Feynman Propagator for Free Particles

• Feynman propagator for free particles is a central quantum field theory object defined as the amplitude for a particle to transition between spacetime points while ensuring causal ordering.
• It is constructed using multiple frameworks such as Minkowski space formalism, discrete causal sets, and path integral techniques, which validate its analytic and geometric structure.
• Advanced operator theory and microlocal methods provide a rigorous framework for incorporating regularization and handling causal anomalies in various spacetime backgrounds.

The Feynman propagator for free particles is a central object in quantum mechanics and quantum field theory, encapsulating the amplitude for a particle to propagate from one spacetime point to another. In the free case, the propagator can be rigorously defined and constructed in a variety of frameworks, each highlighting different aspects of its analytic or geometric structure, its operator-theoretic properties, and its role in regularization, causal structure, and quantum field theory in both flat and curved spacetime backgrounds.

1. Formalism and Definition in Continuum Minkowski Spacetime

In the standard relativistic quantum field theory of a free scalar particle, the Feynman propagator $G_F(x-y)$ is the vacuum expectation value of the time-ordered product of fields,

$$G_F(x-y) = i \langle 0| T\{ \phi(x)\phi(y)\} |0\rangle,$$

where $\phi(x)$ is the Hermitian field operator obeying the Klein–Gordon equation,

$(\Box + m^2)\phi(x) = 0,$

and the commutator

$[\phi(x), \phi(y)] = i\Delta(y-x),$

with $\Delta(x)$ the Pauli–Jordan function, i.e., the difference of the retarded and advanced Green's functions. In momentum space, this gives

$G_F(p) = \frac{1}{p^2 - m^2 + i\epsilon}.$

This prescription (the “+iε” in the denominator) ensures appropriate analytic properties and causal time-ordering. It is equivalently the boundary value of the resolvent of the Klein–Gordon operator at real frequencies, defining the propagator as the boundary limit in the sense of the limiting absorption principle (Dereziński et al., 2016).

2. Discrete and Regularized Constructions: Causal Sets and Lattice Analogs

In approaches motivated by quantum gravity and Lorentz-invariant regularization, the Feynman propagator is recast in a fundamentally discrete spacetime setting. In causal set theory, spacetime events are generated by a Poissonian “sprinkling” into a Lorentzian manifold, leading to a locally finite partially ordered set that captures causal structure without privileging any reference frame (0909.0944). The propagator is then encoded as a $p \times p$ matrix (for $p$ sprinkled points) constructed via discrete analogs of the retarded and advanced propagators: $$K_R = a C (I - abC)^{-1}, \quad \text{with} \; a = \frac{1}{2}, \; b = -\frac{m^2}{\rho}.$$ The Pauli–Jordan function is $i\Delta = K_R - K_A$, and, after diagonalization and vacuum construction, the discrete Feynman propagator reads

$K_F = K_R + iQ,$

where $Q$ is built from eigenvectors/eigenvalues of $i\Delta$. This matches continuum expressions in appropriate statistical/continuum limits and provides a Lorentz-invariant discretization that can serve as a non-perturbative regulator (0909.0944).

3. Propagators in Curved and Torsionful Spacetimes

Extensions to curved and torsionful backgrounds require more sophisticated geometrical construction. For spacetimes with torsion (Riemann–Cartan geometry), the momentum-space representation of the Feynman propagator is expressed using generalized Riemann–normal coordinates constructed from autoparallels rather than geodesics: $$G(x, x') = \int \frac{d^n k}{(2\pi)^n} e^{ik \cdot (x - x')} \left[ \frac{1}{k^2 + m^2} + \sum_{i} G^{(i)}(k) \right],$$ where higher order corrections $G^{(i)}(k)$ encode curvature and torsion effects. The proper-time (Schwinger) representation is

$$G(x,x') = i \int_0^\infty ds \, \frac{\Delta(x, x')}{(4\pi i s)^{n/2}} \exp\left(-i m^2 s - \frac{\sigma(x,x')}{2is}\right) F(x, x'; s),$$

with $\sigma(x, x')$ the autoparallel distance and $F$ containing the expansion in torsion/curvature. When torsion vanishes, the standard DeWitt–Schwinger result is recovered (Wu et al., 2010).

