nLab
Feynman diagram

Context

Algebraic Qunantum Field Theory

Measure and probability theory

Contents

Idea

General

For μ S\mu_S a Gaussian probability measure and exp(I)μ S\exp(-I)\mu_S a perturbation with II polynomial at least of degree 3, there is a combinatorial expression for the moments/expectation values of exp(I)μ S\exp(-I)\mu_S as a sum over certain graphs whose kk-ary vertices are labeled by the monomials of degree kk in II. (This is such that for I=0I = 0 it reduces to Wick's lemma.)

These graphs are called Feynman diagrams.

In quantum field theory

Feynman graphs play a central role in perturbation theory in quantum field theory where exp(I)μ S\exp(I)\mu_S plays the role of an action functional on a space of fields, μ S\mu_S is the exponentiazed kinetic action and hence the measure for free fields, while exp(I)\exp(I) is the interaction part of the action functional: the order-kk monomials in II encode an interaction of kk fields. In this context the corresponding Feynman diagrams are traditionally thought of as depicting interaction processes of quanta of these fields, with propagation along the edges and interaction at the vertices. But this interpretation has its limits, which is partly reflected in speaking of “virtual particles”.

Details

For finitely many degrees of freedoms

We discuss Feynman diagrams for a single real scalar field on a discrete space of kk\in \mathbb{N} points. This contains in it already all the aspects of real Feynman diagrams in perturbative quantum field theory and in this context everything is easily well-defined. The generalization to more field components is immediate and simply obtained by thinking of all “ϕ\phi” in the following as taking values in some appropriate representation space and all products of ϕ\phis as given by suitable intertwiners and inner products, otherwise the form of the formulas remains the same. Similarly, in generalization to continuous space (non-finitely many degrees of freedom) all the diagrammatics remains the same, the only issue now is to make sense (namely via renormalization) of the numerical value that is assigned to any one Feynman diagram.

So a field ϕ\phi here is a map [k][k]\to \mathbb{R} from the kk-element finite set to the real numbers, and hence the space of all field configurations is k\mathbb{R}^k.

Fix then a k×kk \times k real-valued matrix A(A xy)Mat k×k()A \coloneqq (A_{x y}) \in Mat_{k\times k}(\mathbb{R}) of non-vanishing determinant detA0det A \neq 0.

For standard applications this AA is a discretized version of the Laplacian and then the expression

S kin=E kin12 x,y=1 kϕ xA xyϕ y S_{kin} = E_{kin} \coloneqq \tfrac{1}{2}\sum_{x,y = 1}^k \phi_x A_{x y} \phi_y

is the kinetic energy and kinetic action of the field configuration ϕ\phi. More concretely, think of the set of kk space points as being a discretization of a circle and write (ϕ p)(\phi_p) for the corresponding (discrete) Fourier transform of (ϕ x)(\phi_x). Then the kinetic term of the free scalar field on this space is given by AA which is the diagonal matrix in the pp-basis with

A p,p=p 2m 2, A_{p,p} = p^2 - m^2 \,,

where mm is a constant called the mass of the field ϕ\phi.

A sum over all values of ϕ\phi is the finite (and hence well-defined) analog of a path integral. The Gaussian integral of AA is called the partition function:

Z 0 exp(S kin(ϕ))Dϕ kexp(12 x,y=1 kϕ xA x,yϕ y)dϕ 1dϕ k =(2π) k/2(detA) 1/2 \begin{aligned} Z_0 & \coloneqq \int \exp(- S_{kin}(\phi)) \,D\phi \\ & \coloneqq \int_{\mathbb{R}^k} \, \exp(- \tfrac{1}{2}\sum_{x,y = 1}^k \phi_{x} A_{x,y} \phi_y ) \; d\phi_{1} \cdots d\phi_{k} \\ & = (2\pi)^{k/2} (det A)^{-1/2} \end{aligned}

An n-point function ϕ x 1ϕ x 2ϕ x n\langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle is an nnth moment of this Gaussian distribution:

