 
Summary: UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Prof. S. B. Giddings Fall 2010
TA: Erik Perkins
Physics 221A
Relativistic Quantum Field Theory
ASSIGNMENT #2 SOLUTIONS
Problem 1
In the following, U(t) = eiHt denotes the time evolution operator, and U(x, t; x ) =
xU(t)x is the propagator. First let's note that since ^x(t) is a Heisenberg operator,
so must be (^x(t)  x). Then x, t x, t must also be a Heisenberg operator. This precludes
interpreting x, t x, t as (U(t)x ) xU(t) , since this has the opposite time evolution
of a Heisenberg operator. Instead, x, t x, t = U(t) (x x) U(t), the Heisenberg picture
version of x x.
(a)
Let's act on an arbitrary state  = dy (y)y with each operator. First we have
(^x(t)  x) =
d
2
dy (y)ei(^x(t)x)
