nLab A first idea of quantum field theory -- Feynman diagrams

Feynman diagrams

So far we considered only the axioms on a consistent perturbative S-matrix /time-ordered products and its formal consequences. Now we discuss the actual construction of time-ordered products, hence of perturbative S-matrices, by the process called renormalization of Feynman diagrams.

We first discuss how time-ordered product, and hence the perturbative S-matrix above, is uniquely determined away from the locus where interaction points coincide (prop. below). Moreover, we discuss how on that locus the time-ordered product is naturally expressed as a sum of products of distributions of Feynman propagators that are labeled by Feynman diagrams: the Feynman perturbation series (prop. below).

This means that the full time-ordered product is an extension of distributions of these scattering amplitudes- to the locus of coinciding vertices. The space of possible such extensions turns out to be finite-dimensional in each order of $g/\hbar, j/\hbar$, parameterizing the choice of point-supported distributions at the interaction points whose scaling degree is bounded by the given Feynman propagators.

Definition

For $k \in \mathbb{N}$, write

$$\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds} \hookrightarrow \left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}$$

for the subspace of the $k$-fold tensor product of the space of compactly supported polynomial local densities (def. ) on those tuples which have pairwise disjoint spacetime support.

Proposition

(time-ordered product away from the diagonal)

Restricted to $\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}$ (def. ) there is a unique time-ordered product (def. ), given by the star product that is induced by the Feynman propagator $\omega_F$

$$F \star_{\omega_F} G \;\coloneqq\; prod \circ \exp\left( \hbar \left\langle \omega_F , \frac{\delta}{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) (F \otimes G)$$

in that

$$T( L_1 \cdots L_k ) = L_1 \star_{\omega_F} L_2 \star_{\omega_F} \cdots \star_{\omega_F} L_k \,.$$

Proof

Since the singular support of the Feynman propagator is on the diagonal, and since the support of elements in $\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}$ is by definition in the complement of the diagonal, the star product $\star_{\omega_F}$ is well defined. By construction it satisfies the axioms “peturbation” and “normalization” in def. . The only non-trivial point to check is that it indeed satisfies “causal factorization”:

Unwinding the definition of the Hadamard state $\omega$ and the Feynman propagator $\omega_F$, we have

$$\begin{aligned} \omega & = \tfrac{i}{2}( \Delta_R - \Delta_A ) + H \\ \omega_F & = \tfrac{i}{2}( \Delta_R + \Delta_A ) + H \end{aligned}$$

where the propagators on the right have, in particular, the following properties:

1. the advanced propagator vanishes when its first argument is not in the causal past of its second argument:

   $$(supp(F) \geq supp(G)) \;\Rightarrow\; \left( \left\langle \Delta_A , \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = 0 \right) \,.$$

2. the retarded propagator equals the advanced propagator with arguments switched:

   $$\left\langle \Delta_R , \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = \left\langle \Delta_A , \frac{\delta G}{\delta \phi} \otimes \frac{\delta F}{\delta \phi} \right\rangle$$

3. $H$ is symmetric:

   $$\left\langle H, \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = \left\langle H, \frac{\delta G}{\delta \phi} \otimes \frac{\delta F}{\delta \phi} \right\rangle$$

It follows for causal ordering $supp(F) \geq supp(G)$ (def. ) that

$$\begin{aligned} F \star_{\omega_F} G & = prod \circ \exp\left( \hbar \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2}( \Delta_R + \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2}\Delta_R + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2}( \Delta_R - \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \omega , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = F \star_{\omega} G \end{aligned}$$

and for $supp(G) \geq supp(F)$ that

$$\begin{aligned} F \star_{\omega_F} G & = prod \circ \exp\left( \hbar \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2}( \Delta_R + \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2} \Delta_A + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2} \Delta_R + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( G \otimes F ) \\ & = prod \circ \exp\left( \hbar \left\langle \tfrac{i}{2} (\Delta_R - \Delta_A) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( G \otimes F ) \\ & = G \star_{\omega} F \,. \end{aligned}$$

This shows that $\star_F$ is a consistent time-ordered product on the subspace of functionals with disjoint support. It is immediate from the above that it is the unique solution on this subspace.