ϕ x 1ϕ x 2ϕ x n 1Z 0(exp(S kin(ϕ))ϕ x 1ϕ x 2ϕ x n)Dϕ 1Z 0 k(exp(12ϕ xA x,yϕ y)ϕ x 1ϕ x 2ϕ x n)dϕ 1dϕ k \begin{aligned} \langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle & \coloneqq \frac{1}{Z_0} \int \left( \exp(-S_{kin}(\phi)) \phi_{x_1}\phi_{x_2}\cdots \phi_{x_n} \right) D \phi \\ & \coloneqq \frac{1}{Z_0} \int_{\mathbb{R}^k} \left( \exp(-\tfrac{1}{2} \phi_x A_{x,y}\phi_y) \, \phi_{x_1}\phi_{x_2} \cdots\phi_{x_n} \right) d\phi_1 \cdots d\phi_k \end{aligned}

In order to compute these conveniently, pass to the generating function obtained by adding a source variable J=(J x)J = (J_x):

Z(J) (exp(S kin(ϕ)+Jϕ))Dϕ kexp(12 x=1 kϕ xA x,yϕ y+ x=1 kJ xϕ x)dϕ 1dϕ k =Z 0exp(12 x=1 kJ xA x,y 1J y), \begin{aligned} Z(J) & \coloneqq \int( \exp(- S_{kin}(\phi) + J \cdot\phi) ) D\phi \\ & \coloneqq \int_{\mathbb{R}^k} \exp(- \tfrac{1}{2}\sum_{x = 1}^k \phi_{x} A_{x,y} \phi_y + \sum_{x = 1}^k J_x \phi_x ) \; d\phi_{1} \cdots d\phi_{k} \\ & = Z_0 \exp(\tfrac{1}{2} \sum_{x = 1}^k J_x A^{-1}_{x,y} J_y) \end{aligned} \,,

where A 1=(A x,y 1)A^{-1} = (A^{-1}_{x,y}) is the inverse matrix of AA, which appears by computing the new integral here again as a Gaussian integral after completing the square in the exponent.

In applications to field theory this A 1A^{-1} is called the Feynman propagator. In the standard example where AA is the kinetic term of the free field of mass mm then in Fourier-transformed components A 1A^{-1} is diagonal with components

A p,p 1=1p 2m 2. A^{-1}_{p,p} = \frac{1}{p^2 - m^2} \,.

This means, incidentally. that in the non-finite case of interest in physics, AA is not actually naively invertible after all, as p 2=m 2p^2 = m^2 on the mass shell. In this case one has to replace the naive A 1A^{-1} with its zeta-function regularized version.

The way this works is insightful even when the naive A 1A^{-1} does exist. Notice that using “Schwinger parameterization” the propagator is equivalently rewritten as a Mellin transform integral:

A p,p 1= 0 exp(τA p,p)dτ. A^{-1}_{p,p} = \int_0^\infty \exp(- \tau A_{p,p}) d\tau \,.

Again in the example of the standard scalar field kinetic term expressed in Fourier diagonalization A p,p=p 2m 2A_{p,p}= p^2 - m^2 then with X:[0,τ]X \colon [0,\tau] \to \mathbb{R} a parameterization of the straight line with slope pp, the exponent is equivalently

τA p,p =τ(p 2m 2) = 0 τ(X˙ 2m 2). \begin{aligned} \tau A_{p,p} & = \tau (p^2 - m^2) \\ & = \int_0^\tau (\dot X^2 - m^2) \end{aligned} \,.

This now happens to be the standard action functional (Polyakov action) for a sigma model describing the propagation of a particle along its worldline. This means that the propagator of the scalar field may be thought of as coming from the path integral of a scalar particle along its worldline. This perspective is called the “worldline formalism”, it is a formalization of second quantization, expressing the dynamics of fields in terms of that of their particle “quanta” running along worldlines of the form of the corresponding Feynman diagrams (to which we finally come in a moment).

