Spin-Statistics Theorem

Prerequisites: Spin, Canonical Quantum Field Theory. Full path (22 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

Intuitive

All particles come in two kinds. Bosons, like photons, are happy to crowd into the same state; this is what makes a laser work. Fermions, like electrons, refuse: no two identical fermions can occupy the same state (the Pauli exclusion principle). This is why atoms have shells, why there is a periodic table, and why matter does not collapse.

Which kind a particle is turns out to be fixed by its spin. Particles with whole-number spin (0, 1, 2 and so on, in units of Planck's constant) are always bosons, and particles with half-integer spin (one half, three halves and so on) are always fermions. In nonrelativistic quantum mechanics this rule is simply put in by hand, and nothing would go wrong mathematically if electrons were bosons.

In relativistic quantum field theory the rule follows from basic requirements. If we try to describe spin-1/2 particles as bosons, the energy is not bounded from below. If we try to describe spin-0 particles as fermions, measurements at points that cannot communicate with each other (spacelike separated points) start to influence each other. Only the combination observed in nature is consistent with causality, positive energy and Lorentz invariance together.

There is also a picture that captures part of the reason. Swapping two identical particles can be done by moving them around each other, and this motion is related to rotating one of them by a full turn. A particle with half-integer spin picks up a minus sign under a full rotation, and the same minus sign appears when two of them are exchanged. This picture is a heuristic, not a proof, but it shows why spin and exchange are connected at all.

Concrete

• Spin, Statistics, CPT and All That Jazz
• The spin-statistics connection: Some pedagogical remarks in response to Neuenschwander’s question by A. S. Wightman
• A nice book on the topic is Pauli And The Spin-Statistics Theorem by Ian Duck, E.C.G. Sudarshan

Abstract

In the Wightman framework, consider a field $\phi$ transforming under the irreducible representation $(j,k)$ of $SL(2,\mathbb{C})$, together with its adjoint $\phi^*$, in a theory with positive-energy representation of the Poincaré group, a unique invariant vacuum $|0\rangle$ and a positive-definite Hilbert space. A field of type $(j,k)$ has integer spin if $j+k$ is an integer and half-integer spin otherwise. The theorem (Pauli for free fields, Burgoyne and Lüders-Zumino in general) states: if $\phi$ has integer spin and anticommutes with $\phi^*$ at spacelike separation, or if $\phi$ has half-integer spin and commutes with $\phi^*$ at spacelike separation, then $\phi = 0$.

Idea of the proof. By the spectrum condition, the two-point functions $\langle 0|\phi(x)\phi^*(y)|0\rangle$ and $\langle 0|\phi^*(y)\phi(x)|0\rangle$ are boundary values of functions holomorphic in the forward and backward tubes. Lorentz covariance extends them to the extended tube, with invariance under the complexified Lorentz group, which in four dimensions contains the total inversion $x \mapsto -x$. Applied to the product of $\phi$ (type $(j,k)$) and $\phi^*$ (type $(k,j)$), this inversion produces the factor $(-1)^{2(j+k)}$. At the Jost points (real, spacelike $x-y$) this relates one ordering to the other with this sign. Combined with the assumed wrong (anti)commutation relation, the identity theorem gives, after smearing with a test function $f$, $$\| \phi(f)^*|0\rangle \|^2 + \| \phi(f)|0\rangle \|^2 = 0 .$$ Positivity of the Hilbert space norm then forces $\phi(f)|0\rangle = 0$, and a local field that annihilates the vacuum vanishes. The same structure that gives the CPT theorem is used, and the two theorems are usually proved together.

Structure. The theorem identifies two a priori independent $\mathbb{Z}_2$ gradings: the rotation by $2\pi$, $U(R(2\pi))$, which is $(-1)^{2s}$ on states of spin $s$, and the fermion parity $(-1)^F$ defined by statistics. On every state, $U(R(2\pi)) = (-1)^F$. The assumptions are necessary: in theories with indefinite metric (such as Faddeev-Popov ghosts in covariant gauge quantisation, which are anticommuting scalars) or without Lorentz invariance, the connection can fail.

The possible statistics themselves are constrained by topology. In three or more spatial dimensions the fundamental group of the configuration space of $N$ identical point particles is the permutation group $S_N$, and its one-dimensional unitary representations give bosons and fermions (higher-dimensional ones give parastatistics, which can be rewritten in terms of ordinary fields with additional internal symmetry). In two spatial dimensions the fundamental group is the braid group, which allows anyons with arbitrary exchange phases; a spin-statistics relation $e^{i\theta} = e^{2\pi i s}$ also holds there in relativistic theories. In algebraic quantum field theory, the Doplicher-Haag-Roberts analysis of superselection sectors derives the statistics of a sector from locality alone, and the spin-statistics connection has been proven for sectors with finite statistics.

• The standard reference is the book PCT, Spin and Statistics, and All That by Raymond F. Streater,

Why is it interesting?

The spin-statistics theorem tells us that there can be never two identical fermions in the same state (they anticommute), while there can be arbitrarily many identical bosons.

FAQ

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Angular Momentum
12. Configuration Space
13. Euler-Lagrange Equations
14. Special Relativity
15. Lagrangian Formalism
16. Phase Space
17. Classical Field Theory
18. Hamiltonian Formalism
19. Schrödinger Equation
20. Canonical Quantum Mechanics
21. Canonical Quantum Field Theory
22. Spin