Homework Set No. 1, Physics 880.02
Deadline – Thursday, January 22, 2009
1.
Consider a free (real) scalar theory with the Lagrangian density
L
=
1
2
∂
μ
ϕ∂
μ
ϕ

m
2
2
ϕ
2
.
Define the Hamiltonian by
H
(
t
) =
Z
d
3
x
[
π
(
~x, t
) ˙
ϕ
(
~x, t
)
 L
]
.
a.
(3 pts) Show that for classical field configurations
d
dt
H
(
t
) = 0
.
b.
(2 pts) Write
H
(
t
) in terms of
π
and
ϕ
with no ˙
ϕ
’s appearing.
c.
(5 pts) Now imagine that the field is quantized. Use canonical quantization commu
tators
[
ϕ
(
~x, t
)
, π
(
~x
0
, t
)] =
iδ
(
~x

~x
0
)
(with all other commutators being zero) to show that
H
(
t
) (now an operator) generates time
translations, i.e., show that
i ∂
0
ϕ
= [
ϕ, H
(
t
)]
i ∂
0
π
= [
π, H
(
t
)]
.
2.
The same as in problem 1, but now for Dirac field: start with a theory with Lagrangian
density
L
=
¯
ψ
(
iγ
μ
∂
μ

m
)
ψ.
a.
(3 pts) Construct a Hamiltonian
H
(
t
) and show that for classical field configurations
d
dt
H
(
t
) = 0
.
b.
(2 pts) Write
H
(
t
) in terms of
π
and
ψ
.
c.
(5 pts) For quantized field
ψ
use the anticommutation relations
n
ψ
α
(
~x, t
)
, ψ
†
β
(
~x
0
, t
)
o
=
δ
α β
δ
(
~x

~x
0
)
to show that
i ∂
0
ψ
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 Winter '09
 Kovchegov
 Physics, Work, Quantum Field Theory, pts, classical ﬁeld conﬁgurations

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