algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
For $\mu_S$ a Gaussian probability measure and $\exp(-I)\mu_S$ a perturbation with $I$ polynomial at least of degree 3, there is a combinatorial expression for the moments/expectation values of $\exp(-I)\mu_S$ as a sum over certain graphs whose $k$-ary vertices are labeled by the monomials of degree $k$ in $I$. (This is such that for $I = 0$ it reduces to Wick's lemma.)
These graphs are called Feynman diagrams.
Feynman graphs play a central role in perturbation theory in quantum field theory where $\exp(I)\mu_S$ plays the role of an action functional on a space of fields, $\mu_S$ is the exponentiazed kinetic action and hence the measure for free fields, while $\exp(I)$ is the interaction part of the action functional: the order-$k$ monomials in $I$ encode an interaction of $k$ 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”.
We discuss Feynman diagrams for a single real scalar field on a discrete space of $k\in \mathbb{N}$ 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 “$\phi$” in the following as taking values in some appropriate representation space and all products of $\phi$s 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 $\phi$ here is a map $[k]\to \mathbb{R}$ from the $k$-element finite set to the real numbers, and hence the space of all field configurations is $\mathbb{R}^k$.
Fix then a $k \times k$ real-valued matrix $A \coloneqq (A_{x y}) \in Mat_{k\times k}(\mathbb{R})$ of non-vanishing determinant $det A \neq 0$.
For standard applications this $A$ is a discretized version of the Laplacian and then the expression
is the kinetic energy and kinetic action of the field configuration $\phi$. More concretely, think of the set of $k$ space points as being a discretization of a circle and write $(\phi_p)$ for the corresponding (discrete) Fourier transform of $(\phi_x)$. Then the kinetic term of the free scalar field on this space is given by $A$ which is the diagonal matrix in the $p$-basis with
where $m$ is a constant called the mass of the field $\phi$.
A sum over all values of $\phi$ is the finite (and hence well-defined) analog of a path integral. The Gaussian integral of $A$ is called the partition function:
An n-point function $\langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle$ is an $n$th moment of this Gaussian distribution:
In order to compute these conveniently, pass to the generating function obtained by adding a source variable $J = (J_x)$:
where $A^{-1} = (A^{-1}_{x,y})$ is the inverse matrix of $A$, 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 $A^{-1}$ is called the Feynman propagator. In the standard example where $A$ is the kinetic term of the free field of mass $m$ then in Fourier-transformed components $A^{-1}$ is diagonal with components
This means, incidentally. that in the non-finite case of interest in physics, $A$ is not actually naively invertible after all, as $p^2 = m^2$ on the mass shell. In this case one has to replace the naive $A^{-1}$ with its zeta-function regularized version.
The way this works is insightful even when the naive $A^{-1}$ does exist. Notice that using “Schwinger parameterization” the propagator is equivalently rewritten as a Mellin transform integral:
Again in the example of the standard scalar field kinetic term expressed in Fourier diagonalization $A_{p,p}= p^2 - m^2$ then with $X \colon [0,\tau] \to \mathbb{R}$ a parameterization of the straight line with slope $p$, the exponent is equivalently
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 $n$-point function. By construction, it is now equally expressed by partial derivatives of the generating function with respect to the source variable $J$ and evaluated at $J = 0$, and this in turn is a combinatorial expression just in products of the propagator:
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 $J = 0$ appears for even $n$ and are those where all derivatives hit the monomial $\left(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y\right)^{n/2}$.
Thinking of $A^{-1}_{x y}$ here as labeling an edge (a “worldline”) from vertex $x$ to vertex $y$ This is the source of all Feynman digrammatics.
Now consider a polynomial $V(\phi)$ of degree $\geq 3$. 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
The prefactor $g$ 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 $g$ of the moments with respect to the kinetic action of the powers of the interaction term:
By Wick's theorem stated above, each $\langle V(\phi)^\ell\rangle$ is equivalently expressed as a sum over products of components of the propagator $A^{-1}_{x y}$. 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
then
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
.
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 $L_{int}$
in terms of the time-ordered products $T(\cdots)$ given by the star product with the Feynman propagator $\omega_F$:
Each edge in the Feynman diagram graph corresponds to a factor of a Feynman propagator $\omega_F$ in $T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )$, being a distribution of two variables; and each vertex corresponds to a factor of the interaction Lagrangian density $L_{int}$ evaluated at one of the integration-variables $x_i$.
For example a typical Feynman diagram as in quantum electrodynamics, where $L_{int} \propto \underset{\text{fermionic}}{\underbrace{\overline{\psi}}} \underset{\text{bosonic}}{\underbrace{A}} \underset{\text{fermionic}}{\underbrace{\psi}}$ looks as follows (with internal indices notationally suppressed):
Here the Feynman propagators for fermion fields $\omega_F(-,-) = \Delta(-,-)$ are drawn as solid lines…
…while the Feynman propagators $\omega_F(-,-) = G(-,-)$ 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: $x_i \neq x_j$. The definition at coincident vertices $x_i = x_j$ requires a choice of extension of distributions to the diagonal locus. This choice is the “renormalization” of the Feynman diagram.
For general accounts see at
Alain Connes, Matilde Marcolli, section 3 of Noncommutative Geometry, Quantum Fields and Motives
Wikipedia, Feynman diagram
Discussion of Feynman diagrams in the rigorous formulation of causal perturbation theory and perturbative AQFT is due to
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
In terms of motives (see also motives in physics)
Spencer Bloch, Helene Esnault, Dirk Kreimer, On motives associated to graph polynomials, (pdf)
Spencer Bloch, Pierre Vanhove, The elliptic dilogarithm for the sunset graph, IHES preprint P-13-24, pdf
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
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