Continental drift

The theory of continental drift is now a well-established part of orthodox explanations of how the Earth got to look how it does. Not so when I was growing up in the early 1960s. From a very early age I was fascinated by my father’s globe, and the amazing amount of information contained on a piece of paper of area , with r somewhere in the region of 15 cm. My best birthday present ever was an atlas my grandmother gave me when I was 5, and I loved doing jigsaw puzzles, especially when the picture was a map. With this background, it is impossible not to notice that the East Coast of South America fits into to the West Coast of Africa like a hand in a glove. When I was told, it’s just a coincidence, it doesn’t mean anything, I realised for the first time that grown-ups don’t know everything. I knew deep inside me that it could not be a coincidence.

Fast forward half a century. I approach particle physics as a 55-year-old child, wanting to learn everything. But I am still a child, I want to know the truth, I am not interested in answers that say it’s too hard for you to understand, because it is *not* too hard for me to understand. Continental drift is easy for a child to understand. Things that grown-ups say are fixed and have always been that way, children can understand might not always have been that way. I looked at the masses of the elementary particles the way I looked at the coast of South America, and I looked at the Solar System the way I looked at the coast of Africa, and I saw that they fitted together. Everyone says “coincidence”, but the five-year-old child that still lives within me says no, this cannot be. This is not a coincidence. This is the explanation for why the world is as it is.

 

Group theory in physics, 2

Group theorists like to trace the history of the subject back to Plato and the five Platonic solids, but a more realistic beginning lies in the work of Lagrange on solutions of polynomial equations in the 18th century, when the idea of permutations of the roots of the equations started to take hold. Another beginning lies in the work of Galois around 1830, in which a whole new level of abstraction appears, and the word `group’ is used for the first time in this context. But Galois was perhaps a lone genius, and group theory developed very slowly until 1870, when it started to become a separate subject in its own right. By the 1920s, group theory had become a substantial theory, and a large part of this was due to Klein and Lie, who had made the subject into a formidable tool for analysing differential equations in physics. By 1960, the subject had reached the point where a full classification of finite simple groups seemed within reach. A whole generation of highly qualified specialist group-theorists suddenly appeared.

At this time, Murray Gell-Mann was investigating the symmetries of the particle zoo that experiment was producing. The story is often told (at least by physicists who want to prove that physicists are cleverer than mathematicians) that he invented group theory for himself. It is a pity he did not consult any of the army of group theorists available. The group theory that has been passed down from his work into the textbooks is, unfortunately, riddled with schoolboy howlers that make it completely nonsensical. The same is true for much of the rest of the group theory in the standard model. It is full of `pattern-matching’ – that is guessing which groups might explain the symmetries physicists think they see. Most of these groups are wrong, which is why the standard model requires a whole heap of arbitrary parameters in order to paper over the fundamental errors.

All the gauge groups in the standard model are symmetry groups of non-physical invented concepts, and are specifically designed to divorce the nuclear forces and electromagnetism from gravity. Then physicists complain that they can’t unify these forces with gravity. Can’t they see that they have specifically designed their theory in order to prevent such a unification?

In order to move on from this absurd situation, it is time that physicists started to listen to group theorists. You cannot change the experimental results, and you cannot change the facts of group theory, but you can change your assumptions about which groups describe the symmetries of the physical universe. There is only one possible group that can be used in this situation, and that is the group SL(4,R), and its Lie algebra sl(4,R) that describes all possible infinitesimal distortions of spacetime. Any group used in the standard model that is not a subgroup of SL(4,R) or PSL(4,R) is,  ipso facto, rubbish and must be thrown away. This includes all the various versions of SU(3) that occur in the eightfold way and in quantum chromodynamics. They have nothing whatever to do with physical reality.

Trust me, I am a group theorist.

Inertial frames

It is more than five years since I first tried to explain to theoretical particle physicists that the frame of reference in which the particle experiments are done is not an inertial frame, and that assuming it is an inertial frame is therefore a MISTAKE. Anyone with the meanest intelligence can understand that what I say is obviously true. Why is it that no theoretical physicists can understand this absolutely obvious fact? Mathematicians understand that mistakes have to be corrected. Theoretical physicists apparently regard mistakes as something to be ignored. Perhaps I am very stupid, but this is an attitude I simply cannot understand.

 

Acorns

It is interesting how the coronavirus pandemic has changed human civilisation in such fundamental ways in just a few weeks. Three months ago it would have been unthinkable that we should be forced to behave in the ways we are now behaving. Huge sections of society are being forced to consider new ways of living, and new ways of staying alive. Although serious food shortages are not yet threatening the rich part of the world, people do seem to be seriously considering the options for growing more of their own food, and relying less on going out shopping in supermarkets.

