| Feynman propagator | |
|---|---|
| Name | Feynman propagator |
| Field | Quantum field theory |
| Introduced | 1940s |
| Introduced by | Richard Feynman |
Feynman propagator
The Feynman propagator is a two-point Green's function used in quantum field theory to describe the amplitude for a particle or field excitation to propagate between spacetime points. It encodes causality and time-ordering in perturbative calculations, appears as an internal line in Feynman diagram expansions, and provides the link between operator formalisms and the path integral formulation of quantum mechanics. Its proper definition and analytic structure are central to renormalization, scattering theory, and the computation of correlation functions in interacting quantum electrodynamics and other gauge theories.
The Feynman propagator is defined as the vacuum expectation value of a time-ordered product of field operators. For a scalar field φ(x) in the Heisenberg picture one commonly writes the propagator as D_F(x-y) = ⟨0|T φ(x) φ(y)|0⟩, where T denotes time-ordering and |0⟩ is the vacuum state. Physically, D_F gives the amplitude for a quantum of the field to be created at y and annihilated at x, with contributions that reflect propagation forward and backward in time; backward-in-time components are interpreted in perturbation theory as antiparticle propagation, a viewpoint developed by Richard Feynman and related to work by Freeman Dyson and Julian Schwinger.
The propagator enforces causal ordering through its support structure and singularity structure in complex momentum space, distinguishing it from other Green's functions such as the retarded or advanced propagators used in classical electrodynamics or linear response theory.
In momentum space the scalar Feynman propagator for mass m is typically expressed as G_F(p) = i/(p^2 - m^2 + iε), with p^2 = p_μ p^μ and an infinitesimal prescription iε that selects the contour consistent with causal boundary conditions. For relativistic fields this object transforms covariantly under representations of the Lorentz group and arises from Fourier transform of the position-space two-point function. For interacting theories, the full propagator is the two-point 1PI-resummed Green's function related to the self-energy Σ(p) by Dyson's equation G(p) = i/(p^2 - m^2 - Σ(p) + iε), a cornerstone of renormalization and perturbation theory.
The Feynman propagator satisfies the inhomogeneous Klein–Gordon equation with a delta-function source, (□ + m^2)D_F(x) = -i δ^{(4)}(x), which identifies it as a particular Green's function among solutions to linear wave equations.
Different field types require distinct Feynman propagators reflecting spin and gauge structure. For a real scalar field φ one has D_F(x-y) = ⟨0|T φ(x)φ(y)|0⟩ as above. For a Dirac spinor ψ(x) the propagator S_F(x-y) = ⟨0|T ψ(x) \barψ(y)|0⟩ carries spinor indices and in momentum space is S_F(p) = i(γ^μ p_μ + m)/(p^2 - m^2 + iε), where the Dirac gamma matrices γ^μ encode the Clifford algebra. For vector gauge fields such as the photon in QED or gluons in QCD, gauge fixing is necessary: the photon Feynman propagator in covariant gauges is D_F^{μν}(p) = -i η^{μν}/(p^2 + iε) plus gauge-dependent terms; for nonabelian gauge fields, Faddeev–Popov ghosts and gauge-fixing terms affect internal propagators in perturbative expansions.
Each propagator carries index structure, polarization projection, and mass/gauge parameters appropriate to the field and determines the numerator factors in corresponding Feynman rules.
The Feynman propagator is one of several Green's functions of linear wave operators; others include the retarded Green's function G_R and the advanced Green's function G_A. While G_R enforces causality by vanishing for x^0 < y^0, D_F implements time-ordering that mixes causal and acausal contributions when interpreted as amplitudes. In thermal field theory or curved spacetime contexts (e.g., in studies by Stephen Hawking or in cosmology), different boundary conditions and vacuum choices (such as the Hartle–Hawking state or the Bunch–Davies vacuum) lead to modified propagators. The relationship between these Green's functions is clarified by spectral representations and dispersion relations such as the Källén–Lehmann spectral representation, which expresses the interacting propagator in terms of a positive spectral density function.
In the path integral formalism introduced by Richard Feynman, the propagator arises from functional integrals over field configurations weighted by e^{iS}, where S is the action. The time-ordered correlator ⟨T φ(x)φ(y)⟩ is computed by inserting fields into the generating functional Z[J] = ∫ Dφ exp(iS[φ] + i∫ Jφ) and differentiating with respect to sources J. The iε prescription emerges naturally by the Feynman boundary conditions on the contour of integration (e.g., adding a small imaginary part to the action or using the Wick rotation to Euclidean time), which enforces convergence and selects the vacuum state used in perturbation theory.
Path integral methods make manifest the diagrammatic expansion of propagators as internal lines and permit systematic derivation of Feynman rules for both abelian and nonabelian theories, connecting to methods developed in texts by Peskin and Schroeder and others.
Feynman propagators are fundamental ingredients in computing S-matrix elements for scattering processes via the LSZ reduction formula, constructing loop integrals in perturbation theory, and evaluating correlation functions in many-body physics. They appear in calculations of radiative corrections in quantum electrodynamics, in self-energy and vacuum polarization diagrams, and in nonperturbative studies such as Schwinger–Dyson equations. In condensed matter physics, analogous propagators (Green's functions) are used to study quasiparticles and response functions in systems modeled by many-body theory.
Practical computations require regularization and renormalization techniques (e.g., dimensional regularization, MS-bar scheme) to handle divergences in loop integrals built from propagators and vertices.
The analytic structure of the Feynman propagator in complex energy or momentum plane encodes causal propagation and determines contour prescriptions for integrals. The ubiquitous iε term shifts poles off the real axis, prescribing how to perform integrals via contour deformation and ensuring the correct time-ordering. Analytic continuation between Minkowski and Euclidean propagators is performed by Wick rotation, converting oscillatory integrals to exponentially damped ones used in nonperturbative lattice computations often carried out by collaborations such as those at CERN or national laboratories. Unitarity and causality impose constraints on discontinuities across branch cuts of the propagator, which are used in deriving cutting rules like the Cutkosky rules and the optical theorem.