About 50 years ago, I read a book called “How to lie with statistics”. Some of the tricks presented in that book have stuck with me ever since. There are, of course, many ways to lie with statistics, but one of the most widespread, and also one of the most dangerous, is to confuse the probability of B given A, with the probability of A given B. This confusion is behind many miscarriages of justice, because it is easy for lawyers to persuade juries that the two are the same, when they are not. What is the probability that 500 subpostmasters had their hands in the till, given that Horizon found the numbers didn’t add up? It is not the same as the probability that Horizon finds the numbers don’t add up, assuming that the subpostmasters had their hands in the till. The latter probability is essentially 1, i.e. it is a near certainty. The mistake that the innumerate people at the top of the Post Office made is to conclude that the former probability is also a near certainty, instead of calculating the probability that Horizon has made a mistake, given that such a huge number of errors were found. So you can see that this type of statistical lie is very very dangerous.
There is an analogous method of lying with mathematics, which you may have learnt in primary school. You remember those “proofs” that -1 = 1? They relied on the fact that the squares are equal: that is (-1)^2 = 1^2, which is a true fact, of course. Then you take the square root of both sides to get -1 = 1, which of course is not true. The argument is presented with some added complications, to distract you from the main point, so you don’t notice the sleight of hand, but this is the essence. It is true that if x=y, then x^2=y^2, but it is not true that if x^2=y^2 then x=y. So the method of lying here is to prove that statement A implies statement B, and then assert that statement B implies statement A.
But there is an extra subtlety to this example, which is the hidden assumption about what x and y represent. If x and y are assumed to be positive numbers (for example if they are the lengths of sides of triangles, and we are trying to apply Pythagoras’s theorem), then it is true that x^2=y^2 implies x=y. But if we are doing algebra, rather than geometry, then we have to take the negative numbers into account, and it is no longer true that x^2=y^2 implies x=y. This is important for physics, because physicists tend to think in geometrical terms, rather than algebraic terms, and the meaning of negative numbers in geometry is somewhat obscure.
For example, in quantum mechanics, x and y might represent spinors, and their squares then represent vectors. It is tempting to think that if the vectors are equal, then the spinors must be equal also, but this isn’t true. It was Dirac who first exploited this loophole to predict the existence of antiparticles. I am oversimplifying a bit, because the trick here requires complex numbers rather than real numbers, and complex conjugation rather than a sign change, but this is just window-dressing on the same basic method of (not) lying. Dirac’s antiparticles do exist, and have been detected over and over again. So he was right to point out that x^2=y^2 does not imply that x=y. The possibility that x=-y is not only a mathematical possibility, it is also physically real in this instance.
Unfortunately, the mathematical window-dressing, and the geometrical point of view, conspire to create the illusion of a “symmetry” between particles and anti-particles. This symmetry is clearly not a physical symmetry – we are made of particles, not anti-particles – but it appears to be a symmetry in the mathematics. I say “appears to be”, because it is like the symmetry between -1 and 1 in the primary school example. The latter is a perfectly good symmetry, as long as you are only doing addition. Or subtraction. But when you progress to multiplication, it is no longer a symmetry, because negative times negative equals positive. For symmetry, you would need negative times negative equals negative.
To translate to a more familiar example, let’s talk about money. In double-entry book-keeping, there is a symmetry between money and anti-money, because every transaction has to be entered in the books twice, once as a sum of money moving in one direction, and also as a sum of anti-money moving in the opposite direction. But this is a mathematical symmetry only, because physical money (coins, say) is always positive. If I have an overdraft of one pound, I have minus one pound in my bank account. Particle physicists would say I have an anti-pound in my account. But if I go to the bank and try to withdraw that anti-pound, the bank will not give me an anti-pound coin. No, the bank demands a pound coin from me, so that the pound and the anti-pound annihilate, and nothing is left except a couple of photons (tiny round zeros) that keep a record of the transaction in the double-entry book of all transactions in the universe.
