# A reading list to build up to the spin statistics theorem

Wikipedia's article on the spin-statistics theorem sums it up thusly:

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum (that is, the contribution to the total angular momentum which is not due to the orbital motion of the particle). All particles have either integer spin or half-integer spin (in units of the reduced Planck constant ħ).

The theorem states that:

• the wave function of a system of identical integer-spin particles has the same value when the positions of any two particles are swapped. Particles with wavefunctions symmetric under exchange are called bosons;
• the wave function of a system of identical half-integer spin particles changes sign when two particles are swapped. Particles with wavefunctions anti-symmetric under exchange are called fermions.

In other words, the spin-statistics theorem states that integer spin particles are bosons, while half-integer spin particles are fermions.

Question. What would you recommend as the best means to understanding the proof of the spin statistics theorem? To put it another way: if you took (for instance) the proof-sketch for the spin-statistics theorem given later in that same Wikipedia page, and wanted to add enough material to it to create a textbook whose entire purpose was to take someone from a third- or fourth-year undergrad level of physics up to understanding the spin-statistics theorem inside-out, what references would you use to flesh out the material of that book?

Context. I'm a researcher in quantum computation, coming from the computer science end of things: I have a very solid grasp on the basic mathematical framework of non-relativistic QM, and some special relativity, if not all applicable techniques. I have very basic familiarity with fermionic and bosonic operator algebras, as algebras generated by creation/annihilation operators satisfying certain axioms, though I have not had occasion to use them much myself.

I take for granted that understanding the proof of the spin-statistics theorem will involve learning a non-trivial amount of physics (and also likely mathematical) background.

Any recommendations?

Edited to add: If it is sufficient to learn quantum field theory, please recommend a suitable text as an answer. For instance, if you know of a good book on QFT that does not assume much of a background in particular topics such as E&M, and which definitely covers all of the concepts pertinent to the proof-sketch given on the Wikipedia page, and/or itself has a good self-contained proof of the spin-statistics theorem — in short, a book which can take me from Schrödinger and Einstein all the way to the spin-statistics theorem — then please recommend the book in question as an answer.

• You will definitely want to make your self familiar with relativity and QFT since this theorem requires that the relevant particle operators (fields, more precisely) transform as representation of the Poincaré group. In particular, there is nothing to prevent spin 1 fermions and spin 1/2 bosons in the classical QM. Also, the theorem requires dimension greater than two for group theoretic and topological reasons. In two dimensions you get parastatistics and anyons -- I guess you might be familiar with this fact since it is very relevant for (topological) quantum computing. – Marek Aug 20 '11 at 12:51
• There's an old book called PCT, Spin and Statistics, and All That by Streater and Wightman. I haven't read it myself, but long ago I heard good things about it. – Ted Bunn Aug 20 '11 at 15:47
• @Marek: I have passing familiarity with most of the content of special relativity, but know I have to move onwards to QFT (which seems to be directly necessary for the Theorem, as it is the setting for it). I am indeed interested in the special case of 3+1 dimensions, although the related concepts for lower dimensions are also interesting. – Niel de Beaudrap Aug 20 '11 at 18:05
• @Ted Bunn: thanks for the reference, I'll include this in my investigations among whatever other answers I get. – Niel de Beaudrap Aug 20 '11 at 18:05
• @Ron: Quoting {with editorializing}: "The rotation plane includes time {why do we consider such a rotation?}, and a rotation in a plane involving time in the Euclidean theory defines a CPT transformation in the Minkowski theory {why?}. If the theory is described by a path integral {as opposed to...?}, a CPT transformation takes states to their conjugates {how come?}, so that the correlation function ⟨0|Rϕ(x)ϕ(−x)|0⟩ must be positive definite at x=0 by [the assumption that the particle is a 'real excitation] {what is the correspondance between the assumption and the consequence?}..." And so on. – Niel de Beaudrap Aug 20 '11 at 22:52

There is a recent textbook by Banks which covers the spin/statistics theorem pretty good. I hope it is ok. The main difficulty is that there is no quantum field theory book that covers analytic continuation to Euclidean space, and this is the essential thing.

This is worked out by each person on their own, as far as I know. The problem is that it is very easy to say "plug in i times t everywhere you see t" and get 90% of everything right, without understanding anything. Streater and Whitman do it, that's most of their book, but they are too formal to be comprehensible. Schwinger is too long ago (and ideosyncratic). Perhaps the statistical section of Feynman and Hibbs (Path integrals), where they actually rederive the path integral in imaginary time, will allow you to extrapolate to the general bosonic fields.

The Fermionic case requires the Euclidean continuation of Majorana spinors, and this was in the literature more recently: http://arxiv.org/abs/hep-th/9608174. This stuff is covered in none of the textbooks, and unfortunately I can't recommend any of them with a good conscience.

Later Edit: If you don't want to go to Euclidean space, you should avoid anything past Feynman/Schwinger. The best path then is possibly to work through Pauli's argument:W. Pauli, The Connection Between Spin and Statistics, Phys. Rev. 58, 716- 722(1940).

• The explanation on Wikipedia looks uncomplicated and clearly written, I just don't understand any of the arguments on which it relies, due to a lack of background. That, of course, is why I ask. —— So analytic continuation is not the same as what mathematicians refer to by that name? Is the substitution 't → it' meant as a trick to account for the non-Euclidean signature of Minkowski space? Or is it the same mathematical device as the 'Wick rotation', which is used to evaluate e.g. partition functions in terms of Schrödinger-style evolution under a Hamiltonian? – Niel de Beaudrap Aug 21 '11 at 10:26
• "Perhaps the statistical section of Feynman and Hibbs [...] will allow you to extrapolate to the general bosonic fields. The Fermionic case requires the Euclidean continuation of Majorana spinors, and this was in the literature more recently" — I'm confused: hopefully I wouldn't need to work out bosonic and fermionic path integrals in order to prove that bosonic and fermionic particles are the only ones possible. Or are these descriptions for understanding how best to evaluate the path integrals for these particle-types, having discovered that they're the only ones to worry about in 3+1 D? – Niel de Beaudrap Aug 21 '11 at 10:31
• The spin-statistics theorem doesn't prove that bosonic/fermionic fields are the only ones possible (although it strongly suggests that any other type of field in three dimensions could be turned into one of these two possibilities). It says that the statistics (fermion/boson) is determined by the spin (angular momentum). – Ron Maimon Aug 21 '11 at 13:57
• @Neil: The analytic continuation is exactly the mathematicians' analytic continuation for the correlation functions of the theory. The problem is that you want to continue the path integral itself, and this is not developed by mathematicians. The key is to expand $\exp(-tH)$ all complex values of t with real part >0 in a path integral, and show that you have a theory defined on a half-complexified space time which is just Euclidean space, heurisically because Minkowski space turns Euclidian under t->it. – Ron Maimon Aug 21 '11 at 14:00
• the wikipedia quotation seems to describe a spin-symmetry theorem and not a spin-statistics theorem – propaganda Jan 22 '12 at 22:58