Srednicki — Quantum Field Theory · Problem 21.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · quantum-action-background-field

Problem

Consider performing the path integral in the presence of a background field

\bar{\varphi}(x)

; we define

\exp[iW(J;\bar{\varphi})] \equiv \int \mathcal{D}\varphi\, \exp\left[iS(\varphi+\bar{\varphi}) + i\int d^d x\, J\varphi\right] . \tag{21.24}

Then

W(J;0)

is the original

W(J)

of eq. (21.3). We also define the quantum action in the presence of the background field,

\Gamma(\varphi;\bar{\varphi}) \equiv W(J_\varphi;\bar{\varphi}) - \int d^d x\, J_\varphi \varphi , \tag{21.25}

where

J_\varphi(x)

is the solution of

\frac{\delta}{\delta J(x)} W(J;\bar{\varphi}) = \varphi(x) \tag{21.26}

for a specified

\varphi(x)

. Show that

\Gamma(\varphi;\bar{\varphi}) = \Gamma(\varphi+\bar{\varphi};0) , \tag{21.27}

where

\Gamma(\varphi;0)

is the original quantum action of eq. (21.1).