Lecture 5.

Feynman graphs

David Kazhdan

5.1Feynman graph expansion. The technique of Feynman graphs allows one to write down an asymptotic series for functional integral of a QFT in a neighborhood of a free QFT (see Witten’s lecture 1). Nevertheless, the technique itself can be applied to a purely finite dimensional integral. In this lecture we will discuss this application only.

Let V be a finite dimensional real vector space, let V′ be its dual. We will view elements of symmetric algebra Sym(V′) as polynomial functions on V. We fix a nondegenerate positive definite quadratic form b ∈ Sym^{2}(V′). Let b^{−1} ∈ Sym^{2}(V) be the corresponding quadratic form on V′.

Let μ_{0} = e^{−b^{−1}/2}dv be the Gaussian measure on V′; we have ℱ(μ_{0}) = e^{−b/2}.

Let P be a polynomial function on V′. We want to study the “perturbed” measure μ = e^{εP} μ_{0}. This is a well-defined measure if ε > 0, and P is negative.

We are going to write down an asymptotic expansion for the Fourier transform ℱ(μ). More precisely, we will obtain a formal Taylor series μ̃ ∈ ℂ[[V′, ε]] in the following way. Let D(P) be the differential operator with constant coefficients on V corresponding to P.

More precisely, D is a homomorphism of the algebra of polynomial functions on V′ to the algebra of differential operators with constant coefficients on V, such that D(x) = i∂/∂x for x ∈ V.

We write:

ℱ(e^{εP} μ_{0}) = e^{εD(P)}(e^{−b/2}) = ∑_{n=0}^{∞} \frac{1}{n!}ε^{n}D(P)^{n}(e^{−b/2}) .

This is obviously an element of ℂ[[V′, ε]].

This formal power series is connected with the original analytical problem via the notion of Borel summability. However in this lecture we will be interested in the formal expression only.

It is notationally convenient to perform “Wick rotation”. So we consider the series

μ̃(v) = ℱ(e^{εP} μ_{0})(iv) = ∑_{n=0}^{∞} \frac{1}{n!}ε^{n}P̂^{n}(e^{b/2}) .

where P̂ = D(P (−ix)), so that x̂ = ∂/∂x.

The series μ̃ is defined for arbitrary nondegenerate quadratic form b, and a polynomial function P over any field of characteristic 0. (The condition on b to be positive definite is of course irrelevant in this setting.)

There exists a combinatorial way to “compute” μ̃. To describe it we first fix some notations.

By a graph we will mean the data consisting of: two sets Γ_{1} (edges) and Γ_{0} (vertices) and a map j: Γ_{1} → Γ_{0} × Γ_{0}. We also assume an involution σ of Γ_{1} is fixed such that j(γ) = (x, y) ⇔ j(σ(γ)) = (y, x), and σ has no fixed points. We will only consider graphs with no isolated vertices (i.e. pr_{1} ∘ j is surjective). An edge of a graph is an element of Γ_{1}/σ. For a vertex γ_{0} ∈ Γ_{0} we define its “star” as Γ(γ_{0}) = {γ_{1} ∈ Γ_{1}| pr_{2} ∘j(γ_{1}) = γ_{0}}. The set of external vertices is Γ_{ex} = {γ_{0} ∈ Γ_{0}|#(Γ(γ_{0})) = 1}, and the set of inner vertices is Γ_{in} = Γ_{0} − Γ_{ex}. We put: L(γ_{0}) = Sym^{#(Γ(γ_{0}))}(V); L̃(Γ) = ⊗_{γ_{0}∈Γ_{0}} L(γ_{0}); L(Γ) = ⊗_{γ∈Γ_{in}} L(γ).

Assume that for any γ_{0} ∈ Γ_{0} we are given an element s(γ_{0}) ∈ L(γ_{0}). We then take ℓ_{s} = ⊗_{γ_{0}∈Γ_{0}} s(γ_{0}) ∈ L̃(Γ). For any map s̃: Γ_{0} → V we define ss̃(γ_{0}) = (s̃(γ_{0}))^{#(Γ(γ_{0}))} ∈ L(γ_{0}), and denote ℓs̃ = ℓ_{ss̃}.

Lemma 1. There exists a unique linear function τ_{b}^{Γ} on L̃(Γ) such that for any map s̃: Γ_{0} → V we have:

τ_{b}^{Γ}(ℓs̃) = \frac{1}{#Aut(Γ)} ∏_{γ_{1}∈Γ_{1}/σ} b (s̃(pr_{1} ∘ j(γ_{1})), s̃(pr_{2} ∘ j(γ_{1}))) .