Back to the computation of the nn-point function. By construction, it is now equally expressed by partial derivatives of the generating function with respect to the source variable JJ and evaluated at J=0J = 0, and this in turn is a combinatorial expression just in products of the propagator:

ϕ x 1ϕ x n =1Z 0(J x 1J x nZ(J)) |J=0 =(J x 1J x nexp(12 x=1 kJ xA xy 1J y)) |J=0 =pairingsA x j 1x j 2 1A x j 3x j 4 1A x j n1x j n 1. \begin{aligned} \langle \phi_{x_1} \cdots \phi_{x_n} \rangle &= \frac{1}{Z_0} \left( \frac{\partial}{\partial J_{x_1}} \cdots \frac{\partial}{\partial J_{x_n}} Z(J) \right)_{\vert J = 0} \\ & = \left( \frac{\partial}{\partial J_{x_1}} \cdots \frac{\partial}{\partial J_{x_n}} \exp(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y) \right)_{\vert J = 0} \\ & = \underset{pairings}{\sum} A^{-1}_{x_{j_1} x_{j_2}} \cdot A^{-1}_{x_{j_3} x_{j_4}} \cdots A^{-1}_{x_{j_{n-1}}x_{j_n}} \end{aligned} \,.

Here the last equality – known as Wick's theorem – comes from simple inspection: take the derivatives inside the exponential series and observe that then the only summands non-vanishing at J=0J = 0 appears for even nn and are those where all derivatives hit the monomial (12 x=1 kJ xA xy 1J y) n/2\left(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y\right)^{n/2}.

Thinking of A xy 1A^{-1}_{x y} here as labeling an edge (a “worldline”) from vertex xx to vertex yy This is the source of all Feynman digrammatics.

Now consider a polynomial V(ϕ)V(\phi) of degree 3\geq 3. In applications to field theory this represents the potential energy or (self)interaction of the field configuration. The difference of the kinetic energy and the potential energy is called the (here: “Wick rotated”/“Euclidean”) action

S =S kin+S int =12 x,y=1 kϕ xA xyϕ ygV(ϕ). \begin{aligned} S & = S_{kin} + S_{int} \\ & = \tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y - g V(\phi) \end{aligned} \,.

The prefactor gg is called the coupling constant.

Putting everything together, the integral over the full action may be expressed as a power series in the coupling constant gg of the moments with respect to the kinetic action of the powers of the interaction term:

Z(g) =1Z 0exp(S(ϕ))Dϕ 1Z 0(exp(12 x,y=1 kϕ xA xyϕ y+gV(ϕ)))dϕ 1dϕ k =exp(gV(ϕ)) =1+gV(ϕ)+g 22V(ϕ) 2+ \begin{aligned} Z(g) & = \frac{1}{Z_0}\int \exp(- S(\phi)) \, D\phi \\ & \coloneqq \frac{1}{Z_0} \int \left(\exp(-\tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y + g V(\phi)) \right) \, d\phi_1 \cdots d\phi_k \\ & = \langle \exp(g V(\phi)) \rangle \\ & = 1 + g \langle V(\phi)\rangle + \frac{g^2}{2} \langle V(\phi) ^2\rangle + \cdots \end{aligned}

By Wick's theorem stated above, each V(ϕ) \langle V(\phi)^\ell\rangle is equivalently expressed as a sum over products of components of the propagator A xy 1A^{-1}_{x y}. Thinking of each such propagator term as an edge produces a diagram, this is the corresponding Feynman diagram.

For instance, for a cubic point interaction

V(ϕ)= xϕ x 3 V(\phi) = \sum_x \phi_x^3

then

V(ϕ) 2 ( x=1 kϕ x 3) 2 = x 1,x 2=1 kϕ x 1ϕ x 1ϕ x 1ϕ x 2ϕ x 2ϕ x 2 =prefactor x 1,x 2=1 kA x 1x 1 1A x 1x 2 1A x 2x 2 1dumbbelldiagram+prefactor x 1,x 2=1 kA x 1x 2 1A x 1x 2 1A x 1x 2 1thetadiagram \begin{aligned} \langle V(\phi)^2 \rangle &\coloneqq \langle (\sum_{x= 1}^k \phi_x^3)^2 \rangle \\ & = \sum_{x_1, x_2 = 1}^k \langle \phi_{x_1}\phi_{x_1} \phi_{x_1}\phi_{x_2}\phi_{x_2}\phi_{x_2}\rangle \\ & = prefactor \; \sum_{x_1,x_2 = 1}^k \underset{dumbbell\, diagram}{\underbrace{A^{-1}_{x_1 x_1} A^{-1}_{x_1 x_2} A^{-1}_{x_2 x_2}}} + prefactor \; \sum_{x_1, x_2 = 1}^k \underset{theta\, diagram}{\underbrace{A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2}}} \end{aligned}

Here the first summand corresponds to the “dumbbell” Feynman diagram of the form

and the second summand corresponds to the “theta” Feynman diagram of the form

.

