Feynman diagram

In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles.

According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations.

"[2] A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory.

Within the canonical formulation of quantum field theory, a Feynman diagram represents a term in the Wick's expansion of the perturbative S-matrix.

The Dyson series can be alternatively rewritten as a sum over Feynman diagrams, where at each vertex both the energy and momentum are conserved, but where the length of the energy-momentum four-vector is not necessarily equal to the mass, i.e. the intermediate particles are so-called off-shell.

Each Feynman diagram is the sum of exponentially many old-fashioned terms, because each internal line can separately represent either a particle or an antiparticle.

In addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight into the nature of particle interactions.

The technique of renormalization, suggested by Ernst Stueckelberg and Hans Bethe and implemented by Dyson, Feynman, Schwinger, and Tomonaga compensates for this effect and eliminates the troublesome infinities.

Stueckelberg was motivated by the need for a manifestly covariant formalism for quantum field theory, but did not provide as automated a way to handle symmetry factors and loops, although he was first to find the correct physical interpretation in terms of forward and backward in time particle paths, all without the path-integral.

Their motivations are consistent with the convictions of James Daniel Bjorken and Sidney Drell:[10] The Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand.

Dyson's formula expands the time-ordered matrix exponential into a perturbation series in the powers of the interaction Hamiltonian density, Equivalently, with the interaction Lagrangian LV, it is A Feynman diagram is a graphical representation of a single summand in the Wick's expansion of the time-ordered product in the nth-order term S(n) of the Dyson series of the S-matrix, where N signifies the normal-ordered product of the operators and (±) takes care of the possible sign change when commuting the fermionic operators to bring them together for a contraction (a propagator) and A represents all possible contractions.

For the QED interaction Lagrangian describing the interaction of a fermionic field ψ with a bosonic gauge field Aμ, the Feynman rules can be formulated in coordinate space as follows: The second order perturbation term in the S-matrix is The Wick's expansion of the integrand gives (among others) the following term where is the electromagnetic contraction (propagator) in the Feynman gauge.

A simple example is the free relativistic scalar field in d dimensions, whose action integral is: The probability amplitude for a process is: where A and B are space-like hypersurfaces that define the boundary conditions.

On a lattice, (i), the field can be expanded in Fourier modes: Here the integration domain is over k restricted to a cube of side length ⁠2π/a⁠, so that large values of k are not allowed.

This is not strictly as necessary as the space-lattice limit, because interactions in k are not localized, but it is convenient for keeping track of the factors in front of the k-integrals and the momentum-conserving delta functions that will arise.

But the Monte Carlo method also works well for bosonic interacting field theories where there is no closed form for the correlation functions.

From the Lagrangian, the equation of motion is: and in an expectation value, this says: Where the derivatives act on x, and the identity is true everywhere except when x and y coincide, and the operator order matters.

[11] An immediate consequence of the linked-cluster theorem is that all vacuum bubbles, diagrams without external lines, cancel when calculating correlation functions.

Repeating this process eventually erases all the Fermionic lines: this is the Euler algorithm to 2-color a graph, which works whenever each vertex has even degree.

The Euclidean scalar propagator has a suggestive representation: The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space.

The Jacobian for this transformation of variables is easy to work out from the identities: and "wedging" gives This allows the u integral to be evaluated explicitly: leaving only the v-integral.

Abstractly, it is the elementary identity: But this form does not provide the physical motivation for introducing v; v is the proportion of proper time on one of the legs of the loop.

Once the denominators are combined, a shift in k to k′ = k + vp symmetrizes everything: This form shows that the moment that p2 is more negative than four times the mass of the particle in the loop, which happens in a physical region of Lorentz space, the integral has a cut.

The Jacobian for the coordinate transformation can be worked out as before: Wedging all these equations together, one obtains This gives the integral: where the simplex is the region defined by the conditions as well as Performing the u integral gives the general prescription for combining denominators: Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs.

To compute the scattering amplitude for single particle states only requires a careful limit, sending the fields to infinity and integrating over space to get rid of the higher-order corrections.

For example, for the λφ4 interaction of the previous section, the order λ contribution to the (Lorentz) correlation function is: Stripping off the external propagators, that is, removing the factors of ⁠i/k2⁠, gives the invariant scattering amplitude M: which is a constant, independent of the incoming and outgoing momentum.

The lowest order scattering of the theory reveals the non-relativistic interpretation of this theory—it describes a collection of particles with a delta-function repulsion.

They encode not only asymptotic processes like particle scattering, they also describe the multiplication rules for fields, the operator product expansion.

But because Feynman propagators are nonlocal in time, translating a field process to a coherent particle language is not completely intuitive, and has only been explicitly worked out in certain special cases.

For quantum chromodynamics, the Shifman–Vainshtein–Zakharov sum rules describe non-perturbatively excited long-wavelength field modes in particle language, but only in a phenomenological way.