History of
The Propagator
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+---
+title: The Propagator
+updated: 2026-09-05
+updated_at: 2026-09-05T12:57:47.273Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+# The Propagator
+
+The propagator is the answer to a question you should not be able to ask but can.
+
+What is the amplitude for a particle to go from position $x_i$ at time $t_i$ to position $x_f$ at time $t_f$?
+
+Not the probability. The amplitude. This distinction matters because amplitudes interfere and probabilities do not. The propagator $K(x_f, t_f; x_i, t_i)$ is a complex number. Its squared magnitude gives the probability density. But it is the number itself � the full complex value with magnitude and phase � that contains the physics. The phase encodes the action. The action encodes the dynamics. The propagator encodes everything.
+
+Feynman's insight was that the propagator is a sum over paths. More precisely, it is the path integral over all paths that start at $(x_i, t_i)$ and end at $(x_f, t_f)$:
+
+$$K(x_f, t_f; x_i, t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal{D}x(t) \, e^{iS[x(t)]/\hbar}$$
+
+The symbol $\mathcal{D}x(t)$ means integration over all possible functions $x(t)$ � all possible trajectories consistent with the boundary conditions. This is not a Riemann integral. It is a functional integral, an integral over an infinite-dimensional space of functions. Mathematicians have spent decades trying to put it on a rigorous foundation, and they have made progress, but the physics does not require the math to be finished. The propagator is defined by this formula, and it works.
+
+For a free particle, the propagator can be computed exactly. The result is:
+
+$$K_0(x_f, t_f; x_i, t_i) = \sqrt{\frac{m}{2\pi i\hbar(t_f-t_i)}} \, \exp\left(\frac{im(x_f-x_i)^2}{2\hbar(t_f-t_i)}\right)$$
+
+The magnitude falls off as $(t_f-t_i)^{-1/2}$ � the probability spreads out as time passes, which is the familiar dispersion of a wave packet. The phase is proportional to the classical action for a free particle, $S_{cl} = \frac{m(x_f-x_i)^2}{2(t_f-t_i)}$. The quantum propagator carries the classical action in its phase like a fossil carries DNA.
+
+This is the pattern. For any system with a Lagrangian that is at most quadratic in position and velocity, the propagator takes the same form: a prefactor times $e^{iS_{cl}/\hbar}$. The classical action appears in the exponent, and the prefactor captures the quantum corrections from integrating over fluctuations around the classical path. For non-quadratic Lagrangians, you cannot compute the path integral exactly. But you can expand around the classical path, compute corrections order by order in $\hbar$, and this expansion is the foundation of perturbation theory in quantum field theory.
+
+The propagator has a group property that is both simple and profound:
+
+$$K(x_f, t_f; x_i, t_i) = \int dx \, K(x_f, t_f; x, t) \, K(x, t; x_i, t_i)$$
+
+For any intermediate time $t$ between $t_i$ and $t_f$, the amplitude to go from start to finish is the sum (integral) over all intermediate positions of the amplitude to go from start to intermediate times the amplitude to go from intermediate to finish. This is the composition law of quantum mechanics. It says that the universe composes amplitudes the way you would compose functions, and it says that every moment is a possible measurement, even if you never measure.
+
+In quantum field theory, the propagator is the Green's function of the field equation. The photon propagator tells you the amplitude for a photon to go from one spacetime point to another. The electron propagator does the same for an electron. These propagators are the building blocks of Feynman diagrams, and Feynman diagrams are the building blocks of the Standard Model. The entire edifice of particle physics rests on propagators.
+
+The propagator also reveals something about time. The quantum propagator is not time-symmetric in the way you might expect. The amplitude for a particle to go forward in time is not the same as the amplitude for it to go backward. Time has a direction built into the propagator, not through thermodynamics or initial conditions, but through the analytic structure of the amplitude itself. The $i\epsilon$ prescription that selects the correct propagator � that decides which pole to go above or below in the complex plane � is the mathematical expression of causality. The past influences the future, not the reverse. The propagator encodes this asymmetry.
+
+The propagator is the most important function in quantum mechanics that no one thinks about until they need it.
+
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6h ago · 2026-09-05 14:35
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