2.3.3
Other Interaction Terms in Perturbation Theory, and Scaling
Let us look at what sort of effect we would have from the interaction terms we ignored. For example,
we discussed a possible term
g
4
φ
i

φ
i
+1
a
4
→
g
4
(
∂
x
φ
)
4
(2.3.28)
in the continuum limit. Let us consider a situation where the effect of this term is extremely small,
so it can be treated in perturbation theory in
g
4
. In such a case it is just a small perturbative
correction to the Hamiltonain. Schematically, since
∂
x
∼
k
, this correction looks like
H
I
∼
g
4
X
k
i
δ
(
k
1
+
k
2
+
k
3
+
k
4
)
k
1
k
2
k
3
k
4
(
a

k
1
+
a
†
k
1
)(
a

k
2
+
a
†
k
2
)(
a

k
3
+
a
†
k
3
)(
a

k
4
+
a
†
k
4
)(2.3.29)
The integral over
x
gave a momentum conserving delta function, the
∂
x
factors became
k
i
, but most
importantly, this is
an interaction among 4 sound waves
. If we included it, they wouldn’t just pass
through each other anymore, but could scatter. Note that although it appears that it could create or
destroy four sound waves out of nothing, that actually can’t happen, due to momentum and energy
conservation.
Notice that this term is of order (momentum)
4
. Another example of such a term would have
been
(
∂
2
x
φ
)
2
∼
k
4
φ
2
k
(2.3.30)
Since we are studying the long wavelength limit,
λ
→ ∞
, we want
k
→
0 (or more precisely
ka
→
0),
and so higher powers of the momenta are smaller. So by ignoring such terms we are not really
leaving anything out –
we are just making a longdistance approximation
.
16
2.4
Special Relativity and Antiparticles
For our purposes, special relativity is tantamount to the statement that space and time have certain
symmetries.
2.4.1
Translations
The most obvious part of those symmetries is translation invariance – the fact that the laws of
physics are the same everywhere, and at all times. Let us consider what this means in an extremely
simple context, namely just the space of functions of various spatial variables
x
i
labeling a bunch of
points scattered in space.
If a function
f
(
x
1
, x
2
,
· · ·
, x
n
) is translation invariant, then
f
=
f
(
x
i

x
j
)
(2.4.1)
Translation invariance is the reason that momentum space (and the Fourier Transform) is a useful
idea. We have that
f
(
x
+
y
) =
f
(
x
) +
y∂f
(
x
) +
· · ·
=
e
y∂
x
f
(
x
)
(2.4.2)
so we translate a function using spatial derivatives. But in Fourier space
f
(
x
i
) =
Z
∞
∞
n
Y
i
=1
dp
i
2
π
!
˜
f
(
p
i
)
e

ip
i
x
i
(2.4.3)
which means that
∂
∂x
i
f
(
x
i
) =
ip
i
˜
f
(
p
i
)
(2.4.4)
so momentum space
linearizes the action of translations
. Instead of having to compute a (spatial)
derivative, in momentum space we can differentiate by simply multiplying by
ip
i
. This exponentiates,
so moving
f
(
x
)
→
f
(
x
+
y
) just requires multiplication by the phase
e
ipy
.
Translation invariance implies that
f
(
x
i

x
j
) =
Z
n
Y
i
=1
dp
i
2
π
!
˜
f
(
p
i
)
e

ip
i
x
i
δ
(
p
1
+
p
2
+
· · ·
+
p
n
)
(2.4.5)
In other words, translation invariance implies momentum conservation, and vice versa. Momentum
and energy are conserved because the laws of physics are invariant under translations in space and
time, respectively. We will explain this again in a more sophisticated way very soon. In general, the
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 Fall '12
 Halperin
 Physics, mechanics, Quantum Field Theory