Proof is clear.

We can view τ_{b}^{Γ} as a map τ_{b}^{Γ}: L(Γ) → Sym^{#Γ_{ex}}(V′).

Example. Γ_{n} is a disjoint union of n copies of the graph: • • . (The nonoriented edge on the picture corresponds to two elements of Γ_{1} permuted by σ.) Then L(Γ) = ℂ canonically, because Γ_{in} is empty. We have:

τ_{b}^{Γ_{n}}(1) = \frac{b^{n}}{2^{n}n!} .

Note the equality ∑_{n=0}^{∞} τ_{b}^{Γ_{n}}(1) = e^{b/2} where Γ_{0} is the empty graph.

Let P, μ̃ be as above. We write P = ∑_{n ≥ 2} P_{n}/n! where P_{n} is homogeneous of degree n (thus we assume for convenience that P contains no constant or linear term). Theorem. We have an equality of formal series:

μ̃ = ( ∑_{Γ∈𝔤} ε^{#Γ_{in}}τ_{b}^{Γ}(s_{P}))e^{b/2}

Here 𝔤 is the set of isomorphism classes of graphs and s_{P} (γ_{0}) = P_{#Γ(γ_{0})}. Proof. [Another proof appears in §1.3 of Witten’s lecture 1].

The idea of the proof is as follows. We want to prove an equality between two expressions, which are power series in ε and in P_{n}’s. We will show that both the left hand side and the right hand side satisfy the same system of linear differential equations in P_{n}’s as variables, and then we will check that the initial conditions coincide.

So, let Γ be a graph and let λ ⊂ Γ_{ex}, be such that |λ| = n. We denote by Γ^{λ} the “cone over λ”. This means that Γ^{λ} is a graph, such that Γ_{0}^{λ} = Γ_{0} ∪ {∗} − λ and Γ_{1}^{λ} = Γ_{1} − {all edges of Γ with an end in λ}∪{all edges from ∗ to vertices connected with λ}. (We add all edges connecting ∗ to a vertex in λ, and then erase the vertices belonging to λ.) Assume now that n > 1. Then ∗ is an internal vertex of Γ^{λ}.

For any d ∈ S^{n}(V) and s ∈ L(Γ) we define s_{λ}^{d} = d ⊗ s ∈ L(Γ^{λ}). Lemma 2. For any d ∈ S^{n}(V), Γ ∈ 𝔤 and s ∈ L(Γ) we have

\frac{d̂}{n!}(τ_{b}(Γ)(s)) = ∑_{λ⊂Γ_{ex},|λ|=n} τ_{b}(Γ^{λ})(s_{λ}^{d})

(Here d̂ is the differential operator with constant coefficients on V corresponding to d.)

Now we can prove the theorem. Let l(P) denote the left hand side and let r(P) be the right hand side of the equality. We want to write down a certain system of differential equations in P_{n}’s as variables, which will be satisfied by both sides. It is easy to see from the definitions that l(P) satisfies the system of equations

\frac{∂l(P)}{∂P_{n}}(Q) = (εQ̂/n!)l(P)

where Q is an arbitrary polynomial of degree n on V′ and Q̂ is the corresponding differential operator on V; in the left hand side Q is considered as a tangent vector to the space of polynomials of degree n.

But now Lemma 2 tells us precisely that the right hand side of this expression is equal to \frac{∂r(P)}{∂P_{n}}(Q).

Therefore l(P) and r(P) satisfy the same system of differential equations. Let’s now check the initial conditions. By the definition, l _{P =0} = e^{b/2}. On the other hand, one also has r _{P =0} = e^{b/2} by the above Example. Hence we have l(P) = r(P) for any P, so the theorem is proved. Remark 1. The theorem and its proof along with all the statements below can be immediately generalized in the following way. We can take P to be an element of ℂ[V][[ε]] rather than a polynomial, i.e. P = ∑_{m=0}^{∞} ε^{m}P_{(m)} is a Taylor series in ε with polynomial functions on V′ as coefficients. Remark 2. This way of writing an asymptotic expansion for a measure uses Fourier transform, and hence the linear structure of V heavily. For example when one applies the Feynman graph expansion to the gauge theory, V being the space of connections modulo gauge transformation, then V has no natural linear structure. In this case one still can write an asymptotic expansion for functional integral, but an individual term assigned to a particular graph is not canonically defined.

