the fact that charge can be quantised, but current cannot, implies that Q is mathematically a circle, not a straight line, which means it is antisymmetric, not symmetric.

So there we have it. The ultimate question is “What is 2+3?” The reason why Einstein failed to answer this question is because he thought the question was 4+1. So what I want to do now is explain what Einstein would have done in 1915 if he had realised the question was 2+3. To be fair to Einstein, there was no real reason to think the question was 2+3 until much later, when quantum theory was sufficiently developed to make it clear that the gauge group of a quantised entity had to be compact. It could in principle have been deduced from the Dirac model of the electron in 1928, but it wasn’t really until the 1950s that there was enough evidence to reconsider the question, by which time it was too late, as 4+1 had been set in stone.

As you know, the biggest obstacle to answering any question is asking the wrong question. And if you are fixated on answering 4+1, when the correct question is 2+3, you will just go round and round in circles, bashing your head against a brick wall, if that’s not a mixed metaphor. Once you know the question is 2+3, and the answer is 5 Maxwell equations, then it isn’t hard to imagine that to include momentum and energy as well as current and charge, you just have to extend the equations from real numbers to complex numbers, so that the imaginary part gives you five Einstein equations for gravity.

These equations are rather different from the Maxwell equations, because the symmetric/antisymmetric dichotomy is reversed by the extension to Hermitian/anti-Hermitian matrices. In particular we get some anti-Hermitian diagonal matrices, which make things more complicated, since there are no anti-symmetric diagonal matrices in the Maxwell case. The first equation to consider is conservation of energy, which is div(p) + d(p0)/dt = m, where p is the momentum, p0 in the energy (I can’t use E again, because that will cause confusion), and m is a type of mass. If this equation doesn’t make sense to you, then don’t try to think of m as being something you already know, think of this equation as being a definition of m, whose properties we then have to work out as we go along. It’s a Lorentz scalar, so mass is the obvious interpretation.

The second equation is div(g) = p0 + d(m0)/dt, where g is the Newtonian gravitational field. In Einstein’s language, E=mc^2, so that p0 replaces the Newtonian mass as the source term for gravity. But what on earth is this extra term d(m0)/dt doing here? It is an extra source of gravity that actually does exist in Einstein’s theory, although it is not easy to interpret it physically. It is not just a mass term, but a change-of-mass term. In standard gravitational theory, mass cannot change except by moving. But this is not true in reality, because mass is not conserved in the weak interaction. So the d(m0)/dt term cannot be ignored, and creates a genuine mixing between the weak force and gravity.

The importance of this remark cannot be overstated. I’ll pause a while to let it sink in.

The two diagonal terms m and m0 are two different types of mass, one of which is constant, while the other changes over time. Newtonian gravity assumes they are equal, which is of course logically impossible. Einsteinian gravity does not assume they are equal, but still assumes that there is only one type of mass, which is still logically impossible. Einsteinian gravity and the weak force are incompatible.

But if (as is obvious) we do not understand what type of “mass” m0 is, we can take the equation as a definition of m0 in terms of the gravitational field. That is, m0 is the time-integral of div(g)-p0, so it can be calculated from the gravitational field and the energy. That means it is a Machian type of inertia. But m is a Newtonian type of mass.

Now listen carefully – m0 is only defined up to a constant of integration. So you can choose your constant of integration to ensure that m=m0 and there is only one type of mass. However, m is constant (by definition) and m0 is not (also by definition), so over time m0 will drift away from m, and they won’t be equal any more. By reading the literature carefully I have worked out that the calibration m=m0 was carried out in the period between 1971 and 1973, and I have worked out how far m0 has drifted away from m in the past half-century or so.

And I also know why nobody has noticed this drift. It is because it only affects gravitational measurements, and does not affect the standard model of particle physics in any way. The Dirac equation edits out the gravitational field, and shunts this variability (which is physically real) off into some variables that cannot even in principle be measured. So it is logically impossible to detect this drift in particle physics experiments. It can only be detected in gravitational experiments.

And once you know this drift is physically real, you can detect it in the measurements if you look closely enough. You can detect it in Cavendish-type measurements of the Newtonian gravitational constant G. You can detect it in inconsistent measurements of the gravitational mass of copies of the International Prototype Kilogram. You can detect it in pre-1969 experiments to measure the electron/proton mass ratio, before the ruthless dictatorship of the Dirac equation came into force, and suppressed all opposition from the Machian gravitational party.

And just because I live in Birmingham I can’t resist mentioning the Lunar Society, which was a major force in the advancement of science, technology and philosophy in the second half of the 18th century. They regulated their activities by the phase of the moon, because it was only safe to walk around Birmingham at night if the moon was full. But they made progress in many endeavours by communicating across boundaries, sharing ideas, and thinking outside the box. They were not crackpots or lunatics, they were lunar luminaries.

Progress in science today also depends on communicating across boundaries, sharing ideas and thinking outside the box. Preferably by the light of the moon. And it was the light of the moon that shone on a very faint trace of a 19-year oscillation in the m0/m relationship. And it shone a gravitational light on the neutral kaon (weak) eigenstates. And it shone clearly on the charged/neutral kaon mass ratio. And on the charged/neutral pion mass ratio. By the light of the moon, walking fearlessly across boundaries and thinking outside the box, I have seen things you wouldn’t believe. I know, you don’t believe them. You think I am crackpot or a lunatic. But I am not. I am a lunar luminary.

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A new paradigm for unification

January 19, 2026

Last week I gave an online talk to the “causal fermion systems” group in Regensburg, as they had expressed some interest in my group-theoretical perspective on whether spacetime has signature (3,1) or (1,3). My perspective, in a nutshell, is that this is a meaningless question, because it is a question about Clifford algebras, and what is actually required for quantum physics is not Clifford algebras, but Lie algebras. Anyway, the slides are at https://robwilson1.wordpress.com/wp-content/uploads/2026/01/regensburg.pdf in case you want to look at them. They contain quite a bit more than I was able to get through in the time available.

