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PHY5667 Problem Set #2 (due Tue Sep 7)
(1)
In quantum mechanics, the operators ˆ
x
and ˆ
p
satisﬁes a commutation relation [ˆ
x,
ˆ
p
] =
i
(in the
~
= 1 units as usual). Then, in the basis spanned by the eigenstates of ˆ
x
(deﬁned as
ˆ
x

x
i
=
x

x
i
), the representation ˆ
p
=

i∂/∂x
realizes this commutator.
Now, consider a QFT for a ﬁeld operator Ψ(
~x
) with a commutation relation [Ψ(
~x
)
,
Ψ
†
(
~
y
)] =
δ
(
~x

~
y
). In the basis spanned by the eigenstates of Ψ(
~x
) (deﬁned as Ψ(
~x
)

ψ
i
=
ψ
(
~x
)

ψ
i
),
ﬁnd a representation of the operator Ψ
†
(
~x
) that realizes the commutation relation.
[Hint: Follow the analogy with quantum mechanics!]
(2)
Consider a QFT in 1+1 dimensions for a Hermitian ﬁeld Φ and its conjugate momentum
Π described by the Hamiltonian density
H
(
x
) =
v
2
2
[Π(
x
)]
2
+
1
2
[
∂
x
Φ(
x
)]
2
,
with commutation relations [Φ(
x
)
,
Π(
y
)] = i
δ
(
x

y
) and [Φ(
x
)
,
Φ(
y
)] = [Π(
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 Fall '10
 Okui
 mechanics

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