Essay: Space–Time Approach to Quantum Electrodynamics

Overview
Richard P. Feynman reconceives quantum electrodynamics through a space–time, or "sum-over-histories," viewpoint that transforms how scattering amplitudes are calculated and interpreted. The core idea replaces operator algebra and perturbative expansions in Hilbert space with a pictorial and calculational scheme in which contributions to amplitudes are represented by space–time graphs. Each graph corresponds to a term in a perturbation expansion, and the rules for translating graphs into mathematical expressions streamline previously opaque manipulations.
The exposition shifts focus from abstract field operators to propagating particles and their interactions at space–time points. This yields an intuitive picture in which electrons and photons trace lines through space–time, meet at vertices, and give rise to observable processes. The approach neatly incorporates relativity, causality, and quantum interference in a unified computational framework.

Path-integral and action principle
A central conceptual move replaces canonical quantization by a path-integral-like formulation: amplitudes result from summing over possible paths of charged particles weighted by exp(iS/ħ), where S is the classical action. While not a full technical development of the modern functional integral, the essay emphasizes the action as the organizing quantity and uses it to motivate the rules for propagators and interaction factors.
This viewpoint clarifies how classical trajectories emerge in semiclassical limits and why interference among different histories yields quantum corrections. It also provides a transparent way to incorporate relativistic propagation and boundary conditions through Green's functions defined in space–time.

Propagators and space–time Green's functions
The work introduces the space–time Green's functions that propagate electron and photon amplitudes between space–time points. The electron propagator embodies the amplitude for an electron to travel from one point to another, carrying spinor structure and relativistic phase factors. The photon propagator encodes the propagation of the electromagnetic field with its gauge structure and causal prescription.
A small imaginary term in the energy denominator is used to enforce causal boundary conditions, giving the propagators their familiar "Feynman" form. These propagators become the basic building blocks of calculations: internal lines in a graph correspond to propagators connecting vertices.

Diagrammatic rules and Feynman diagrams
A concise set of diagrammatic rules emerges for translating a space–time graph into a mathematical amplitude: assign propagators to internal lines, wavefunctions or polarization vectors to external lines, vertex factors tied to the charge and Dirac matrices at each interaction point, and integrate over the space–time positions of vertices. Symmetry and combinatorial factors account for identical configurations and closed loops.
The diagrams serve both as intuitive pictures of processes and as precise bookkeeping devices that greatly simplify perturbative calculations. They make manifest conservation laws at vertices and provide a straightforward route to compute scattering matrices, cross sections, and radiative corrections.

Antiparticles, loops, and renormalization
A striking interpretive insight identifies antiparticles as particles propagating backward in time, a simple space–time rule that explains crossing symmetry and certain sign conventions. Closed electron loops and multiple-vertex configurations naturally appear in the diagrammatic expansion and produce divergent integrals when summed over intermediate momenta.
The necessity of redefining physical parameters, mass and charge, becomes apparent, and the method accommodates renormalization procedures to absorb infinities into measured quantities. Concrete calculations using the new rules reproduce and organize previously puzzling radiative corrections, including contributions relevant to the Lamb shift and the electron magnetic moment.

Legacy and impact
The space–time approach crystallizes a method that becomes foundational in quantum field theory: Feynman diagrams and propagator techniques turn intractable operator calculations into tractable algebraic integrals tied to simple pictures. The combination of intuitive clarity and calculational power reshaped particle physics, enabling systematic higher-order computations and informing the development of gauge theories and modern renormalization methods.
Beyond practical computation, the conceptual shift toward summing histories and reading antiparticles as time-reversed particles deepened understanding of relativistic quantum processes and remains central to how quantum field theory is taught and applied.