At the same time, I was inspired to write a more detailed version, which, in an ideal world, I would have submitted to the arXiv. However, I know they would reject it, and probably use it as a pretext to cancel me completely. In the post-modern post-truth world, reasoned debate has been completely replaced by loud-mouthed idiots who want to be in charge but haven’t got the wherewithal to win a fair argument. Soon, I have no doubt, it will become the post-modern post-truth post-world.

Anyway, I’ve put the paper up here so you can read it at https://robwilson1.wordpress.com/wp-content/uploads/2026/01/spunif4.pdf, and judge for yourself whether you think it is post-modern, post-truth or just post-ed.

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A new paradigm for gravity

January 8, 2026

There have been two paradigm shifts in our understanding of the motions of the celestial bodies in the past 1000 years. Both were the culmination of long and arduous struggles to understand the data. Both resulted in a new mathematical model, based on new physical principles. The first was Newton’s theory of gravity, developed in the late 17th century. The second was Einstein’s theory of gravity, developed in the early 20th century.

To put Newton’s revolution into context, the ancient system was based on the assumption that circular motion was natural, so that all orbits had to be circular. The Ptolemaic system was a vast array of superimposed circular motions (called epicycles, meaning cycles on top (of other cycles)), which got more and more complicated as the data got more and more accurate. The Copernican revolution did not in fact change this one iota – it merely changed the interpretation of the same system of epicycles.

It was not until Kepler studied the incredible data collected by Tycho Brahe that it started to become clear that ellipses ruled the skies, not circles. But nothing really changed until Newton proved mathematically that an inverse square law of gravity gives rise to elliptical motion. At that point the paradigm shift became inevitable.

Newton’s theory of gravity held sway for more than two centuries, and explained practically everything incredibly well. But eventually some tiny details started to show that it wasn’t perfect. The data in question concerned the precession of the perihelion of Mercury. What this means is that the elliptical orbit of Mercury around the Sun does not stay in a fixed place, but the ellipse itself moves slowly backwards around the Sun. Newton’s theory predicts this phenomenon, so that’s not the problem.

The problem was that Newton’s theory got the wrong answer. There was more precession than Newton’s theory said there should be. Now it is quite difficult to build an entire new paradigm on one data point, but that is more or less what Einstein did. It helped that he had already built a new paradigm for electromagnetism, based on the physical principle of relativity – i.e. that physical reality is the same for all observers, even though different observers will naturally use different coordinate systems to describe this reality.

In Newton’s case, the principle says that the theory is the same in all places and at all times, and once you have picked a place and a time, the theory is the same in all directions. Mathematically, this says you can change your x,y,z coordinate system by any rotation, and these rotations form a group SO(3), which is called the point group of the theory. Einstein (aided and abetted by Lorentz, Minkowski etc.) had extended this point group to SO(3,1), which introduced three more transformations (called Lorentz transformations) that explained how moving observers measure time and space differently. By doing so, he ensured that the laws of physics remained the same for an observer moving at constant speed in a straight line.

What he wanted to do, but didn’t do, was to ensure that the laws of physics remain the same for observers that are rotating. For that, he would have had to base his theory on Mach’s principle, which in a nutshell says that physics looks different if you are rotating, but is fundamentally the same. To incorporate this principle into physics you need to consider two different observers, rotating in different ways relative to the phenomenon they are trying to observe. Each rotation requires three parameters for its description, so the transformation between them requires nine.

If you follow through the mathematics, then you can work out what these transformations look like from the point of view of the Lorentz group SO(3,1), and you can work out what the extended group (which obviously has 6+9=15 degrees of freedom in total) looks like. Actually the mathematics only tells you it is one of two possibilities. You have to use physics to tell you which is correct. Einstein picked one of them. I don’t know why he picked that one. I suspect it was because he relied on mathematicians to do the mathematics for him, and they weren’t thinking in terms of Mach’s Principle. Perhaps they didn’t realise that there were two possibilities. Perhaps they just assumed that because space and time are described by real numbers, the transformations must also be described by real numbers.

Anyway, he picked one, and used it to correct the precession of the perihelion of Mercury. And then he used it to predict that gravity bends light waves. Which was confirmed in 1919, by observing stars behind the Sun in a total eclipse. So the paradigm shift began.

This paradigm has lasted over 100 years, and has made a number of successful predictions, including a correction for atomic clocks running at a different rate in geostationary satellites and on the ground, which you rely on for your electronic gadgets to know where you are.

But by 1980, things weren’t looking so good. Galaxies weren’t behaving the way Einstein said they should. In the early 1980s, Mordehai Milgrom played the role of Kepler, by finding empirical laws that described the rotation of galaxies better than Einstein. But there is still no Newton to drive the paradigm shift. So although it is long overdue, Ptolemy’s epicycles (nowadays called “dark matter”) are still in charge. In fact, epicycles are a very good analogy for dark matter, because in both cases there is limitless scope for varying the geometry in order to fit observations. Dark matter isn’t a theory, it is a fudge factor to fit observations without having to deal with the fact (and fact it is) that the paradigm has been falsified.

We need a new paradigm, as Pavel Kroupa never tires of saying. Well, Pavel, I’ve got one right here. Do you want it? As I said, when Einstein built his theory of gravity, there were really two options. He chose one. If that one is falsified, as Pavel and many others believe, then we must choose the other one. All we have to do is multiply the Einstein tensor by a square root of -1. That’s it. You might think this is just a mathematical nicety that makes no practical difference, but that is not the case at all. It changes the point group from SL(4,R), which Einstein used, to SU(1,3), which I use.