In quantum field theory

Summary

Feynman diagrams in causal perturbation theory

In perturbative quantum field theory, Feynman diagrams are labeled graphs that encode products of Feynman propagators as they arise in the expansion – the Feynman perturbation series– of the S-matrix of a given interaction Lagrangian density L intL_{int}

S(gL int)=kg k kk!T(L intL intkfactors) 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}}} )

in terms of the time-ordered products T()T(\cdots) given by the star product with the Feynman propagator ω F\omega_F:

T(L intL int)=prodexp(ω F(x,y)δδϕ(x)δδϕ(y))(L intL int). 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} ) \,.

Each edge in the Feynman diagram graph corresponds to a factor of a Feynman propagator ω F\omega_F in T(L intL intkfactors)T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} ), being a distribution of two variables; and each vertex corresponds to a factor of the interaction Lagrangian density L intL_{int} evaluated at one of the integration-variables x ix_i.

For example a typical Feynman diagram as in quantum electrodynamics, where L intψ¯fermionicAbosonicψfermionicL_{int} \propto \underset{\text{fermionic}}{\underbrace{\overline{\psi}}} \underset{\text{bosonic}}{\underbrace{A}} \underset{\text{fermionic}}{\underbrace{\psi}} looks as follows (with internal indices notationally suppressed):

Here the Feynman propagators for fermion fields ω F(,)=Δ(,)\omega_F(-,-) = \Delta(-,-) are drawn as solid lines…

…while the Feynman propagators ω F(,)=G(,)\omega_F(-,-) = G(-,-) for bosons are drawn as wiggly lines:

This way each part of the Feynman diagram graph corresponds to a product of distributions:

graphics grabbed from Brouder 10

A priori this product of distributions is defined away from coincident vertices: x ix jx_i \neq x_j. The definition at coincident vertices x i=x jx_i = x_j requires a choice of extension of distributions to the diagonal locus. This choice is the “renormalization” of the Feynman diagram.

References

For general accounts see at

In causal perturbation theory

Discussion of Feynman diagrams in the rigorous formulation of causal perturbation theory and perturbative AQFT is due to

  • Kai Keller, chapter IV of Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization, PhD thesis (arXxiv:1006.2148)

parts of which also appears as

An exposition of this is in

Relation to the Hopf algebra structure on Feynman diagrams due to Kreimer 97 is discussed in

Relation to periods (in the sense here) is discussed in

See also

Review is in

  • Katarzyna Rejzner, section 6.5.2 of Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (pdf)

In terms of motivic structures

In terms of motives (see also motives in physics)

In homological BV-quantization

A clean derivation of the Feynman rules for finite-dimensional spaces of fields in terms of quantization by passing to cochain cohomology of BV-complexes (see at BV-formalism – Homological quantization) is in

with a review in the broader context of factorization algebras of observables in section 2.3 of

As string diagrams (morphisms in monoidal categories)

It has been observed that Feynman diagrams, notably in gauge theory, are in particular string diagrams (in the sense of category theory, not here in any sense of string theory!) in the given category of representations: the edges are labelled by particle species, hence by Wigner classification by irreps, and the vertices are labeled by representation homomorphisms (“intertwiners”) which, indeed, label the interaction of particles in the Feynman diagram.

A review of this formulation is in

Moreover, one may think of string diagrams in monoidal categories as providing categorical semantics for proof nets (see there for more) in multiplicative linear logic. Under this identification then Feynman diagrams have a relation to proof nets. Something like this is discussed in

Revised on October 30, 2017 14:52:40 by Urs Schreiber (94.220.69.162)