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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) = e-iHt denotes the time evolution operator, and U(x, t; x ) =
x|U(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 ) x|U(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)
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