Srednicki — Quantum Field Theory · Problem 39.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spin-statistics-theorem

Problem

The spin-statistics theorem for spin-one-half particles. We will follow the proof for spin-zero particles in section 4. We start with

b_s(\mathbf{p})

and

b_s^\dagger(\mathbf{p})

as the fundamental objects; we take them to have either commutation

(-)

or anticommutation

(+)

relations of the form

[b_s(\mathbf{p}),b_{s'}(\mathbf{p}')]_\mp = 0\,,

[b_s^\dagger(\mathbf{p}),b_{s'}^\dagger(\mathbf{p}')]_\mp = 0\,,

[b_s(\mathbf{p}),b_{s'}^\dagger(\mathbf{p}')]_\mp = (2\pi)^3 2\omega\,\delta^3(\mathbf{p}-\mathbf{p}')\delta_{ss'}\,. \tag{39.42}

Define

\Psi^+(x) \equiv \sum_{s=\pm}\int\widetilde{dp}\,b_s(\mathbf{p})u_s(\mathbf{p})e^{ipx}\,,

\Psi^-(x) \equiv \sum_{s=\pm}\int\widetilde{dp}\,b_s^\dagger(\mathbf{p})v_s(\mathbf{p})e^{-ipx}\,. \tag{39.43}

a) Show that

U(\Lambda)^{-1}\Psi^\pm(x)U(\Lambda) = D(\Lambda)\Psi^\pm(\Lambda^{-1}x)

. b) Show that

[\Psi^+(x)]^\dagger = [\Psi^-(x)]^{\mathrm{T}}\mathcal{C}\beta

. Thus a hermitian interaction term in the lagrangian must involve both

\Psi^+(x)

and

\Psi^-(x)

. c) Show that

[\Psi^+_\alpha(x),\Psi^-_\beta(y)]_\mp \neq 0

for

(x-y)^2 > 0

. d) Show that

[\Psi^+_\alpha(x),\Psi^-_\beta(y)]_\mp = -[\Psi^+_\beta(y),\Psi^-_\alpha(x)]_\mp

for

(x-y)^2 > 0

. e) Consider

\Psi(x) \equiv \Psi^+(x) + \lambda\Psi^-(x)

, where

\lambda

is an arbitrary complex number, and evaluate both

[\Psi_\alpha(x),\Psi_\beta(y)]_\mp

and

[\Psi_\alpha(x),\overline{\Psi}_\beta(y)]_\mp

for

(x-y)^2 > 0

. Show these can both vanish if and only if

|\lambda|=1

and we use anticommutators.