Srednicki — Quantum Field Theory · Problem 3.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · canonical-quantization-scalar-fields

Problem

Recall that

T(a)^{-1}\varphi(x)T(a) = \varphi(x-a)

, where

T(a) \equiv \exp(-iP^\mu a_\mu)

is the spacetime translation operator, and

P^0

is identified as the hamiltonian

H

. a) Let

a^\mu

be infinitesimal, and derive an expression for

[\varphi(x), P^\mu]

. b) Show that the time component of your result is equivalent to the Heisenberg equation of motion

i\dot{\varphi} = [\varphi, H]

. c) For a free field, use the Heisenberg equation to derive the Klein-Gordon equation. d) Define a spatial momentum operator

\mathbf{P} \equiv -\int d^3x\,\Pi(x)\nabla\varphi(x)\,. \tag{3.36}

Use the canonical commutation relations to show that

\mathbf{P}

obeys the relation you derived in part (a). e) Express

\mathbf{P}

in terms of

a(\mathbf{k})

and

a^\dagger(\mathbf{k})