Srednicki — Quantum Field Theory · Problem 8.8

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · path-integral-free-field-theory

Problem

A harmonic oscillator (in units with

m = \hbar = 1

) has a ground-state wave function

\langle q|0\rangle \propto e^{-\omega q^2/2}

. Now consider a real scalar field

\varphi(x)

, and define a field eigenstate

|A\rangle

that obeys

\varphi(\mathbf{x},0)|A\rangle = A(\mathbf{x})|A\rangle\,, \tag{8.18}

where the function

A(\mathbf{x})

is everywhere real. For a free-field theory specified by the hamiltonian of eq. (8.1), show that the ground-state wave functional is

\langle A|0\rangle \propto \exp\left[-\frac{1}{2}\int\frac{d^3k}{(2\pi)^3}\,\omega(\mathbf{k})\tilde{A}(\mathbf{k})\tilde{A}(-\mathbf{k})\right]\,, \tag{8.19}

where

\tilde{A}(\mathbf{k}) \equiv \int d^3x\,e^{-i\mathbf{k}\cdot\mathbf{x}}A(\mathbf{x})

and

\omega(\mathbf{k}) \equiv (\mathbf{k}^2+m^2)^{1/2}