2QFT in one dimension (i.e. QM)

2.3 Effective quantum field theory
We now see what happens when we try to obtain effective field theories in 1
dimension. Suppose we have two real-valued fields
x, y
:
S
1
R
. We pick the
circle as our universe so that we won’t have to worry about boundary conditions.
We pick the action
S[x, y] =
Z
S
1
1
2
˙x
2
+
1
2
˙y
2
+
1
2
m
2
x
2
+
1
2
M
2
y
2
+
λ
4
x
2
y
2
dt.
As in the zero-dimensional case, we have Feynman rules
1/(k
2
+ m
2
) 1/(k
2
+ M
2
)
λ
As in the case of zero-dimensional QFT, if we are only interested in the correla-
tions involving
x
(
t
), then we can integrate out the field
y
(
t
) first. The effective
potential can be written as
Z
Dy exp
1
2
Z
S
1
y
d
2
dt
2
+ M
2
+
λx
2
2
y dt
,
where we integrated by parts to turn ˙y
2
to y¨y.
We start doing dubious things. Recall that we previously found that for a
bilinear operator M : R
n
× R
n
R, we have
Z
R
n
d
n
x exp
1
2
M(x, x)
=
(2π)
n/2
det M
.
Now, we can view our previous integral just as a Gaussian integral over the
operator
(y, ˜y) 7→
Z
S
1
y
d
2
dt
2
+ M
2
+
λx
2
2
˜y dt ()
on the vector space of fields. Thus, (ignoring the factors of (2
π
)
n/2
) we can
formally write the integral as
det
d
2
dt
2
+ M
2
+
λx
2
2
1/2
.
S
eff
[x] thus looks like
S
eff
[x] =
Z
S
1
1
2
( ˙x
2
+ m
2
x
2
) dt +
1
2
log det
d
2
dt
2
+ M
2
+
λx
2
2
We now continue with our formal manipulations. Note that
log det
=
tr log
,
since
det
is the product of eigenvalues and
tr
is the sum of them. Then if we
factor our operators as
d
2
dt
2
+ M
2
+
λx
2
2
=
d
2
dt
2
+ M
2
1 λ
d
2
dt
2
+ M
2
1
x
2
2
!
,
then we can write the last term in the effective potential as
1
2
tr log
d
2
dt
2
+ M
2
+
1
2
tr log
1 λ
d
2
dt
2
M
2
1
x
2
2
!
The first term is field independent, so we might as well drop it. We now look
carefully at the second term. The next dodgy step to take is to realize we know
what the inverse of the differential operator
d
2
dt
2
M
2
is. It is just the Green’s function! More precisely, it is the convolution with the
Green’s function. In other words, it is given by the function G(t, t
0
) such that
d
2
dt
2
M
2
G(t, t
0
) = δ(t t
0
).
Equivalently, this is the propagator of the
y
field. If we actually try to solve this,
we find that we have
G(t, t
0
) =
1
2M
X
nZ
exp
M
t t
0
+
k
T
.
We don’t actually need this formula. The part that will be important is that it
is
1
M
.
We now try to evaluate the effective potential. When we expand
log
1 λG(t, t
0
)
x
2
2
,
the first term in the expansion is
λG(t, t
0
)
x
2
2
.
What does it mean to take the trace of this? We pick a basis for the space we
are working on, say {δ(t t
0
) : t
0
S
1
}. Then the trace is given by
Z
t
0
S
1
dt
0
Z
tS
1
dt δ(t t
0
)
Z
t
0
S
1
dt
0
(λ)G(t, t
0
)
x
2
(t
0
)
2
δ(t
0
t
0
)
.
We can dissect this slowly. The rightmost integral is nothing but the definition
of how
G
acts by convolution. Then the next
t
integral is the definition of how
bilinear forms act, as in (
). Finally, the integral over
t
0
is summing over all
basis vectors, which is what gives us the trace. This simplifies rather significantly
to
λ
2
Z
tS
1
G(t, t)x
2
(t) dt.
In general, we find that we obtain
tr log
1 λG(t, t
0
)
x
2
2
=
λ
2
Z
S
1
G(t, t)x
2
(t) dt
λ
2
8
Z
S
1
×S
1
dt dt
0
G(t
0
, t)x
2
(t)G(t, t
0
)x
2
(t
0
) ···
These terms in the effective field theory are non-local ! It involves integrating
over many different points in
S
1
. In fact, we should have expected this non-
locality from the corresponding Feynman diagrams. The first term corresponds
to
x(t)
x(t)
Here G(t, t) corresponds to the y field propagator, and the
λ
2
comes from the
vertex.
The second diagram we have looks like this:
x(t)
x(t)
x(t
0
)
x(t
0
)
We see that the first diagram is local, as there is just one vertex at time
t
. But
in the second diagram, we use the propagators to allow the
x
at time
t
to talk
to x at time t
0
. This is non-local!
Non-locality is generic. Whenever we integrate out our fields, we get non-local
terms. But non-locality is terrible in physics. It means that the equations of
motion we get, even in the classical limit, are going to be integral differential
equations, not just normal differential equations. For a particle to figure out
what it should do here, it needs to know what is happening in the far side of the
universe!
To make progress, we note that if
M
is very large, then we would expect
G
(
t, t
0
) could be highly suppressed for
t 6
=
t
0
. So we can try to expand around
t = t
0
. Recall that the second term is given by
Z
dt dt
0
G(t, t
0
)
2
x
2
(t)x
2
(t
0
)
We can write out x
0
(t
2
) as
x
0
(t
2
) = x
2
(t) + 2x(t) ˙x(t)(t
0
t) +
˙x
2
(t) +
1
2
x(t) ˙x(t)
(t t
0
)
2
+ ··· .
Using the fact that
G
(
t, t
0
) depends on
t
0
only through
M
(
t
0
t
), by dimensional
analysis, we get an expansion that looks like
1
M
2
Z
dt
α
M
x
4
(t) +
β
M
3
x
2
˙x
2
+
1
2
x
2
¨x
+
γ
M
5
(4-derivative terms) + ···
Here α, β, γ are dimensionless quantities.
Thus, we know that every extra derivative is accompanied by a further power
of
1
M
. Thus, provided
x
(
t
) is slowly varying on scales of order
1
M
, we may hope
to truncate the series.
Thus, at energies
E M
, our theory looks approximately local. So as long
as we only use our low-energy approximation to answer low-energy questions, we
are fine. However, if we try to take our low-energy theory and try to extrapolate
it to higher and higher energies, up to
E M
, it is going to be nonsense. In
particular, it becomes non-unitary, and probability is not preserved.
This makes sense. By truncating the series at the first term, we are ignoring
all the higher interactions governed by the
y
fields. By ignoring them, we are
ignoring some events that have non-zero probability of happening, and thus we
would expect probability not to be conserved.
There are two famous examples of this. The first is weak interactions. At
very low energies, weak interactions are responsible for
β
-decay. The effective
action contains a quartic interaction
Z
d
4
x
¯
ψ
e
e
p G
weak
.
This coupling constant
G
weak
has mass dimensional
1. At low energies, this is
a perfectly well description of beta decay. However, this is suspicious. The fact
that we have a coupling constant with negative mass dimension suggests this
came from integrating some fields out.
At high energies, we find that this 4-Fermi theory becomes non-unitary, and
G
weak
is revealed as an approximation to a
W