The theoretical reason why we know SU(1,3) is correct is because it scales all the way down to the scale of a proton, and describes the behaviour of the quarks inside the proton in exactly the same way that the standard theory (quantum chromodynamics) does. Einstein’s SL(4,R) does not scale. We know it doesn’t scale, because it’s been tried, and it doesn’t work. Experimentally, we notice things going wrong even when we scale down to the kilogram scale, although it is still just about possible to put that down to experimental error.

Anyway, the SU(1,3) model scales from the mass of the solar system to the mass of the proton, which is 57 orders of magnitude. Can we also scale up another 23 orders of magnitude, to the mass of the whole observable universe? I believe we can. The paradigm shift is on its way.

Call me Isaac.

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The source of the “probabilities” in quantum mechanics

January 3, 2026

By re-writing the general theory of relativity in its logical form as a generalisation of special relativity, instead of the Einstein form which is a completely separate theory of gravity, I have achieved two things that Einstein failed to achieve. First of all, the new theory of gravity incorporates Mach’s Principle, and therefore has a complete theory of transformations between non-inertial frames of reference. Secondly, and relatedly, it unifies the electromagnetic and gravitational fields into a single mathematical construct, and shows how the dividing line between the two forces is observer-dependent, in exactly the same way that special relativity explains the movable boundary between electricity and magnetism.

In particular, the Lorentz group appears as an observer-dependent subgroup of SU(2,3) via SU(1,3) and U(1,3). Similarly, the local gravitational fields of the Earth, the Moon and the Sun, and the rotations and revolutions of these bodies, break the SU(3) symmetry of the spacelike coordinates down to nothing at all. In particular, the three coordinates x,y,z of macroscopic space cannot be chosen arbitrarily, because x,y,z symmetry does not respect the gravitational field. There is nothing you can do about this, because there is simply no way that you can shield your experiment from quantum gravity. In principle, your experiment could be contaminated by quantum gravity in multiple different ways, and you cannot ignore this possibility, so you should be alert to it.

Now in quantum mechanics a group SL(2,C) is used locally for a description of the “spin” of electrons, protons, neutrons and so on. It cannot be related to the Lorentz group SO(1,3) as a double cover, because this is mathematically impossible, so the only possibility is to embed it in SU(2,2) inside SU(2,3). That means that the three “directions” of spin are projected onto a two-dimensional space of macroscopic directions. The third macroscopic dimension has absolutely no meaning in quantum mechanics. But now it matters how the two dimensions that are actually used, relate to the three gravitational directions. Since this relationship is completely ignored in quantum mechanics, some important physical variables are simply thrown away, and replaced by a random number generator.

That is where the probabilities come from, and why they are there. Einstein was right – God does not play dice. But we do – we throw away one third of the information, and try to guess what happens based on the other two-thirds.

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CPR for physics

December 31, 2025

It can’t have escaped your notice that theoretical high energy physics is in crisis, and has been for decades. It is in desperate need of cardio-pulmonary resuscitation. The heart, or CORE (Classical Origin of Relativistic Electromagnetism, that is, Maxwell’s equations) has been injected with a Particle Source (PS) of electricity (i.e. an electron), and is in serious danger of becoming a CORPSE (Classical Origin of Relativistic Particle-Sourced Electricity).

Now as I explained in the previous post, the main problem in theoretical physics is to unify the four standard theories – Classical (C), Quantum (Q), and Particle (P) theories and Relativity (R). Quantum theory is always a problem: QC (foundations of quantum mechanics) is in a mess, to put it politely, QR (quantum gravity) shows no signs of progress, and QP (grand unified theories) has been in the doldrums for 50 years. So let’s forget about Q for now, and concentrate on the rest (CPR).

We start from C (Maxwell’s equations). These are differential equations with four (partial) differential operators, with respect to time t and three directions x,y,z in space, plus a constant “source” term for the electric charge and current. The equations therefore have five terms, so can be expressed in terms of 5×5 matrices. Usually only four of these five equations are called Maxwell’s equations, but the other one is equally important (conservation of electric charge). To remind you how it works, I’ll write the operator as (1,d/dt,del), where del is the usual shorthand for space derivatives, and use the charge (density) rho and current J. To avoid having to use the constants epsilon_0 and mu_0 I’ll use D = epsilon_0.E for the electric field and H = B/mu_0 for the magnetic field.

Conservation of charge is the equation d(rho)/dt + del.J = 0, that is d(rho)/dt + div(J) = 0, so I put (0,rho,J) in the first column of the matrix. The scalar equation is div D = rho, that is del D – rho = 0, so I put (-rho,0,D) in the second column. The vector equation is curl H = dD/dt + J, so I put (J,D,-H) in the remaining three columns, where -H here is a 3×3 anti-symmetric matrix with the three components of H in the off-diagonal entries.

Now you ought to be asking me the question, how do I know to put (-rho,0,D) in the second column, instead of (rho,0,-D)? And similarly for the last three columns: (J,D,-H) or (-J,-D,H)? How do I know that rho is antisymmetric, and J and D are symmetric? If I change the sign of the second column, I get D antisymmetric, and rho and J symmetric. If I change the sign of the first column, I get J antisymmetric and rho and D symmetric. If I change the sign of the last three columns, I get all of rho, J and D antisymmetric. These are the only possibilities, because if I change the sign of the whole matrix, the symmetry properties do not change.

  1. In the cases when D is antisymmetric, spacetime (i.e. the last four coordinates) is Euclidean, so we can rule out these two cases.
  2. In the case when rho is symmetric and J is antisymmetric, the Lie algebra is so(1,4), and we get the de Sitter model of spacetime which was used in the early days to unify electromagnetism and gravity.
  3. In the case when rho is anti-symmetric and J is symmetric, the Lie algebra is so(2,3), and we get the anti-de Sitter model of spacetime, giving an alternative model of unification of electromagnetism and gravity.