So when physicists ask for an “explanation” of the asymmetry between particles and antiparticles, they are asking a meaningless question. They are asking for an explanation of the difference between money and anti-money. Dirac himself was quite clear about this distinction, and his mental picture of a “Dirac sea” of negative energy “particles” describes the situation quite well, as long as it isn’t taken too literally. But the modern geometrical way of thinking has introduced a particle/antiparticle symmetry that isn’t really there. To understand why there is no such symmetry, we have to think a bit more algebraically, and a bit less geometrically.
Let’s start at the beginning, and go back to primary school. When you first start to arrange the numbers in a number line: 1, 2, 3, 4, … you inevitably start to ask, what happens if you go in the opposite direction along this line? What are those numbers? And then you learn about negative numbers, and subtraction, and zero, and so on, and you have this beautifully symmetric number line, and you start to think geometrically. First of all, why is the zero here, why couldn’t it be there? Well, of course, it could, if you are thinking geometrically. But if you are thinking algebraically, zero is defined by addition: 0 + 0 = 0. No other number has the property that x+x=x, so you can’t put the zero anywhere else but here. Second of all, why is the one on this side of the zero, and not on the other side? And how far away is it anyway? If you are thinking geometrically, there is no difference between the two sides, and there is nothing to say what the scale is. But if you are thinking algebraically, one is defined by multiplication: 1 * 1 = 1. No other number, apart from zero, has the property that x * x = x. We certainly know that one is not equal to zero, so one has to go precisely there, and nowhere else.
Now let’s come back to the problem in hand. Classical mechanics is purely additive and geometrical. Why? Because multiplication implies quantisation, since it tells you where one is on the real line. Quantum mechanics is based on this fundamental insight, because it is based on a spinor being a square root of a vector, so you have to multiply a spinor by itself to get a quantised vector. But, as I have pointed out, this is not so much an experimental fact (which it is, of course) as a mathematical necessity. Quantisation requires multiplication, and multiplication implies quantisation. In the primary school example, multiplication distinguishes -1 from +1, and the multiplicative group here has order 2 – which just means it has exactly two elements, +1 and -1. In Dirac’s example, the symmetry group again has order 2, but instead of -1 he uses the operation of complex conjugation. This makes things much more complicated, but doesn’t change the symmetry group – it still has order 2.
But if you’re doing quantisation of spinors properly, you need to quantise not only the real numbers and the complex numbers (which is partly done in the Dirac equation), you also need to quantise the quaternions. I’ve explained in several recent posts why and how you need to do this. There are only two possibilities that have the required generation symmetry of order 3, and only one of these symmetry groups has an irreducible three-dimensional representation, which seems to be a requirement of any theory, imposed by the three-quark structure of the proton. The group is the binary tetrahedral group, which quantises the (non-commutative) algebra of quaternions. This is the algebra you need if you want to quantise everything in physics at once. There is no other possibility.
How you use this algebra, is another question entirely. But you do have to use the algebraic (multiplicative) structure, or else you do not get the quantisation. Over the past few years, I have made some suggestions, which have generally been shown to be too naive in their physical interpretations. Nevertheless, the mathematics is non-negotiable, and the only questions lie at the interface between the mathematics and the physics. There are three things you need to take into account when buying a new theory of physics: algebra, algebra, and algebra. I can’t sell my theories, because people don’t like the colour of the wallpaper.
There is a school of thought in quantum gravity that all you need is non-commutative geometry, but as I have demonstrated, you do not get the quantisation unless you first quantise the algebra. As far as I can tell, they don’t do that. It goes back, I imagine, to a long-standing prejudice against algebra, exemplified by such derogatory expressions as “Gruppenpest”, and Atiyah’s aggressive anti-algebra, pro-geometry stance. There is another school of thought among physicists that they need new mathematics, because the old mathematics they know isn’t good enough to describe what they think they see physically. What they actually need is old mathematics that they don’t know, not “new” mathematics that mathematicians don’t know. There is no need for advanced mathematical “string theories”, there is only a need to understand the algebra of quantised quaternions.
There are three kinds of lies: lies, Donald Trump, and string theory.