## The Lagrangian in Quantum Mechanics

*"Dirac's idea served as an explosive release in Feynman's imagination…"*

FIRST PRINTING IN ORIGINAL WRAPPERS of Dirac's critically important paper; the inspiration for Richard Feynman's path-integral formalism. SCARCE.

In 1941, "Feynman went to a beer party at the Nassau Tavern. He sat with a physicist lately arrived from Europe, Herbert Jehle… Jehle asked Feynman what he was working on. Feynman explained and asked in turn whether Jehle knew of any application of the least-action principle in quantum mechanics.

"Jehle certainly did. He pointed out that Feynman's own hero, Dirac, had published a paper on just that subject eight years before. The next day Jehle and Feynman looked at it together in the library. It was short. They found it, "The Lagrangian in Quantum Mechanics," in the bound volumes of *Physikalische Zeitschrift der Sowjetunion*, not the best-read of journals. Dirac had worked out the beginnings of a least-action approach in just the style Feynman was seeking, a way of treating the probability of a particle's entire path over time. Dirac considered only one detail, a piece of mathematics for carrying the wave function… forward in time by an infinitesimal amount, a mere instant. Infinitesimal time did not amount to much, but it was the starting point of the calculus...

"In the past eight years neither Dirac nor any other physicist had been able to follow up on the notion of a Lagrangian in quantum mechanics-- a way of expressing a particle's history in terms of quantity of action. Now Dirac's idea served as an explosive release in Feynman's imagination. The uneasy elements of quantum mechanics broke loose and rearranged themselves into a radically new formulation. Where Dirac had pointed the way to calculating how the wave function would evolve in an infinitesimal slice of time, Feynman needed to carry the wave function farther, through finite time. A considerable barrier separated the infinitesimal from the finite. Making use of Dirac's infinitesimal slice required a piling up of many steps--infinitely many of them. Each step required an integration, a summing of algebraic quantities. In Feynman's mind a sequence of multiplications and compounded integrals took form. He considered the coordinates that specify a particle's position. They churned through his compound integral. The quantity that emerged was, once again, a form of the action. To produce it, Feynman realized, he had to make a complex integral encompassing every possible coordinate through which a particle could move. The result was a kind of sum of probabilities… Feynman summed the contributions of every conceivable path from the starting position to the final position… [He] realized that he had burrowed back to first principles and found a new formulation of quantum mechanics" (Gleick, *Genius*, 128- 132).

Dirac's paper, particularly in original wrappers and without any institutional stamps is extremely rare. Dirac chose to submit it to the new Soviet journal *Physikalische Zeitschrift der Sowjetunion* to show his support for Soviet physics but in doing so ensured that his paper would receive only a very limited circulation.

IN: Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1, 1933. Octavo, original wrappers; custom box. Some chipping to spine; a remarkable survival in original wrappers.

Price: $4,500 .