So we need a good reason to choose between de Sitter (dS) and anti-de Sitter (AdS). In Einstein’s day, only CR unification was being discussed (i.e. the foundations of relativity), because there wasn’t enough particle physics (P) known at that time. And in the CR regime, the only reason to choose one or the other is Einstein’s mass equation m^2c^4 = E^2 – p^2c^2, which points towards so(1,4) as being the correct signature. So have I got it wrong, or did they get it wrong?

Either way, the electric field must be a symmetric matrix, if it is to be united with the magnetic field (anti-symmetric) by the Lorentz group SO(3,1). The standard assumption that it is anti-symmetric is just plain wrong, I’m afraid. It is not just a “convention”, as physicists tend to argue when they get a sign wrong. The question is, whether the current should also be a symmetric matrix (AdS) or whether the standard assumption that it is antisymmetric (dS) is correct. This is not just a convention, it is a question of getting the sign right. We must not guess. This is a question of life or death. Either CPR works or it doesn’t. The evidence is that CPR using the standard (dS) signs does not work. So we had better try the other (AdS) signs before the patient becomes a CORPSE.

Either way, Einstein’s theory of gravity is obtained by adding energy and momentum to the charge/current 4-vector, and adding a “symmetric” tensor to the “anti-symmetric” D and H fields. I’ve put these words in quotes, because they are tensors of SO(3,1), not SO(4). The words symmetric and anti-symmetric, taken out of context, usually refer to matrices, which means SO(4) tensors, not SO(3,1). So what we need to do is generalise Maxwell’s equations to complex numbers, and replace “symmetric” and “anti-symmetric” by Hermitian and anti-Hermitian respectively (applied to matrices, not tensors). So now the Newtonian gravitational field is the imaginary part of D, and the Einstein correction is the imaginary part of H (three degrees of freedom), plus some imaginary diagonal matrices (four more degrees of freedom).

The Lie algebra then becomes su(1,4) in the dS case and su(2,3) in the AdS case. Either way, the spacetime part of the algebra (i.e. the vacuum, without any mass or charge, momentum or current, or energy) is su(1,3), splitting into a real part (electromagnetic field) and an imaginary part (gravitational field). The three “scalar” degrees of freedom (mass, charge and energy) are then related by a three-dimensional Lie algebra, which is compact su(2) in the AdS case, and split sl(2,R) in the dS case. Which is it? It is not just a convention. We must not guess. This is a matter of life or death. CPR either works or it doesn’t.

According to particle physics, the (weak) force that unites mass, energy and charge has gauge group SU(2). But can we trust them? Can we be sure that it really is compact and not split? In this instance, I think we can trust them, because they have found the three gauge bosons (W+, W- and Z), and bosons are always anti-Hermitian. This is a fundamental tenet of Yang-Mills theory also, that gauge bosons always live in a compact Lie group or Lie algebra. So I would say we can proceed with caution, on the assumption that SU(2) is correct. This means that CPR works with su(2,3), and does not work with su(1,4). We still need to test this hypothesis with experimental data, but it looks like the correct path to follow if we want full unified CPR.

So now we can look at the first two coordinates (1, d/dt) in isolation, and note that the charge (rho) is attached to the matrix (0,-1;1,0), which generates a U(1) gauge group for electromagnetism. The energy is attached to the gravitational analogue (0,i;i,0), and the mass to the diagonal matrix (i,0;0,-i). Together these generate the weak gauge group SU(2)_L, and it is chiral because it matters whether you define the mass in terms of energy x charge or charge x energy. If mass is always positive, then it must be one way round and not the other.

What about the space coordinates? If we treat them in isolation, without time, energy, mass or charge, we are left with SU(3). This group has no measurable properties of its own, but it is essential for the existence of the three-dimensional space we live in. In particle physics (P) it is the gauge group of the strong force, and the three things it acts on are called colours (and anti-colours, because they are complex variables, not real). But in our model of CR, it is a gauge group of space, that ensures the real (electromagnetic) definition of space is compatible with the imaginary (gravitational) definition of space. In order to do that, it has to ensure that the electromagnetic definitions of the electron and proton masses agree with the gravitational definition of the mass of the hydrogen atom.

At last! The pulmonary (P) part of CPR is coming to life! The patient is starting to breathe! The lungs are not just a vacuum, they are full of aether! Einstein gravity has gluons in it! And Newtonian gravity is made of neutrinos! (Or should I call them Newtrinos?) Now we can see how C and P work together to make R. It is no good having a heartbeat (Cardio) if you’re not breathing (Pulmonary). It is no good breathing if your heart has stopped. CPR needs a strong (spatial) force, and the timing (weak force) has to be right. Gravity needs massive protons and neutrons, and it is the strong force that generates that mass.

So, you see, I was right all along. The key is to unify C (Maxwell’s equations, with SO(2,3) symmetry) with P (Yang-Mills gauge groups SU(2), SU(3) and U(1)) into SU(2,3) for CP unification, and then interpret SU(2,3) as relativity (R). The patient lives. CPR is successful.

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Fifty shades of unification

December 30, 2025

Unification means different things to different physicists, but ideally we want to unify classical (C), relativistic (R), quantum (Q) and particle (P) theories of physics. There is a widespread view that CR unification was already achieved by Einstein in his general theory of relativity, but there is room for doubt on that score, because classical physics obeys Mach’s Principle, but relativistic physics does not. So there is a school of thought that this bit of unification needs more work, in order to incorporate Mach’s Principle into relativity theory. There is a widespread view that PQ unification was already achieved in the Standard Model of particle physics, but in reality P and Q have been just thrown together, without any attempt at unification beyond the cobbling together of electromagnetism and the weak force into an unintelligible mess. So there is a school of thought that this bit of unification needs more work, and a “Grand Unified Theory” (GUT) is required. In practice, such theories are far too Grand and not nearly Unified enough.

