1QFT in zero dimensions 1 QFT in zero dimensions
We start from the simplest case we can think of, namely quantum field theory in
zero dimensions. This might seem like an absurd thing to study the universe
is far from being zero-dimensional. However, it turns out this is the case where
we can make sense of the theory mathematically. Thus, it is important to study
0-dimensional field theories and understand what is going on.
There are two reasons for this. As mentioned, we cannot actually define the
path integral in higher dimensions. Thus, if we were to do this “properly”, we will
have to define it as the limit of something that is effectively a zero-dimensional
quantum field theory. The other reason is that in this course, we are not going
to study higher-dimensional path integrals “rigorously”. What we are going
to do is that we will study zero-dimensional field theories rigorously, and then
assume that analogous results hold for higher-dimensional field theories.
One drawback of this approach is that what we do in this section will have
little physical content or motivation. We will just assume that the partition
function is something we are interested in, without actually relating it to any
physical processes. In the next chapter, on one-dimensional field theories, we
are going see why this is an interesting thing to study.
Let’s begin. In
d
= 0, if our universe
M
is connected, then the only choice of
M
is
{pt}
. There is a no possibility for a field to have spin, because the Lorentz
group is trivial. Our fields are scalar, and the simplest choice is just a single
field
φ
:
{pt} R
, i.e. just a real variable. Similarly, we simply have
C
=
R
.
This is not an infinite-dimensional space.
The action is just a normal function
S
:
C
=
R R
of one real variable.
The path integral measure D
φ
can be taken to just be the standard (Lebesgue)
measure dφ on R. So our partition function is just
Z =
Z
R
dφ e
S(φ)/~
,
where we assume
S
is chosen so that this converges. This happens if
S
grows
sufficiently quickly as φ ±∞.
More generally, we may wish to compute correlation functions, i.e. we pick
another function f(φ) and compute the expectation
hf(φ)i =
1
Z
Z
dφ f(φ)e
S(φ)/~
.
Again, we can pick whatever
f
we like as long as the integral converges. In this
case,
1
Z
e
S(φ)/~
is a probability distribution on
R
, and as the name suggests,
hf
(
φ
)
i
is just the expectation value of
f
in this distribution. Later on, when we
study quantum field theory in higher dimensions, we can define more complicated
f
by evaluating
φ
at different points, and we can use this to figure out how the
field at different points relate to each other.
Our action is taken to have a series expansion in
φ
, so in particular we can
write
S(φ) =
m
2
φ
2
2
+
N
X
n=3
g
n
φ
n
n!
.
We didn’t put in a constant term, as it would just give us a constant factor in
Z
.
We could have included linear terms, but we shall not. The important thing is
that
N
has to be even, so that the asymptotic behaviour of
S
is symmetric in
both sides.
Now the partition function is a function of all these terms:
Z
=
Z
(
m
2
, g
n
).
Similarly,
hfi
is again a function of
m
2
and
g
n
, and possibly other things used
to define f itself.
Note that nothing depends on the field, because we are integrating over all
possible fields.