# How to describe spin and statistics

I’m in the midst of writing a couple of sections on how spin really works. Not coincidentally, I’m also discussing these topics in my Quantum Field Theory class, where we get to do all of the math.

In the process, I’ve found myself at a bit of an intuitive impasse. I’m trying to relate the following three properties about Fermions in a conceptual way:

1. They have an intrinsic angular momentum (spin) of $hbar/2$.
2. If you turn the coordinates around one full turn, the wave-function gets a factor of -1.
3. Exchanging particles in identical states multiplies your wave-function by -1, ultimately giving rise to the Pauli exclusion principle.

The first can almost be seen as a fact of the universe. It doesn’t have to be the case that spin 1/2 particles exist, but they happen to. Given that, I think I have a pretty good way of describing why it is that you need to turn them around twice to get the same wave-function out that you started with.

However, I’m at a loss to find an intuitive way to relate the fact that fermions (electrons, quarks, neutrinos and the like) obey the Pauli exclusion principle, while bosons (photons and so forth) like being in the same state. I am not hopeful. Here’s what Feynman has to say:

Why is it that particles with half-integral spin are Fermi particles whose amplitudes add with the minus sign, whereas particles with integral spin are Bose particles whose amplitudes add with the positive sign? We apologize for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way of reproducing his arguments on an elementary level. It appears to be one of few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved. For the moment, you will just have to take it as one of the rules of the world.

If Feynman can’t think of a simplifying explanation, I don’t know who can. On the other hand, it’s been 50 years since the Feynman lectures, so maybe some clever person has come up with a useful way of thinking about it in the interim. If you’ve heard a great intuitive explanation, by all means, point me to it.

-Dave

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### 2 Responses to How to describe spin and statistics

1. Guy says:

I’ve recently came across this:
http://arxiv.org/pdf/hep-ph/9405255v1.pdf
Although it is far from simple 😛
As a grad student I can understand it all right after going through 4 quantum mechanics courses two of which discussing QFT… but if I were an under-grad I would not have a clue about what the hell he’s talking about 😛

2. Gnomic says:

Help me visualize a spin.