If, however, you really do believe that CR and PQ unification is already adequate, then you believe, as Einstein did, that QR unification is the issue. This union of quantum theory and relativity is also called Quantum Gravity, and there are various schools of thought as to how best to achieve this goal. Many people are working on this aspect of unification in one way or another. But if CR unification is really done and dusted, then it is enough to do CQ unification instead. This is where Einstein focussed his attack in the 1930s, because he did not believe that the Copenhagen School had done the CQ unification properly. Nor do I. The Danelaw that has been set up as a result of the Viking invasion is not a unification, it is an occupation. So if we can’t do QR unification, then the probable reason is that one or other of CR or CQ unification has been done wrong.

You will have noticed that I have ranted and raved about both CR and CQ unification many times on this blog. They cannot both be correct, because they are mathematically inconsistent (not to mention physically inconsistent). People are generally very tribal about which one they believe in, and which one they attack. But in reality they are both wrong, and it is impossible to construct a theory of quantum gravity without doing the whole CQR unification again from scratch. So that is what I am aiming to do, and I follow Einstein in thinking that the key is to repel the Viking invasion, and re-establish the theory of quantum physics on solid foundations. A Viking longship is not an adequate foundation for a castle. The “standard” view of CQ unification must be challenged.

Now if we approach the problems from a particle physics (P) perspective, QR unification is very far from their concerns. Not relevant to particle physics, they say. But if PQ unification has been achieved (which it hasn’t) then you might as well say that QR unification is equivalent to PR unification (which it isn’t, of course). From 2010 onwards, I was recruited to work on PQ unification (GUT), and not allowed to talk about QR or PR unification. But I could see that it was impossible to understand PQ unification without also discussing PR and QR unification. The whole triangle of PQR unification cannot be separated into separate pieces. So I looked at PR unification, and by 2015 I had found the evidence I needed that PR unification is actually the most interesting part of the whole jigsaw. That is where the experimental evidence is really bounteous, and unification is therefore tightly constrained.

Strangely enough, nobody is interested in PR unification. I find this utterly bizarre. I have talked to many people who are interested in PQ unification, and to many people who are interested in QR unification, but I have never met anyone who is interested in both at the same time. Neither group of people therefore has any interest in PR unification. As far as I can tell, I have the entire field to myself. (That probably isn’t true – there must be lots of other crackpots out there like me.) Ten years ago, it was all about trawling through the experimental evidence to find the links between P and R. Today, it is all about building the model of PR unification. That is what I have achieved in 2025. The model of PR unification is SU(2,3), whose compact part is the gauge group of P, and which splits into real (special) R and imaginary (general) R. Hence P mixes the real (electromagnetic) and imaginary (gravitational) parts of R (and C), while R mixes the different forces and particles in P.

The model therefore makes oodles and oodles of predictions. It fits the experimental data incredibly well. But because it is not in an established field like PQ, QR, CR or CQ it is ignored. Nobody but a crackpot would work in PR unification, would they?

Well, I’ve talked about five flavours of unification, that is CR, CQ, QR, PQ and PR, so what is missing? The “top” unification CP. I call it “top” because it is the top predator in this wilderness (landscape, swampland, whatever you want to call it). Classical physics (C) explains many things in terms of waves, which particle physics (P) explains in terms of particles. So, obviously, particles and waves must be the same thing, right? Particles must be like waves on a tiny vibrating string, right? So string theory was born out of an attempt at CP unification. It was a bold idea, to go straight from C to P without going through either Q or R on the way. Sidestep the problem that Q and R are incompatible by ignoring the problem altogether. Brilliant. Give that man a Nobel Prize for Nothing.

The trouble is, that because string theory completely ignores both Q and R, it is completely divorced from physical reality. It cannot, and never will, predict anything at all about real physics that we can measure in experiments. It is not, and never will be, the “only game in town”. There are, as I have shown, six games in this town. Each of them comes in various shades of grey. Let me remind you:

  1. CP – string theory
  2. QR – quantum gravity, e.g. loop quantum gravity, non-commutative geometry
  3. PQ – grand unified theories
  4. CQ – foundations of quantum mechanics
  5. CR – Machian gravity
  6. PR – my own private Idaho

I have listed them roughly in order on a scale from mainstream to crackpot – although you might order them slightly differently. The first four have been actively pursued by large numbers of people for a very long time, and between them they have absolutely nothing to show for it. No predictions, or else utterly wild predictions, none of which has ever been verified. Extremely complicated and ugly theories that make Ptolemy’s epicycles look like a model of simplicity. Landscapes and swamplands with no clue where to go. So isn’t it time to invest in the other two games?

Machian gravity has a few adherents, Unzicker and McCulloch for example, but they are regarded by the mainstream as crackpots. Revisiting the foundations of relativity (CR) is not regarded by the mainstream as a productive activity. Still, more productive than string theory, I would say. Revisiting the foundations of quantum mechanics (CQ) is more obviously necessary, but still ignored by the mainstream, as being too “philosophical” – what they really mean is, too difficult. But they can’t tell us what a “spinor” is, so where is the connection between quantum mechanics and reality? It’s all a bit too vague, if you ask me.