Remark

(time-ordered product is assocativative)

Prop. implies in particular that the time-ordered product is associative, in that

$$T( T(V_1 \cdots V_{k_1}) \cdots T(V_{k_{n-1}+1} \cdots V_{k_n} ) ) = T( V_1 \cdots V_{k_1} \cdots V_{k_{n-1}+1} \cdots V_{n_n} ) \,.$$

It follows that the problem of constructing time-ordered products, and hence (by prop. ) the perturbative S-matrix, consists of finding compatible extension of the distribution $prod \circ \exp\left( \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right)$ to the diagonal.

Moreover, by the nature of the exponential expression, this means in each order to extend products of Feynman propagators labeled by graphs whose vertices correspond to the polynomial factors in $F$ and $G$ and whose edges indicate over which variables the Feynman propagators are to be multiplied.

Definition

(scalar field Feynman diagram)

A scalar field Feynman diagram $\Gamma$ is

1. a natural number $v \in \mathcal{N}$ (number of vertices);

2. a $v$-tuple of elements $(V_r \in \mathcal{F}_{loc} \langle g,j\rangle)_{r \in \{1, \cdots, v\}}$ (the interaction and external field vertices)

3. for each $a \lt b \in \{1, \cdots, v\}$ a natural number $e_{a,b} \in \mathbb{N}$ (“of edges from the $a$th to the $b$th vertex”).

For a given tuple $(V_j)$ of interaction vertices we write

$$FDiag_{(V_j)}$$

for set of scalar field Feynman diagrams with that tuple of vertices.

Proposition

(Feynman perturbation series away from coinciding vertices)

For $v \in \mathbb{N}$ the $v$-fold time-ordered product away from the diagonal, given by prop.

$$T_v \;\colon\; \left(\mathcal{F}_{loc}\langle g,j\rangle\right)_{pds}^{\otimes^{v}} \longrightarrow \mathcal{W}[ [ g/\hbar, j/\hbar] ]$$

is equal to

$$T_k(V_1 \cdots V_v) \;=\; prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum} \underset{ r \lt s \in \{1, \cdots, v\} }{\prod} \tfrac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle (V_1 \otimes \cdots \otimes V_v) \,,$$

where the edge numbers $e_{r,s} = e_{r,s}(\Gamma)$ are those of the given Feynman diagram $\Gamma$.

(Keller 10, IV.1)

Proof

We proceed by induction over the number of vertices. The statement is trivially true for a single vertex. Assume it is true for $v \geq 1$ vertices. It follows that

$$\begin{aligned} T(V_1 \cdots V_v V_{v+1}) & = T( T(V_1 \cdots V_v) V_{v+1} ) \\ &= prod \circ \exp\left( \left\langle \hbar \omega_F, \frac{\delta}{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) \left( prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum} \underset{ r \gt s \in \{1, \cdots, v\} }{\prod} \frac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle (V_1 \otimes \cdots \otimes V_v) \right) \;\otimes\; V_{v+1} \\ & = prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum} \underset{ r \gt s \in \{1, \cdots, v\} }{\prod} \tfrac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle \left( \underset{e_{1,{v+1}}, \cdots e_{v,v+1} \in \mathbb{N}}{\sum} \underset{t \in \{1, \cdots v\}}{\prod} \tfrac{1}{e_{t,v+1} !} \left( \frac{\delta^{e_{1,v+1}} V_1 }{\delta \phi_{1}^{e_{1,v+1}}} \otimes \cdots \otimes \frac{ \delta^{e_{v,v+1}} V_v}{ \delta \phi_{v}^{e_{v,v+1}} } \right) \;\otimes\; \frac{\delta^{e_{1,v+1} + \cdots + e_{v,v+1}} V_{v+1}}{\delta \phi_{v-1}^{e_{1,v+1} + \cdots + e_{v,v+1}}} \right) \\ &= prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v+1}}}{\sum} \underset{ r \lt s \in \{1, \cdots, v+1\} }{\prod} \tfrac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle (V_1 \otimes \cdots \otimes V_{v+1}) \end{aligned}$$

