The previous post has convinced me there is a need for a more penetrating analysis of the uses of left-handedness and right-handedness in physics. If you ask a particle physicist, they will tell you that there are two distinct sorts of handedness, which they call chirality and helicity. But if you ask them to explain the difference, they will be unable to do so. It is therefore necessary for you to explain it to them. Here is how you do it.
First take a circle. You can go round it either clockwise or anti-clockwise. This distinction is called helicity, because it distinguishes between the helix of a right-handed screw (that goes into the wall when you turn it clockwise) from that of a left-handed screw (that is only ever used in a few specialist applications, such as violin chin-rest attachments and a certain type of corkscrew). If you look at a right-handed screw in a mirror, you always see a left-handed screw. If you look at a clock in the mirror, the hands go anti-clockwise. (Well, that’s true for normal clocks, but there are a few anti-clocks in the world as well – I used to have one myself, until the burden of contrariness became too much for it, and the hands started to object to being pushed the wrong way round the face.)
The equation of a circle is x^2 + y^2 = 1. Now change one sign, to get x^2 – y^2 = 1. This is a hyperbola, and it comes in two pieces, disconnected from each other. This distinction is called chirality. If you put a mirror along the y axis, it reflects the right-handed part of the hyperbola into the left-handed part. If you put a mirror along the x axis, it does not. To be sure of changing the chirality, you must use two mirrors at right angles to each other.
When we move up to higher-dimensional spaces, such as 4-dimensional spacetime, the details are more complicated, but the basic distinction is still the same. If you use an odd number of mirrors, it is a helicity, and if you use an even number it is a chirality. This is an absolutely clear-cut mathematical distinction about which there can be no argument whatsoever. Or can there? Physicists will argue the hind leg off a donkey (I know, I’ve seen them do it) and will argue black is white (I know, I’ve seen them do it). And since you can appear to change the chirality with only one mirror, if you put it in the right place, they will confidently assert that the mathematical distinction is not relevant to physics. But it is. Do anti-clocks tell negative time? No. Do anti-particles have negative mass? No. These facts are important.
Physicists always use complex numbers, so they can multiply by a square root of -1 any time they feel like it. Hence they can convert a circle into a hyperbola, and hence convert a helicity into a chirality. Never mind that that is cheating, or that it destroys the very thing they are trying to describe. They do it anyway. So that when Distler and Garibaldi try to define a chirality by complexifying the Lorentz group, it is not at all clear that they are not actually defining a helicity. Check your mirrors: how many have they used? One. They will argue till they are blue in the face that they have defined a chirality (I know, I have seen one of them do it), but no amount of argument will convince me that one is an even number.
Now let’s describe the handedness of the real universe we actually live in, instead of some fictitious universe described by some invented physical theory. First look at spacetime, with a unit distance defined by t^2 – x^2 – y^2 – z^2 = 1. This equation defines a 4-dimensional hyperboloid in two pieces. It therefore has a chirality. You need an even number of mirrors, that physicists called T (time reversal) and P (parity). The latter is either one mirror in space, that negates one coordinate, or three mirrors, that negate all three coordinates, but it really doesn’t matter. It only matters that T and P individually are odd numbers of mirrors, so describe helicities, but the two together have an even number, so describe a chirality. The TP-chirality is the obvious fact that you can’t go backwards in time, and you can’t convert a left-handed screw into a right-handed screw.
The existence of anti-particles with opposite charge to normal particles introduces a third mirror, that physicists call C (charge conjugation). Or does it introduce two new mirrors? Is charge conjugation a question of helicity, like the anti-clocks, or a question of chirality? I believe it is a chirality, that is a distinction between particles on one branch of the hyperbola, and anti-particles on the other. It is surely too fundamental a distinction to be a helicity. If so, then the basic CPT symmetry of quantum physics is also a chirality. It is this fundamental chirality of physics that models must be able to reproduce, and explain. (It is not enough just to implement the PT chirality, that is chirality of spacetime.)
The reason why all theories of quantum mechanics from Dirac (1928) onwards fail to do this is because by complexifying the Dirac algebra they have converted the fundamental chirality of the Lorentz group SO(3,1) into the helicity of SO(4), and they have converted the fundamental chirality of the hyperbola into the helicity of a circle. In a sense, therefore, Distler and Garibaldi are not to blame – they are only copying what everyone else has done for nearly a century – but still, they should know better. If I were to characterise this problem in one sentence, I would say that one left hand (chirality) doesn’t know what the other left hand (helicity) is doing.
But wait a minute – is the theory actually correct? Is the so-called `chirality’ of the weak interaction in fact a `helicity’ as the standard model actually implements it? Is it really, experimentally, a property of an odd number of mirrors, or an even number? Think about it, analyse the Wu experiment of 1957 for yourself – what does it say? It links the chirality of spin to the parity mirror. It is an odd number of mirrors. It is a violation of P-symmetry (one mirror or three). It is a helicity, not a chirality. Mathematically, this is also obvious, because the distinction between SU(2)_L and SU(2)_R in SO(4) is quaternion conjugation, which requires three mirrors. And because it is a helicity, not a chirality, you can understand it with a bunch of screws. You need three screws – they are called the Sun, the Earth and the Moon. Screw number one: which way does the Earth go round the Sun? Screw number two: which way does the Earth rotate on its axis? Screw number three: which way does the Moon go round the Earth? There is your helicity of the weak interaction.