Srednicki — Quantum Field Theory · Problem 3.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · canonical-quantization-scalar-fields

Problem

Use the commutation relations, eq. (3.29), to show explicitly that a state of the form

|k_1 \ldots k_n\rangle \equiv a^\dagger(\mathbf{k}_1)\ldots a^\dagger(\mathbf{k}_n)|0\rangle \tag{3.33}

is an eigenstate of the hamiltonian, eq. (3.30), with eigenvalue

\omega_1 + \ldots + \omega_n

. The vacuum

|0\rangle

is annihilated by

a(\mathbf{k})

a(\mathbf{k})|0\rangle = 0

, and we take

\Omega_0 = \mathcal{E}_0

in eq. (3.30).