In this context I was reminded of a neglected resource, that is the oak tree.  This tree is highly prized as a source of wood, once it is dead. That is like valuing the cow as a source of beef rather than a source of milk. The acorn was once a staple food throughout Europe, only falling into disuse gradually as the cultivation of cereal crops took over. Cereal crops have now taken over completely, and we have forgotten the acorn. The oaks here in Birmingham are just now exploding into their glorious green spring colours. Come the autumn, they will each produce tens or hundreds of thousands of acorns. When the acorns are ripe, they are dark brown and fall off the tree. Eat them, they are delicious. Don’t leave them for the squirrels.

Mass and weight

Weight is a concept that is as old as humanity, and probably much older. It describes something pressing down, and has two aspects, namely magnitude (how hard it presses) and direction (which way is down). In the 17th century, Isaac Newton separated the concept of weight into two separate concepts, namely the concept of mass and the concept of gravity. The mass takes (some of) the magnitude, and gravity takes the direction (and the rest of the magnitude). Since that time, mass has been considered the fundamental concept in science, and weight has been considered a derived concept. They are related by the force of gravity.

Science has therefore progressed in a way that separates gravity from everything else. The forces of electromagnetism, and the weak and strong nuclear forces,  are all described in terms of mass, rather than weight, and are therefore completely separated from gravity in the theory of physics. Einstein’s theory of special relativity unified the concepts of mass and energy, and ignored weight. Einstein’s general theory of relativity revolutionised the theory of gravity, in such a way that gravity is no longer considered to be a force. As a result, the concept of weight disappeared altogether from physics, leaving energy alone as the fundamental concept.

In the early 20th century, the development of quantum mechanics was based on the concepts of mass and charge of fundamental particles (at that time, the electron and the proton), but by the 1930s it had become clear that there was a problem: quantum mechanics was incompatible with general relativity. Nearly a century of hard work by thousands of very clever people has not solved this problem, and has failed to unify the theory of quantum mechanics with the theory of gravity. Why is this?

Could it be that science has developed in a way that absolutely prevents such unification? Could it be that Newton was wrong, and that weight is after all a more fundamental concept than mass? Could it be that Einstein was wrong, and that energy is not equivalent to mass, but is equivalent to weight? Could it be that Dirac was wrong, and the Dirac equation should have a weight term instead of a mass term?

“NO, OF COURSE NOT!!!” – I hear a chorus of thousands of physicists.

Just wait.

Just weight.

 

 

Theorems in physics

From Euclid onwards, every theorem in mathematics has been of the form “hypothesis implies conclusion”. There is no exception to this rule, under any circumstances, anywhere, anywhen, anybody, anything. Mathematicians and philosophers generally have precisely two ways of interpreting such a theorem: (a) if hypothesis is true, then conclusion is true, and (b) if conclusion is false, then hypothesis is false. Most other people, and unfortunately this includes a lot of physicists, have only one interpretation: (c) hypothesis is true, therefore conclusion is true. This is usually abbreviated to (d) conclusion is true.

The pitfalls of this approach are obvious. As more and more such theorems are proved, one is forced to swallow more and more absurd conclusions. There are quite a number of theorems in which the hypothesis is something like the axioms of quantum field theory, and the conclusion is something like the existence of the multiverse, or many-worlds, or electrons have free will, or any number of other completely absurd conclusions. Many physicists refuse to question the hypotheses, and are therefore compelled to believe these outrageous conclusions. As a mathematician, and an amateur philosopher, my interpretation is the opposite: the conclusions are absurd, therefore the hypotheses are false.

This argument is so old and so well-known that it is called “reductio ad absurdum”. Why can I not get a single mainstream physicist to accept this obvious, even tautological, argument?

The axioms of quantum field theory are false. End of.

The end of an era

My PhD supervisor, John Horton Conway, died on Saturday at the age of 82, apparently of Covid-19, after being taken ill suddenly on Tuesday. It is hard to say anything at this time.

He was an inspiration to a generation of mathematicians. His anti-establishment, anti-authority attitude was refreshing, and helped me and others to think outside the box with regard to problems of all kinds. His approach was that mathematicians make mathematics too complicated, and there is always something simple at the root of it all. If you can only understand the simplicity, then all the rest follows. Like many of his students, I have made a career doing complicated mathematics – but I am not proud of that – what I am proud of is the rare cases where I found the simple underlying principle.

He taught me that you do not need to be an expert in order to make the big breakthroughs – when I asked him to be my supervisor for a PhD in group theory, he told me he wasn’t a group theorist – yet he discovered three of the 26 sporadic simple groups, and showed how the Leech lattice unified 12 of them. A lesson he tried to teach me – I don’t know if he succeeded – is that if your problem divides into a number of cases, then you should always attack the hardest one first. That is the opposite of what most people do.

