Path integrals for dummies

A commenter has urged me to adopt the path integral interpretation of quantum mechanics, and in order to explain why I am not going to do that, I need to explain path integrals in a way that I could understand when I was eight. Let’s start with a grandfather clock. When I was eight, I was fascinated by my grandmother’s grandfather clock, so much so that she left it to me when she died. The slow tick-tock was mesmerising, and if I was allowed to see inside, the swing of the pendulum which exactly coincided with the ticks on swinging one way and the tocks on the other.

I can’t remember which were the ticks, and which were the tocks, because the clock hasn’t worked for 25 years. But if you shut the clock case, and just listen to the tick-tock-tick-tock then you know what’s happening schematically: it’s going round and round an endless cycle of ticks and tocks. So you can draw a little circle on a piece of paper, and label it with a tick on one side and a tock on the opposite side. Now this piece of paper is called “phase space” by physicists, and in quantum mechanics the circle is really really small, and all you know is its area, which is called Planck’s constant, written h. You can’t see the circle, because it’s hidden, all you can hear is the tick-tock. So you have no idea how it gets from the tick to the tock in phase space, all you know is that it does.

The path integral is the way that Feynman gets around this: he considers all possible ways of getting from tick to tock, works out how likely they all are, and takes the average. That is the toy version of the path integral, suitable for an 8-year-old. Now to explain why he does this, I have to delve into the history of quantum mechanics, and the first quantum theory that Planck devised his constant for, in the emission of light from hot things that glow (like light bulbs). Planck heard the ticking, but he only heard tick-tick-tick-tick, not tick-tock-tick-tock, so he knew something was going round in a circle but only put one point on the circle for the ticks. Fermi listened more carefully to the ticking, by looking at the electrons that are the source of the light, and he heard tick-tock-tick-tock. So what to do?

Tick and tock had already been put in the same place on the circle, so he had to go round the circle twice to explain the tick-tock. Well, a child can see that that was a silly thing to do. He should have just put the tick on one side and the tock on the other. But no, he took his circle of paper, gave it a twist and stuck the ends back together after half a twist, so that instead of a circle has had a Moebius band. Well, now, an 8-year-old can have endless (pun not intended) fun with Moebius bands, so let’s leave them to play. “Play” is, of course, another word for “learn”, so we don’t want to disturb them.

Let’s think a bit more about the difference between tick and tock. At the age of 5, you probably think that birds go “tweet! tweet!”, but by the age of 8 you should be more discerning. There are birds that actually do go “tweet! tweet!”, and I hear them sometimes in my garden. But there’s another bird that sings in my garden, which the textbook reliably informs me is a great tit, and the textbook says that great tits sing “Teacher! Teacher!” all day long. Well, this bird obviously hasn’t read the textbook, because he sings “cher-tea! cher-tea!” instead. According to physicists, that means he is an anti-bird, travelling backwards in time. But I am fairly sure that nature has a simpler explanation. He just starts singing at a different point on the circle. He still goes round the circle in the same sense as all the other birds. The standard convention is to use the anti-clockwise direction, but it really doesn’t matter.

The problem of quantisation is to understand how to square the circle. All points on the circle look the same, but the circle has corners labelled tick and tock. Why? How? Well, perhaps it was actually a square? I prefer to think of it as a hexagon, for various reasons that I may try and explain. Basically, this toy model only deals with one-dimensional space, and in three dimensions we really need cubes, which squash flat nicely into little hexagons. This enables you to cut the hexagons up into three squares, stretched sideways, and think of dividing up a proton into three quarks, and other such tricks. It also enables you to think of three generations of electron at once, all going tick-tock with opposite corners of the hexagon. So the whole thing really goes “tickety-tockety-tickety-tockety” like trains used to, in the old days when the rails were still quantised. You remember the nursery rhyme? Hickory, dickory, dock, the mouse ran up the clock? It should really be “Tockety, tickety, tock” of course, but these things get corrupted when people forget the original meaning.

Now for the reason why I am not really interested in the path integral. First, it’s too hard. There must be a simpler way. Second, it puts too much emphasis on the circumference of the circle, instead of its area. The spinor approach confuses the area pi.r^2 with the circumference 2.pi.r, and divides h by 2pi to get what it thinks is the fundamental parameter. Actually, if you want to do it properly with a hexagon, you should quantise pi, and put it equal to 3. The area of the hexagon is 3 r^2, and that factor of 3 is the most important thing you will ever learn about quantum mechanics. The area is the fundamental unit of charge, and if you chop your proton-hexagon up into three pieces, you’ve got to chop the charge up into pieces as well. If you think these pieces keep the same charge as the proton goes “tockety-tickety-tock”, then you have to have two pieces with charge 2/3 and one with charge -1/3. But this is a misunderstanding of what really happens to charge in phase space.

But the reason why I’m not interested in path integrals? The electron does not travel from tick to tock, there is no path. Why should I need something that doesn’t exist in order to calculate something that does? The electron has a tick and a tock. It is meaningless to ask how it gets from tick to tock. All that matters is the symmetry group of order 2 that swaps the tick with the tock. And when I study the three generations of electron, I use the rotation group of the hexagon, a cyclic group of order 6, which contains all the physical information that there is. Nothing else matters. Therefore I can calculate with my hexagons everything that Feynman calculates with his path integrals, and more.

When I set my electrons and protons going tockety-tickety-tock, the electrons click through the three generations, and the protons click through the three quarks, and the path integral calculated with respect to the gravitational field gives me the sum of the masses of all three electrons and three protons. Now I look at this thing from a long way away, so that I can’t feel any electromagnetic effects, only the gravitational effects, and what do I see? It is the same as five neutrons. But I can’t explain that with a pendulum in a grandfather clock. For that I need a gyroscope, not a pendulum, because a gyroscope has three circles that enable you to study phase space in three directions at once, and thereby understand everything about quantum physics. A gyroscope is not a toy.