But even CR and CQ don’t make it as far on the crackpot scale as PR does. Only a lunatic would look seriously at PR. Well, why not, if all the other shades of unification have failed? Why not look at the only remaining possibility? You never know, you might find a link between:

  1. kaon decay CP-violation and curvature of the Earth’s gravitational field;
  2. kaon lifetimes and the gravity of the Moon;
  3. neutron lifetimes and the rotation of the Earth;
  4. the weak equivalence principle and the tilt of the Earth’s axis;
  5. the muon g-2 anomaly and the direction gravity;
  6. the W/Z mass anomaly and the tides;
  7. the weak force and the calibration of time;
  8. the strong force and the calibration of space;
  9. neutrino oscillations and curvature of spacetime;

and so on and so on. And when you’ve got that far, you may find that PR unification has helped you to understand what went wrong with CR unification in 1915, what went wrong with CQ unification in 1925, what went wrong with PQ unification in 1975, what went wrong with CP unification in 1985, and what went wrong with QR unification in 1995.

And put it right.

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Why is the Sun’s atmosphere so hot?

December 25, 2025

I have answered this question before, but I think I understand it a bit better now. The temperature of the Sun’s surface is about 6000 degrees (Kelvin, Celsius or centigrade), but the atmosphere is about 1000000K, which is the same as the surface of a neutron star. This is a clue that the physical processes that generate this extremely high temperature may be the same.

The surface of a neutron star literally boils at this temperature. The process is like nuclear fission, but with an atom that weighs as much as a star. You might think the individual neutrons would boil off one by one, but the process is more complicated than that. In nuclear fission, typically alpha particles (helium nuclei) are ejected, that is two protons and two neutrons stuck together. In a neutron star, the protons are generated by a quantum-gravitational equilibrium between the five-neutron-plus-one-neutrino state and the three-proton-plus-three-flavours-of-electrons state. These two states have exactly the same mass, so the change of state occurs adiabatically (without energy transfer).

But at the surface, this equilibrium is disturbed, because the electrons can boil off, leaving too many protons on the surface, which are then evicted for overcrowding, and then there’s an imbalance between the three electron flavours, which has to be corrected as well. Correcting this imbalance generates an enormous amount of energy, which keeps the temperature up.

Alternatively, one can consider a stable unit of 5n+3p+e+mu+tau getting separated off. It is trying to be an atom of lithium-8 (atomic number 3), but it is horribly unstable, both because lithium usually occurs as lithium-7, and because two of its electrons are in the wrong flavour (mass) eigenstates. Various things can happen, including beta decay to beryllium-8, which then captures a neutron (there are plenty around) to become beryllium-9, losing energy by muon and tau decay to electrons. Whatever happens, the energy is generated by muon and tau decay, and the protons and neutrons end up in some mixture of hydrogen, helium, lithium and beryllium nuclei (possibly including small amounts of heavier nuclei), which then boil off into outer space, together with all the electrons that got separated from them.

Basically, the same thing happens in the solar atmosphere. There’s plenty of hydrogen, helium, lithium and beryllium in the Sun, and it keeps boiling off into the atmosphere. You would think, of course, that all their electrons would be ordinary electrons, but you are forgetting two things. First, the nuclei are completely ionised – they have no electrons. The electrons are left entirely to their own devices. And second, solar neutrinos occur in equal numbers in all three generations. Therefore, so do the electrons.

Hence the solar atmosphere is exactly the same as the atmosphere of a neutron star. Hence it behaves in exactly the same way. Hence it is at the same temperature.

Basically, the same thing is supposed to have happened in the Big Bang. It is called BBN (Big Bang Nucleosynthesis), because it synthesises nuclei. But it doesn’t need a big bang. It happens all the time to neutron stars. And especially it happens when neutron stars collide. Which they often do.

So it turns out that that weird little mass equation that I stumbled upon back in April 2015 is the key to understanding both Big Bang Nucleosynthesis and Dark Matter, not to mention the coronal heating problem and the solar neutrino problem. Who woulda thunk it?

Oh, and by the way, this tells us that exploding neutron stars generate 5 neutrons for every 3 protons. Or you could say the atomic mass to atomic number ratio of a neutron star is 8/3 or about 2.667. Ordinary matter has a ratio of 1 (hydrogen), 2 (helium, carbon, nitrogen, oxygen, etc) rising to 238/92 or about 2.587 for uranium and 244/94 or about 2.596 for plutonium (96% neutron saturation). Anything more than this, and the nucleus will spontaneously decay under 5n -> 3p+e+mu+tau, i.e. the same process as neutron star evaporation and coronal heating.

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The no-singularity theorem

December 23, 2025

The biggest problem with General Relativity (apart from the fact that it is contradicted by astronomical observations, to the extent that many experts consider it to be categorically falsified), is that it predicts gravitational collapse into singularities. Singularities are mathematical constructs of continuous theories, that are never observed in practice, and cannot occur in quantum (i.e. discrete) theories.

What is undeniably observed in practice are regions of spacetime with very high density. The highest density that can be reliably confirmed is that of a neutron star, in which all the atoms of normal matter are squashed down to the size of the nucleus. Higher densities than this are predicted by some theories, but not confirmed by any observations. In particle physics, higher densities than this are forbidden by the Pauli exclusion principle. In general relativity, they are not forbidden, and therefore they are compulsory.

In other words, the no-singularity theorem is a property of quantum theory, and usually goes by the name of the Pauli exclusion principle. In order to construct a quantum theory of gravity, therefore, it is necessary to embrace the Pauli exclusion principle, and prove a no-singularity theorem for gravity. In the standard scenario, with a gauge group SL(4,R) for spacetime, there is no invariant metric, and therefore nothing to prevent gravitational collapse to a singularity. In my proposal, with SU(1,3) as a gauge group for spacetime, there is an invariant metric, which implies that collapse to a singularity is impossible.

