The Beer Party in Princeton

Dirac's Mysterious Phrase and the Sum Over Every Possible Path

Smoke and conversation filled the oak-paneled dining room of the Nassau Tavern in Princeton, New Jersey, during an autumn evening in 1941. Princeton graduate student Richard Feynman had walked down from Fine Hall looking for anyone who knew how to quantize a classical system that lacked a Hamiltonian. Working with his thesis advisor John Archibald Wheeler, Feynman had developed an absorber theory of electrodynamics. In their model, electrons interacted directly across time without an independent electromagnetic field. Their classical action eliminated the infinite self-energy of a point charge by combining half-retarded waves moving into the future with half-advanced waves moving into the past.

The catch was that their action depended on the positions of particles at two different times. Because there was no instantaneous state of the field at a single moment, Feynman could not write down a Hamiltonian operator $H$. Without a Hamiltonian, both Erwin Schrödinger's wave equation and Werner Heisenberg's matrix mechanics were useless.

At the tavern table sat Herbert Jehle, a German physicist who had recently escaped wartime Europe after surviving French internment camps. Feynman sat down next to him and explained his bottleneck. Did anyone know a way to step a quantum wave function forward in time using the Lagrangian $L$ or the action $S$ directly? Jehle raised his hand. He recalled a short nine-page paper published in 1933 by the British theorist Paul Dirac in the Soviet journal Physikalische Zeitschrift der Sowjetunion.

The next morning, Feynman and Jehle pulled the bound volume from the Fine Hall library shelves. In that 1933 paper, Dirac examined how a quantum wave function evolves across a brief time interval $\epsilon$. Dirac observed that the quantum transformation function connecting states at time $t$ and time $t + \epsilon$ was analogous to a specific phase factor. That factor was the complex exponential of the classical Lagrangian multiplied by $\epsilon$ and divided by Planck's reduced constant, $\exp(i\epsilon L/\hbar)$.

Standing at a Fine Hall blackboard with chalk in hand, Feynman decided to test what Dirac's cautious English phrase meant in practice. What if analogous meant strictly proportional? He wrote down the integral carrying a wave function $\psi(x_1, t)$ to $\psi(x_2, t + \epsilon)$ by multiplying by a normalization constant $A$ and the phase factor $\exp\left(\frac{i\epsilon}{\hbar} L\right)$. He substituted the standard Lagrangian $L = \frac{1}{2}m\left(\frac{x_2 - x_1}{\epsilon}\right)^2 - V(x)$ and expanded the integral in powers of $\epsilon$.

Within minutes, the Gaussian integral over the displacement $x_2 - x_1$ fixed the constant $A = \sqrt{\frac{m}{2\pi i\hbar\epsilon}}$. It also reproduced Schrödinger's partial differential equation term for term. Feynman turned toward Jehle in surprise, realizing that what Dirac had cautiously called analogous was in fact strictly equal up to a simple proportionality constant in front.

By slicing a finite time interval $t_b - t_a$ into thousands of infinitesimal steps $\epsilon$ and integrating over every intermediate position $x(t_k)$, Feynman constructed his sum-over-histories formulation of quantum mechanics. The total transition kernel $K(b,a)$ for a particle to travel from spacetime point $a$ to point $b$ is the sum over all continuous paths $x(t)$:

Here the puzzle that Abram Bader had shown him at Far Rockaway High School found its physical mechanism. A quantum electron does not choose a single trajectory in advance. Instead, it travels along every mathematical path simultaneously. Each path carries a tiny unit arrow in the complex plane whose phase angle $\theta = S[x(t)]/\hbar$ rotates at a speed set by the classical action.

Because $\hbar$ is tiny ($1.05 \times 10^{-34}$ joule-seconds), two paths far from the classical parabola have actions that differ by millions of radians. Their phase arrows point in random directions and cancel out by destructive interference. Only inside a narrow bundle of paths around the classical trajectory—where the first variation $\delta S = 0$—do neighboring histories share nearly identical actions. Those aligned arrows add together by constructive interference to build the macroscopic world. While Jehle copied the derivation furiously into his notebook, Dirac himself later remarked at the 1946 Princeton Bicentennial that he had never imagined the phase relation could be turned into a complete calculus of quantum mechanics.