We denote Z = ∑_{Γ∈𝔤} τ_{b}^{Γ}(s_{P})ε^{#Γ_{in}}.

One of the difficulties in using this expansion is that there are “too many” terms; so we try to reduce it somehow. We denote F = ∑_{Γ∈𝔤_{con}} τ_{b}^{Γ}(s_{P})ε^{#Γ_{in}}, where 𝔤_{con} is the set of connected nonempty graphs.

Note that F (and all series that will be introduced below) lie in a subring ℂ[V′][[ε]] ⊂ ℂ[[V′, ε]].

Claim 1. F = log Z.

Proof. Decomposing a graph into union of its irreducible components one can identify the two combinatorial expressions:

Z = ∑_{Γ∈𝔤} τ_{b}^{Γ} = ∑_{Γ_{1},..,Γ_{k}∈𝔤_{con};n_{1},..,n_{k}} \frac{1}{n_{1}!n_{2}!..n_{k}!}τ_{b}^{Γ_{1}} … τ_{b}^{Γ_{k}} = exp(F)

where the summation is over all sets of distinct elements Γ_{i} ∈ 𝔤_{con}.

We next want to study the “quasi-classical approximation” to our integral. So we introduce another variable ℏ and consider the expression:

F_{ℏ}(v) = log ℱ (e^{(b^{−1}/2+εP)/ℏ}dv) (ℏv) .

As an immediate corollary to the Theorem, we get:

Lemma 3. F_{ℏ} = ℏ^{−1} ∑_{Γ∈𝔤_{con}} ℏ^{h^{1}(Γ)}τ_{b}^{Γ}(s_{P})ε^{#Γ_{in}} where h^{1}(Γ) = dim(H^{1}(Γ)) = #Γ_{1}/σ − #Γ_{0}.

Proof. Apply the same formal procedure to the quadratic form b̃ = ℏb and polynomial P̃ = ℏ^{−1}P, and note that for a connected graph Γ we have τ_{ℏb}^{Γ}(s_{P/ℏ}) (ℏv) = ℏ^{h^{1}(Γ)−1}τ_{b}^{Γ}(s_{P})(v).

5.2 Quasi-classical (low-loop) approximations. Recall that the “classical” approximation to Fourier transform is the Legendre transform. An appropriate version of the definition of Legendre transform is as follows.

If f is a function on V′, then we can view its differential df as a map df: V′ → V.

If df is an isomorphism then the Legendre transform of f is defined by

L(f)(v) = ⟨v, (df)^{−1}(v)⟩ − f ((df)^{−1}(v))

i.e. L(f)(v) is the critical value of the function v − f where v is considered as a linear function on V′.

Let now G ∈ ℂ[[V, ε]] be a formal power series. Then its differential dG (in the V′- direction) is an element of Hom(V′, V) ⊗ ℂ[[V, ε]].

Assume that G = g_{0} + g_{1} + εG_{1} where g_{0} is a nondegenerate quadratic form on V′, and g_{1} contains no terms of degree less then 3 in V. Then dG is invertible, i.e. there exists H = (dG)^{−1} ∈ Hom(V, V′) ⊗ ℂ[[V, ε]] with no constant term such that dG ∘ H = id and H ∘ dG = id. In this situation the Legendre transform L(G) ∈ ℂ[[V′, ε]] can be defined by the same formula.

To describe the next term of the asymptotics we define for any function f on V′ as above a new function H(f) on V as follows. Let Hess_{f}(p) = det(\frac{∂^{2}f}{∂v_{i}∂v_{j}} )|_{p} be the Hessian of f (we assume that a constant volume form on V is fixed so that the determinant of the quadratic form is taken with respect to that volume form).

We put: H(f)(v) = Hess ((df)^{−1}(v)).

For G = g_{0}+g_{1}+εG_{1} ∈ ℂ[[V′, ε]] as above we can define Hess(G) and H(G) ∈ ℂ[[V′, ε]] by the same formula.

