III Advanced Quantum Field Theory - QFT in zero dimensions

1QFT in zero dimensions

III Advanced Quantum Field Theory

1.3 Feynman diagrams

We now try to understand the terms in the asymptotic series in terms of Feynman

diagrams. The easy bits are the powers of



~λ



This combination is essentially fixed by dimensional analysis. What we really

want to understand are the combinatorial factors, given by

(4!)

(4n)!

(2n)!

To understand this, we can write the path integral in a different way. We have

Z(~, m, λ) =

dφ exp



−





dφ

∞

n=0



−

4!~



exp



−



∼

∞

n=0

(4!)



−λ



hφ

free

Thus, we can apply our previous Wick’s theorem to compute this! We now see

that there is a factor of

(4!)

coming from expanding the exponential, and

the factor of

(4n)!

(2n)!

came from Wick’s theorem — it is the number of ways to

pair up 4

vertices together. There will be a factor of

coming from Wick’s

theorem when evaluating hφ

i, so the expansion is left with a ~

factor.

Let’s try to compute the order

term. By Wick’s theorem, we should

consider diagrams with 4 external vertices, joined by straight lines. There are

three of these:

But we want to think of these as interaction vertices of valency 4. So instead of

drawing them this way, we “group” the four vertices together. For example, the

first one becomes

⇒

Similarly, at order λ

, we have a diagram

⇒

Note that in the way we are counting diagrams in Wick’s theorem means we

consider the following diagrams to be distinct:

However, topologically, these are the same diagrams, and we don’t want to count

them separately. Combinatorially, these diagrams can be obtained from each

other by permuting the outgoing edges at each vertex, or by permuting the

vertices. In other words, in the “expanded view”, when we group our vertices as

we are allowed to permute the vertices within each block, or permute the blocks

themselves. We let

be the set of all graphs, and

be the group consisting

of these “allowed” permutations. This

is a semi-direct product (

)

o S

Then by some combinatorics,

| =

(4n)!

(2n)!

, |G

| = (4!)

n!.

Recall that

is the number Wick’s theorem gives us, and we now see that

happens to be the factor we obtained from expanding the exponential. Of

course, this isn’t a coincidence — we chose to put

instead of, say,

in front

of the φ

term so that this works out.

We can now write the asymptotic series as

Z(m

, λ)

Z(m

, 0)

∼

n=0



−~λ



It turns out a bit of pure mathematics will allow us to express this in terms

of the graphs up to topological equivalence. By construction, two graphs are

topologically equivalent if they can be related to each other via

. In other

words, the graphs correspond to the set of orbits

under

. For

= 1,

there is only one graph up to topological equivalence, so

= 1. It is not hard

to see that |O

| = 3.

In general, each graph Γ has some automorphisms that fix the graph. For

example, in the following graph:

3 4

we see that the graph is preserved by the permutations (1 3), (2 4) and (1 2)(3 4).

In general, the automorphism group

Aut

(Γ) is the subgroup of

consisting of

all permutations that preserve the graph Γ. In this case,

Aut

(Γ) is generated by

the above three permutations, and |Aut(Γ)| = 8.

Recall from IA Groups that the orbit-stabilizer theorem tells us

Γ∈O

|Aut(Γ)|

Thus, if we write

O =

[

n∈N

then we have

Z(m

, λ)

Z(m

, 0)

∼

Γ∈O

|Aut(Γ)|



−~λ



where the

appearing in the exponent is the number of vertices in the graph Γ.

The last loose end to tie up is the factor of



−~λ



appearing in the sum.

While this is a pretty straightforward thing to come up with in this case, for

more complicated fields with more interactions, we need a better way of figuring

out what factor to put there.

If we look back at how we derived this factor, we had a factor of

−

coming from each interaction vertex, whereas when computing the correlation

functions

hφ

, every edge contributes a factor of

−2

(since, in the language

we expressed Wick’s theorem, we had

). Thus, we can imagine this as

saying we have a propagator

~/m

and a vertex

−λ/~

Note that the negative sign appears because we have

−S

in “Euclidean” QFT.

In Minkowski spacetime, we have a factor of −i instead.

After all this work, we can now expand the partition function as

Z(m

, λ)

Z(m

, 0)

∼

∅

+ + + + + ···

1 +

−λ~

128

+ ···

In generally, if we have a theory with several fields with propagators of value

, and many different interactions with coupling constants λ

, then

Z({λ

})

Z({0})

∼

Γ∈O

|Aut Γ|

(Γ)|

E(Γ)−V (Γ)

where

– e

(Γ) is the number of edges of type i in Γ;

– v

(Γ) is the number of vertices of type α in Γ;

– V (Γ) is the number of vertices in total; and

– E(Γ) the total number of edges.

There are a few comments we can make. Usually, counting all graphs is rather

tedious. Instead, we can consider the quantity

log(Z/Z

This is the sum of all connected graphs. For example, we can see that the term for

the two-figure-of-eight is just the square of the term for a single figure-of-eight

times the factor of

, and one can convince oneself that this generalizes to

arbitrary combinations.

In this case, we can simplify the exponent of

. In general, by Euler’s theorem,

we have

E − V = L −C,

where

is the number of loops in the graph, and

is the number of connected

components. In the case of

log

(

Z/Z

), we have

= 1. Then the factor of

just ~

L−1

. Thus, the more loops we have, the less it contributes to the sum.

It is not difficult to see that if we want to compute correlation functions, say

hφ

, then we should consider graphs with two external vertices. Note that in

this case, when computing the combinatorial factors, the automorphisms of a

graph are required to fix the external vertices. To see this mathematically, note

that the factors in

(4!)

came from Taylor expanding

exp

and the coefficient

. We do not get

these kinds of factors from φ

when computing

dφ φ

−S[φ]

Thus, we should consider the group of automorphisms that fix the external

vertices to get the right factor of

Finally, note that Feynman diagrams don’t let us compute exact answers

in general. Even if we had some magical means to compute all terms in the

expansions using Feynman diagrams, if we try to sum all of them up, it will

diverge! We only have an asymptotic series, not a Taylor series. However, it

turns out in some special theories, the leading term is the exact answer! The tail

terms somehow manage to all cancel each other out. For example, this happens

for supersymmetry theories, but we will not go into details.