The way this works is that the electric and gravitational fields obey Fermi-Dirac statistics, while the magnetic and tidal fields obey Bose-Einstein statistics. The usual assumption of quantum electrodynamics is that the electric field consists of “virtual” photons (whatever that means), and is therefore bosonic. But since the virtual photon cannot be directly observed, there is nothing to say it doesn’t decompose into two fermions, travelling in opposite directions. String theorists ought to know this, because that is how standing waves arise, from superposition of two ordinary waves travelling in opposite directions. So the way that electrons and protons hold each other at arm’s length (so to speak) is by firing neutrinos at each other the whole time. It’s just a microscopic game of three-dimensional tennis, played at such a speed that we don’t see the ball, all we see is a blur.

It is really important to understand that the tennis ball is a fermion, not a boson. In SL(4,R) theory, it is a boson, but in SU(1,3) theory, it is a fermion. And this is true not only for electrical tennis, but also for gravitational tennis. The gravitational field is fermionic, and therefore gravitational fields repel each other. They do not attract each other, as the theory of general relativity predicts. The reason why matter does not collapse under gravity to a singularity is that beyond a certain density the self-repulsion of the gravitational field balances the mutual attraction of the matter. The reason why a proton and an electron do not continue to attract each other and collapse into a neutron, as you might expect by extrapolating the classical theory, is because beyond a certain point the electric fields repel each other, and balance the electrostatic attraction of the matter.

The reason why two electrons do not continue to repel each other once they get cold enough to huddle together is probably the same – and superconductivity is the result. While two neutrinos surely repel each other, and so do two antineutrinos, a neutrino and an antineutrino must logically attract each other. Opposites attract, and all that. The quantum effects here manifest themselves once you reach the scale where you have to separate the individual neutrinos and antineutrinos, and can no longer treat them as a combined “virtual photon”.

The same happens with quantum gravity. On a human scale, with strong gravitational fields generated by bodies in the Solar System, there is a strong mutual repulsion of gravitational fields, which is edited out of the standard theory, so that the gravitational constant G is ameliorated, and we measure a smaller value than if we were in a weaker gravitational field (such as applies to a very wide binary star system). General relativity does not predict this effect, which is genuinely observed.

Since this is a blog post, and not a mathematics paper, I do not propose to go into more detail of the mathematical proof of the no-singularity theorem for SU(1,3). But I hope you can see that it makes sense physically, regardless of the details of the mathematics.

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Never try to repeat a failure

December 22, 2025

One of my father’s many wise sayings was “Never try to repeat a success”. Apparently this saying is attributed to Walt Disney, although I am sure it is much older than that. But it occurred to me that The Trouble With Physics is rather different. The problem with string theory is not that they are trying to repeat a success – they are trying to repeat a failure. And, to be fair to them, they are very, very good at repeating failure. Spectacularly good at it, I would say. But it stretches a point to call this achievement a “success”. Which, of course, they do.

Now, without wishing to brag about it, I think I am actually quite good at failure. I’ve had a lot of practice at it, and if ever I start to feel that I’m not as good at failing as I used to be, the world steps in to show me how wrong I am. But repeating a failure? No, I try to avoid that. Although on occasion I might fail to avoid it, of course. But I prefer to fail in new and more interesting ways. Which is why, when one of my models of physics fails, I move on to another one.

I started out, as you know, working on E8 models of fundamental physics. They failed, as you know. There are many people working in this field who are trying to repeat this failure, but I’m not one of them. There is a mindset among very clever people that says, our failure is just because we are too stupid. So they keep banging their heads against a brick wall, hoping that the brick wall will give way. There is a mindset among very stupid people (like me) that says, our failure is just because we are too clever. You have to be very clever to understand string theory. The failure of string theory is very obviously a failure caused by being too clever. The solution is obvious – just be more stupid. I’ve said it before, and I’ll say it again, there is just not enough creative stupidity in theoretical physics.

If you have enough creative stupidity, then you can be sure that you will always fail to repeat a failure. You may fail in ever new and creative ways, but as my old friend Richard Parker used to say, “Never discount the possibility of success”. Once you have exhausted all possible ways to fail, whatever remains, however unlikely it seems, must be success.

When I had run out of new ways to fail, the only model that was left was the SU(2,3) model. When I had run out of new ways to fail to show that the Standard Model is wrong, I discovered that the Standard Model is completely correct, and is completely described by the Lie algebra su(3,2). For years I had failed to show that the Dirac equation is wrong, and eventually I was forced to accept that the Dirac equation is correct. And because it is an equation in quantum theory, it is formally an equation in a Lie algebra. So I did the calculations to find out which Lie algebra it is, and it turned out to be su(2,3). There are only two possibilities: ether the Dirac equation is wrong, or the gauge group of fundamental physics is SU(2,3).

I spent ten years exploring the first possibility, and I have to admit that that exploration ended in failure. I do not want to repeat that failure, so now I am working on the second possibility.

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The riddle of mass

December 16, 2025

In the SU(3,2) model there is an underlying five-dimensional complex “spacetime”, consisting of three complex (or six real) “spacelike” degrees of freedom and two complex (or four real) “timelike”. But it is really a misnomer to call it “spacetime”, because it is just a mathematical abstraction on which all the theory is based. The physical things that happen lie in the adjoint representation, which has 24 real dimensions, of which four are diagonal matrices. The 20 off-diagonal things include momentum, current, kinetic and electromagnetic energy, electric, magnetic and gravitational fields. Two of the diagonal elements are definitely masses, but what about the other two? In classical gravity, they are part of the tidal field, but what are they in quantum gravity?