Here in the first step we use the associativity of the time-ordered product (remark ), in the second step we use the induction assumption, in the third we pass the outer functional derivatives through the pointwise product using the product rule, and in the fourth step we recognize that this amounts to summing in addition over all possible choices of sets of edges from the first $v$ vertices to the new $v+1$st vertex, which yield in total the sum over all diagrams with $v+1$ vertices.

Remark

(loop order and powers of Planck's constant)

From prop. one deduces that the order in Planck's constant that a (planar) Feynman diagram contributes to the S-matrix is given (up to a possible offset due to external vertices) by the “number of loops” in the diagram.

In the computation of scattering amplitudes for fields/particles via perturbative quantum field theory the scattering matrix (Feynman perturbation series) is a formal power series in (the coupling constant and) Planck's constant $\hbar$ whose contributions may be labeled by Feynman diagrams. Each Feynman diagram $\Gamma$ is a finite labeled graph, and the order in $\hbar$ to which this graph contributes is

$$\hbar^{ E(\Gamma) - V(\Gamma) }$$

where

1. $V(\Gamma) \in \mathbb{N}$ is the number of vertices of the graph

2. $E(\Gamma) \in \mathbb{N}$ is the number of edges in the graph.

This comes about, according to the above, because

1. the explicit $\hbar$-dependence of the S-matrix is

   $$S\left(\tfrac{g}{\hbar} L_{int} \right) = \underset{k \in \mathbb{N}}{\sum} \frac{g^k}{\hbar^k k!} T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )$$

2. the further $\hbar$-dependence of the time-ordered product $T(\cdots)$ is

   $$T(L_{int} L_{int}) = prod \circ \exp\left( \hbar \int \omega_{F}(x,y) \frac{\delta}{\delta \phi(x)} \otimes \frac{\delta}{\delta \phi(y)} \right) ( L_{int} \otimes L_{int} ) \,,$$

where $\omega_F$ denotes the Feynman propagator and $\phi(x)$ the field observable at point $x$ (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as scalar fields, because this is all that affects the counting of the $\hbar$ powers).

The resulting terms of the S-matrix series are thus labeled by

1. the number of factors of the interaction $L_{int}$, these are the vertices of the corresponding Feynman diagram and hence each contibute with $\hbar^{-1}$

2. the number of integrals over the Feynman propagator $\omega_F$, which correspond to the edges of the Feynman diagram, and each contribute with $\hbar^1$.

Now the formula for the Euler characteristic of planar graphs says that the number of regions in a plane that are encircled by edges, the faces here thought of as the number of “loops”, is

$$L(\Gamma) = 1 + E(\Gamma) - V(\Gamma) \,.$$

Hence a planar Feynman diagram $\Gamma$ contributes with

$$\hbar^{L(\Gamma)-1} \,.$$

So far this is the discussion for internal edges. An actual scattering matrix element is of the form

$$\langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right) \vert \psi_{in} \rangle \,,$$

where

$$\vert \psi_{in}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{in}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle$$

is a state of $n_{in}$ free field quanta and similarly

$$\vert \psi_{out}\rangle \propto \tfrac{1}{\sqrt{\hbar^{n_{out}}}} \phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle$$

is a state of $n_{out}$ field quanta. The normalization of these states, in view of the commutation relation $[\phi(k), \phi^\dagger(q)] \propto \hbar$, yields the given powers of $\hbar$.

This means that an actual scattering amplitude given by a Feynman diagram $\Gamma$ with $E_{ext}(\Gamma)$ external vertices scales as

$$\hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 } \,.$$

(For the analogous discussion of the dependence on the actual quantum observables on $\hbar$ given by Bogoliubov's formula, see there.)