Another thing he said was, you should attack six problems at once, at various levels of difficulty from the trivial to the impossible. That way you allow for the possibility of progress on all possible scales at once. When you don’t solve the impossible problems, you get recognition from solving moderately hard problems, and if you can’t solve those then at least you keep your boss happy for a little bit longer by solving the easy problems. And, more importantly, you keep yourself happy by feeling that you have done something worthwhile. But if you solve an impossible problem, then you’ve won the lottery.

I like to think of Conway as embodying Einstein’s dictum that everything should be made as simple as possible, but not simpler. He re-worked all his ideas tirelessly, throwing away yesterday’s version as soon as today’s version was on paper, and simplified, simplified, simplified until no more simplification was possible. The modern generation of mathematicians and physicists could learn a lot from taking this idea more seriously.

Realism in quantum mechanics

There are more than 100 essays in the FQXi essay competition, covering the whole spectrum of craziness from practically mainstream to completely absurd. The point, of course, is to draw together and share lots of crazy ideas and get some discussion going as to which of them show promise for new approaches to old problems. It is counterproductive to dismiss any of them without first engaging with the discussion.

An essay I particularly liked was by Boris Semyonovich Dizhechko in which he argues for a Cartesian view of reality, in which matter and space are really the same thing. This philosophical viewpoint is quite attractive to those who try to explain macroscopic features such as spacetime as `emergent’ from aggregates of more elementary concepts. My view is rather similar, but with a relativistic slant: that is, I do not accept space as a fundamental concept, but insist on using spacetime instead.

The separation of spacetime into space and time is a property of the observer. It is then absolutely critical to maintain clarity as to which parts of a model are observer-independent and which are not. It is a main theme of this blog that this clarity is missing (and always has been) from all discussion of quantum mechanics. Wherever probability enters into quantum mechanics, one can take the view that the experiment is an “observer”, and that the uncertainty arises from an underlying observer-dependence. I have discussed many examples on this blog, including neutrino oscillations, kaon oscillations and others.

The central problem of quantum mechanics, as I see it, is that of interpretation of the spinor. In mainstream quantum mechanics, the spinor is a piece of magic that happens to work to make predictions. As such, I regard it in much the same light as any other piece of magic that claims to predict the future, such as horoscopes, crystal balls and tea-leaves. It may well be argued that quantum mechanics does a better job of prediction than these other methods, but the predictions are on different scales so cannot be directly compared. Either way, the spinor is still magic, and does not exist in the real universe.

It is of course true that a spinor (2-dimensional complex) cannot be embedded in Euclidean 3-space. But there is nothing to prevent it from embedding in 4-dimensional spacetime. There is therefore nothing to prevent us from regarding a fundamental spin 1/2 particle as a fundamental quantum of spacetime. Then the macroscopic structure of spacetime is built up from the local interactions of these spinors with each other. Everything in this blog is really just an elaboration of this idea, with some mathematics to describe space-time on different scales.

In particular, while spacetime on a macroscopic scale looks pretty rigid (as long as you don’t travel so fast that time dilation and Lorentz contraction start to kick in), on a microscopic scale it starts to look distinctly rubbery, or worse. At the smallest possible scales, spacetime effectively loses its structure altogether. But the elementary particles of which spacetime is made are still 4-dimensional objects, and therefore the local symmetry group in which the particle operates cannot be anything other than SL(4,R). This is not the symmetry group of the particle itself, which is obviously finite: it is the symmetry group of the spacetime environment in which the particle finds itself.

In other words, this group SL(4,R) contains within itself all the information required to describe the interactions of the particle with its environment, that is with other particles.  Details can be found in many posts on this blog. For example, the measured value of the mass of an elementary particle contains too much information to be intrinsic to the particle, and instead lies in the group SL(4,R) that describes how the particle relates to its environment. Understanding mass therefore amounts to understanding this group, and how it actually models the relationship of a particle to its environment.

Essay for FQXI competition

The Foundational Questions Institute is running an essay competition on the topic of Undecidability, Unpredictability and Uncomputability, and I decided to submit an essay, entitled “Unpredictable unpredictables”, which you can find on their website at FQXi.org. The theme of my essay is that all of these concepts are intimately bound up with what assumptions are being made. The most obvious false assumption that is made in particle physics is that the rest frame of the typical particle accelerator is an inertial frame. I discuss the possibilities for replacing this assumption by something more plausible, and the implications of such a replacement. I invite you to read the essay in full, and make comments and assign marks out of 10 on the FQXi website.

Updated preprint (INI 19014)

A corrected and significantly extended version of this preprint is now available here.