Srednicki — Quantum Field Theory · Problem 32.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spontaneous-breaking-continuous-symmetries

Problem

Consider the Noether current

j^\mu

for the U(1) symmetry of eq. (32.1), and the corresponding charge

Q

. a) Show that

e^{-i\alpha Q} \varphi e^{+i\alpha Q} = e^{+i\alpha}\varphi

. b) Use eq. (32.5) to show that

e^{-i\alpha Q}|\theta\rangle = |\theta + \alpha\rangle

. c) Show that

Q|0\rangle \neq 0

; that is, the charge does not annihilate the

\theta = 0

vacuum. Contrast this with the case of an unbroken symmetry.