Srednicki — Quantum Field Theory · Problem 21.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · symmetries-quantum-action

Problem

Symmetries of the quantum action. Suppose that we have a set of fields

\varphi_a(x)

, and that both the classical action

S(\varphi)

and the integration measure

\mathcal{D}\varphi

are invariant under

\varphi_a(x) \to \int d^d y\, R_{ab}(x,y)\varphi_b(y) \tag{21.22}

for some particular function

R_{ab}(x,y)

. Typically

R_{ab}(x,y)

is a constant matrix times

\delta^d(x-y)

, or a finite number of derivatives of

\delta^d(x-y)

; see sections 22, 23, and 24 for some examples. a) Show that

W(J)

is invariant under

J_a(x) \to \int d^d y\, J_b(y) R_{ba}(y,x) . \tag{21.23}

b) Use eqs. (21.20) and (21.23) to show that the quantum action

\Gamma(\varphi)

is invariant under eq. (21.22). This is an important result that we will use frequently.