Propagators & Three Calculi

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|amplitude| ≈ — (relative, illustrative scale)

Exchange between two spacetime vertices

Separation |x−y| 4.0

Mass m (electron mode) 0.5

D_F^μν(q) = −i g^μνq² + iε (momentum space, Feynman gauge)

In configuration (position) space the massless photon propagator behaves like 1 / (x−y)², and a massive propagator falls off like e^{−m r}/r at large spacelike r (Yukawa decay). The white curve above the grid plots this falloff; the marker sits at your chosen separation. Slide the mass to watch the electron amplitude die faster — the photon (m = 0) has infinite range.

Why the "product of propagators" is a trap

Position-space propagators are distributions (generalized functions), singular on the light cone x² = 0. A loop diagram multiplies propagators at coincident points — e.g. the electron self-energy involves S_F(x−y)·D_F(x−y). But in Schwartz's theory of distributions, products of distributions at the same point are not defined in general — squaring a delta function is meaningless.

This is precisely why loop integrals in QED diverge.
Physics answers with regularization + renormalization: deform the theory (cutoff, dimensional regularization), absorb infinities into redefined mass and charge.
Rigorous alternatives exist: Epstein–Glaser causal perturbation theory extends the distribution products carefully instead of subtracting infinities.

So the honest answer on any foundation of calculus: the naive product is ill-defined; the physically meaningful object requires an extra prescription.

Three foundations of calculus

Newton's fluxions (1660s–70s): quantities "flow" in time; the fluxion ẋ is a velocity of change, computed via vanishing increments ("moments") — powerful but criticized by Berkeley (1734) as "ghosts of departed quantities" for lacking rigor about the vanishing step.
Limits & ε–δ (Cauchy, Weierstrass, 19th c.): derivative as a limit over the completed real-number continuum; distributions (Schwartz, 1945) extend this to objects like δ(x) — but forbid arbitrary products of them.
Ganita / computational tradition (India): Aryabhata (499 CE) built sine tables by finite differences; the Kerala school — Madhava (c. 1340–1425) — derived infinite series for π (π/4 = 1 − 1/3 + 1/5 − …) and for sine/cosine, centuries before Newton, treating them as computational procedures with explicit correction (error) terms rather than completed limits. Advocates of "zeroism" (notably C. K. Raju) argue such a finite, algorithmic calculus discards negligible terms by convention — a philosophy closer to how computers and lattice field theory actually work.

Quiz: which foundation says what? score 0/3

1. "The derivative is defined by an ε–δ limit over the real continuum."

2. "π/4 = 1 − 1/3 + 1/5 − … was derived here first, with explicit finite correction terms."

3. "Quantities flow in time; derivatives are velocities, computed with vanishing moments."

Answering also highlights the matching era-flag on the 3D grid.

Reading the 3D scene

Grid: a slice of spacetime (one space axis + one time axis).
Glowing spheres: interaction vertices x and y where electron lines meet the photon.
Wavy line: the exchanged photon (standard Feynman-diagram notation); in electron mode it becomes a straight fermion line with an arrow-sense coloring.
White curve: |propagator| vs. separation r — 1/r²-like for the massless photon, e^−mr/r for the massive electron. Values shown are relative estimates for illustration, not absolute amplitudes.
Near r → 0 the curve blows up: that divergence at coincident points is exactly the singular-product problem discussed at left.