Claim 2. a) Let F_{0} = ∑_{Γ∈T} τ_{b}^{Γ}(s_{P})ε^{#Γ_{in}}, where T is the set of (nonempty) trees. Then F_{0} is the Legendre transform of \frac{b^{−1}}{2} − εP.

b) Let F_{1} = ∑_{Γ∈𝔤_{1}} τ_{b}^{Γ}(s_{P})ε^{#Γ_{in}} where 𝔤_{1} is the set of one-loop connected graphs (i.e.

graphs with h^{0}(Γ) = h^{1}(Γ) = 1. Then F_{1} = log(H(\frac{b^{−1}}{2} −εP )^{−1}). (We assume that the background volume form is such that det(\frac{b^{−1}}{2} ) = 1. Then H(\frac{b^{−1}}{2} − εP) ∈ 1 + εℂ[[V′, ε]]+V′ℂ[[V′, ε]], and log(H(\frac{b^{−1}}{2} −εP)) is a well-defined element of ℂ[[V′, ε]]).

Proof of the Claim. From Lemma 3 we see that F_{ℏ}(v) = F_{0}(v)/ℏ + F_{1}(v) + O(ℏ). Thus the claim follows from the stationary phase approximation applied to the integral

ℱ(f)(iv) = \frac{1}{(2π)^{d/2}} ∫_{V′} e^{(v,p)+(εP −b^{−1}/2)(p)}dp

One can also directly show the equality of the two combinatorial expressions:

F_{0} (d(\frac{b^{−1}}{2} − εP)|_{p}) = \frac{b^{−1}}{2} (p) − ε ∑_{n} \frac{P_{n}}{(n − 1)!}(p) + ε ∑_{n} \frac{P_{n}}{n!} (p)

= ⟨p, d(\frac{b^{−1}}{2} − εP )|_{p}⟩ − (b^{−1}/2 − εP)(d(\frac{b^{−1}}{2} − εP)) (5.1)

and

(5.2)  F_{1}(d(\frac{b^{−1}}{2} − εP)|_{p}) = − log Hess(\frac{b^{−1}}{2} − εP )

Here (5.1) is equivalent to F_{0} = L(\frac{b^{−1}}{2} − εP ) and (2) is equivalent to F_{1} = log(H(\frac{b^{−1}}{2} − εP )^{−1}).

Let us sketch the combinatorial proof of (5.1) and (5.2).

Let us identify V and V′ by means of b and use an equality ε^{#Γ_{in}}τ_{b}^{Γ}(s_{P})∘d(\frac{b^{−1}}{2} −εP ) = ∑(−1)^{c_{i}}ε^{#Γ_{in}^{i}}τ_{b}^{Γ^{i}}(s_{P}) where the graph Γ^{i} is obtained from the graph Γ by adding c_{i} vertices of arbitrary valencies, each one of which is connected with an external vertex of Γ in such a way that any external vertex of Γ is connected with no more then one new vertex.

Now it is not hard to check that in the expansion of F_{0} ∘ (d(\frac{b^{−1}}{2} − εP )) all terms with ε^{i} for i ≥ 2 cancel, and identify the sum of remaining terms with the RHS of (5.1).

Likewise in the expansion of F_{1} ∘ (d(\frac{b^{−1}}{2} − εP )) only the terms with one-particle irreducible one-loop graphs (“circles”) do not cancel; the sum of these terms is identified with ∑_{n=1}^{∞} ε^{i}tr(dP)^{n}/n = − log det(Id − dP) which coincides with the RHS of (5.2).

5.3 Effective potential. Let us call a nonempty graph 1-particle irreducible (or just 1irreducible) if it is connected and remains connected after removal of any internal edge. To any connected graph there corresponds a unique tree with a 1-particle irreducible graph assigned to each vertex together with an identification of the set of edges coming to a vertex of a tree with the set of external vertices of the corresponding graph. Thus “computation” of F_{0} can be reduced to summation over trees and over 1-particle irreducible graphs separately. Denote F_{1−irr} = ∑_{Γ∈𝔤_{1−irr}} ε^{#Γ_{in}}τ_{b}^{Γ}(s_{P}) where 𝔤_{1−irr} is the set of 1particle irreducible graphs. (Notice that the graph • • is 1-particle irreducible by the definition).

Claim 3. F_{1−irr} = b − b^{∗}(L(F)) where b is viewed as a map from V to V′ and L is the Legendre transform.

Proof. By Remark 1 after the proof of the theorem we can apply claim 2 to P′ = ε^{−1}(b^{−1})^{∗}(F_{1−irr} − b/2) ∈ ℂ[V′][[ε]].

From the above combinatorial observation we see that F = F_{P} = (F_{0})_{P′}. Hence claim 2 implies that F = L(b^{−1}/2 − εP′) = L ((b^{−1})^{∗}(b − F_{1−irr})). Since Legendre transform is involutive we get the claim.

F_{1−irr} is called the effective potential of the theory.