4. Propagator as Path Integral, Groupoid, and Kernel Constructs

The path integral representation provides an explicit sum-over-histories or worldline picture. In flat or curved backgrounds, the free-particle propagator can be written as

$$G(x_1, x_2; m) = \sum_{\mathrm{paths}} \mathcal{M} \exp[-m \ell(x_1, x_2)],$$

where $\ell$ is the geodesic path length and $\mathcal{M}$ denotes path measure. The formulation in terms of groupoids and Dirac–Feynman–Schwinger (DFS) states recasts the propagator as a reproducing kernel for a family of states derived from a q-Lagrangian function on the groupoid of configurations. This GNS (Gelfand–Naimark–Segal) construction ties the sum-over-histories approach to operator algebraic quantum mechanics (Ciaglia et al., 2021). Moreover, as shown through Aronszajn's theory, the Feynman propagator serves as a reproducing kernel determining the quantum Hilbert space structure and, when UV regularized, encodes features like spacetime granularity (Aubin-Frankowski, 2018).

5. Operator-Theoretic and Microlocal Perspectives

On static or asymptotically Minkowski spacetimes, the Feynman propagator is characterized as the boundary value of the resolvent of the Klein–Gordon operator. The limiting absorption principle provides the necessary analytic framework: for the Klein–Gordon operator $K$,

$G_F = \lim_{\epsilon \to 0^+} (K - i\epsilon)^{-1}$

in the strong operator topology, securing the causal and spectral properties (Dereziński et al., 2016). In the broader geometric context, including asymptotically Minkowski backgrounds, microlocal techniques based on the 3sc-calculus rigorously construct the propagator as the unique inverse of the Klein–Gordon operator acting on Sobolev spaces with prescribed regularity near radial points (i.e., sources and sinks of the Hamiltonian flow at null infinity), and prove that the Schwartz kernel satisfies the microlocal Hadamard condition,

$\widetilde{\mathrm{WF}}(E) = \mathrm{diag}_{T^* X \setminus 0} \cup \bigcup_{s \geq 0} \tilde{\Phi}_s(\mathrm{diag}_{\Char(P_0)}),$

which is essential for the correct singular structure in quantum field theory (Baskin et al., 2 Jul 2025).

6. Path Integral Evaluation, Boundary Conditions, and Advanced Techniques

For non-relativistic free particles and those in topologically nontrivial or bounded domains, the Feynman propagator is efficiently computed via path integral, Fock–Bargmann space, or spectral expansion. In the presence of boundaries or nontrivial topology, auxiliary phase factors must be accounted for—reflected in the method of images and encoded by local rules that depend on boundary conditions (Dirichlet, Neumann, Robin). After Poisson summation, the resulting propagators exactly reproduce those derived from spectral decompositions (Hage-Hassan, 2010, Lee, 2022), up to normalization.

Advanced discretization schemes—such as finite-dimensional Hilbert space approximations and the use of metric ultraproducts—allow for a rigorous approach to approximating infinite-dimensional models, with the observation that naive eigenvector-based definitions may be pathological unless properly averaged to recover the continuum kernel behavior (Hirvonen et al., 2014).

7. Causal, Fakeon, and Wheeler Alternatives

The Feynman propagator is not strictly causal; wave packets can develop support outside the light cone. By contrast, the Wheeler propagator—half retarded plus half advanced—has support strictly inside the light cone and lacks on-shell contributions, a property that can be formalized in terms of real part and principal value prescriptions. This makes it suitable for causal approaches, but at the expense of losing certain unitarity and spectral properties. Fakeon prescriptions are shown to eliminate the problems associated with the Wheeler/Feynman–Wheeler (principal value) propagator, preserving unitarity and locality even in higher-derivative (e.g., quantum gravity) settings by introducing a prescription termed average continuation (Koksma et al., 2010, Bollini et al., 2010, Anselmi, 2020).

8. Summary Table: Major Perspectives on the Free Particle Feynman Propagator

Approach	Mathematical Expression	Key Features
Minkowski continuum	$G_F(p) = \frac{1}{p^2 - m^2 + i\epsilon}$	Ensures causality, time-ordering, proper analytic structure
Causal set formulation	$K_F = K_R + i Q$	Lorentz-invariant discretisation; regulator for divergences
Static/Curved spacetimes	$G_F = \lim_{\epsilon \to 0^+} (K - i\epsilon)^{-1}$	Rigorous via resolvent, limiting absorption principle
Microlocal analysis	$\widetilde{\mathrm{WF}}(E)...$	Propagation of singularities, Hadamard condition
Path integral	$\sum_{\text{paths}} \mathcal{M} e^{-m \ell}$	Geometric, extensible to curved/topologically nontrivial settings
Reproducing kernel	$K((x,t),(y,s)) = G_F(x,t;y,s)$	Hilbert space construction, space-time granularity

The Feynman propagator for a free particle thus unifies operator theory, geometric analysis, and path integral approaches, remains robust under discretization and regularization, and serves as a probe for both foundational and applied quantum theory across a spectrum of physical settings.