So let’s look at the diagonal in detail. The five diagonal entries either add to zero (in the Lie algebra) or multiply to 1 (in the Lie group), so we only need to consider four of them to get the full information. Actually, this isn’t quite true – in the Lie algebra, the only restriction is that the sum is an integer – it doesn’t have to be zero – and this is very important for quantisation. The first (spacelike) entry is the mass in the Dirac equation, represented by (1,i,i,i,i) in the Lie group SU(2,2). In the original Dirac equation, this mass was the mass of the electron. Not a muon, or a quark or a neutrino, only an electron. This is important, because when you generalise the equation to include the muon, you get an equation that looks the same, but is not the same. It cannot be the same, because the mass is different. So the mass of the muon is represented by a different choice of subgroup SU(2,2), and might, for example, be (i,1,i,i,i). Similarly, the mass of the tau particle (third generation electron) might be in (i,i,1,i,i).

Now let’s look at the “timelike” masses. These are Einstein masses, not Dirac masses, and they are masses that are used in the theory of gravity. Basically, they are neutrons, because that is all that astronomers care about. Or hydrogen atoms, which are near enough the same, just about 0.13% lighter than a neutron. But also, the diagonal element (1,1,1,i,-i) represents the third component of some kind of isospin, and distinguishes the proton from the neutron. So we might have (i,i,i,i,1) representing a neutron, and (i,i,i,1,i) representing a proton, say. The details are clearly not quite right, because it is not entirely clear which of the three components of isospin we need to measure. But the principles are surely correct. The diagonal consists of five masses (electron, muon, tau, proton and neutron) with a linear relation between them, to make the diagonal four-dimensional instead of five-dimensional.

So that is a really fundamental reason why there must be a mass equation something like electron+muon+tau+3protons=5neutrons. The model doesn’t immediately tell us the numbers in this equation, but experiment does. Since this is an additive equation, rather than multiplicative, it must be a property of the Lie algebra, not the Lie group. There is some choice for a basis of the four-space of diagonal matrices, but once the basis is chosen, the Lie algebra contains all the quantum numbers that we need for quantum gravity. I have written these quantum numbers down in several papers and blog posts already, so all that is missing is to match them up to the 5×5 diagonal matrices.

One possibility is just to write down the quantum numbers I already used, and add a fifth quantum number to ensure the sum is zero. This puts the three generations of electrons in (1,1,0;-1,-1), (0,1,1;-1,-1) and (1,0,1;-1,-1), the proton in (1,1,1;1,-4) and the neutron in (1,1,1;0,-3). Then the proton is the generator for Einstein mass, rather than the neutron, because additive notation (1/4,/1/4,1/4,1/4,-1) corresponds to multiplicative notation (i,i,i,i,1). But one can also change basis on the weak isospin (time) coordinates, or on the flavour (space) coordinates, and get a variety of alternatives. The neutron, for example, identifies coordinate 4 as the time coordinate, that measures the neutron lifetime, as well as the charge coordinate, which was where I started with the quantum numbers.

But the Dirac equation puts the electron masses in (1,1,0;1,1)/4, (0,1,1;1,1)/4 and (1,0,1;1,1)/4, which is a little bit different. The coordinates no longer add to zero, but to 1, which is important for quantisation of mass: if they add to zero, you can multiply/divide by any scalar, and there is no longer any quantisation. But if they add to 1, you cannot divide by anything, that is an indivisible unit of matter. So where do we put the proton and neutron? The mass equation seems to require the spacelike coordinates to be (1,1,1)/4, which suggests timelike coordinates (1,0)/4 and (0,1)/4 for proton and neutron (in some order). So if the neutron mass is the Einstein mass then we should have (1,1,1;0,1)/4 for the proton and (1,1,1;1,0)/4 for the neutron. Then we calculate the three electrons plus three protons as (5,5,5;3,6)/4, which is no longer equal to five neutrons, that is (5,5,5;5,0)/4, but differs by (0,0,0;-1,3)/2. The latter might represent a neutrino, with zero Dirac (spacelike) mass but nonzero Einstein (timelike) mass, for example. It lives only in the last two coordinates, that is the “left-handed” spinor coordinates. It is a particle (rather than an anti-particle), because the sum of the coordinates is +1 (rather than -1).

A neutrino is in any case required in order to balance the spins on the two sides of the equation. This is independent of whether spin is defined in the usual Fermi-Dirac way, or in the Hermitian/anti-Hermitian way. The neutrinos in that case are responsible for linking the two timelike coordinates together, to provide a classical concept of “time” that works well provided we do not stray too far towards either the relativistic boundary, or the quantum boundary. Or the MOND boundary. The three flavours of neutrinos each provide a different “timescale”: the electron neutrino measures the lifetime of the neutron, and the muon and tau neutrinos measure the lifetimes of the muon and the tau particle respectively. The classical concept of time projects all these three lifetimes onto a single time scale. But in principle, and in quantum mechanics, these lifetimes are three independent parameters. Time is gauged with SU(2), not U(1).

You will have noticed that the three generations of electrons, with spacelike coordinates (1,1,0), (0,1,1) and (1,0,1), can be distinguished by just looking at two of the three coordinates. So if we just keep a single Dirac equation, with mass in the first coordinate, we still have (1,0) (1,1) and (0,1) in the other two coordinates to tell the three particles apart. So the Dirac equation still has enough degrees of freedom to distinguish the three generations of electrons. This is important, because it means that we can break the symmetry of space by imposing a gravitational field, and still distinguish the three generations, even if we do all our experiments perpendicular to the gravitational field. If we couldn’t do that, then the model would be in serious conflict with experiment, and would have to be abandoned. But this does come at a cost – the third (vertical) dimension has to be implemented as a “quantum superposition” of the two horizontal dimensions, instead of being an independent dimension in its own right.

So we come to the same conclusion as before – the concept of “quantum superposition” is not a description of physical reality, it is a description of our failure to distinguish two isomorphic groups, SU(2) and Spin(3).

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