Srednicki — Quantum Field Theory · Problem 6.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · path-integrals-quantum-mechanics

Problem

a) Find an explicit formula for

\mathcal{D}q

in eq. (6.9). Your formula should be of the form

\mathcal{D}q = C\prod_{j=1}^N dq_j

, where

C

is a constant that you should compute. b) For the case of a free particle,

V(Q) = 0

, evaluate the path integral of eq. (6.9) explicitly. Hint: integrate over

q_1

, then

q_2

, etc, and look for a pattern. Express your final answer in terms of

q'

t'

q''

t''

, and

m

. Restore

\hbar

by dimensional analysis. c) Compute

\langle q'',t''|q',t'\rangle = \langle q''|e^{-iH(t''-t')}|q'\rangle

by inserting a complete set of momentum eigenstates, and performing the integral over the momentum. Compare with your result in part (b).