## The Lagrangian in Quantum Mechanics.

Charkow: Technischer Staatsverlag, 1933.

Extremely rare offprint of this seminal paper which, in the hands of Richard Feynman, gave birth to the path-integral formulation of quantum mechanics and Feynman integrals.

In his Nobel Lecture, Feynman described how he discovered a way to quantize a classical theory given by the principle of least action based upon a Lagrangian, rather than the usual approach via Hamiltonians. Feynman explained the problem to Herbert Jehle, whom he met by chance during a visit of Jehle to Princeton in 1947. Jehle pointed out that Dirac, in the present paper, had given “an infinitesimal time development operator involving the classical Lagrangian. Successive applications of this operator to the initial wave function generated the wave function at any later time, and the wave function was equivalent to finding the solution of the Schrödinger equation. To obtain the wave function after a finite time has elapsed, however, [Feynman realised that] one had to integrate over all possible paths containing two arbitrary space-time points. This, in fact, was the path-integral approach of Feynman” (*Biogr. Mems Fell. R. Soc.Lond*. 48 (2002), p. 107).

“In the autumn of 1932, [Dirac] found another way of [developing quantum mechanics by analogy with classical mechanics], by generalising the property of classical physics that enables the path of any object to be calculated, regardless of the nature of the forces acting on it.

“[At the heart of this technique are two quantities.] The first, known as the Lagrangian, is the difference between an object’s energy of motion and the energy it has by virtue of its location. The second, the so-called ‘action’ associated with the object’s path, is calculated by adding the values of the Lagrangian from the beginning of the path to its end. In classical physics, the path taken by any object between two points in any specified time interval turns out... to be the one corresponding to the smallest value of the ‘action’...

“Dirac thought that the concept of ‘action’ might be just as important in the quantum world of electrons and atomic nuclei as it is in the large-scale domain. When he generalised the idea to quantum mechanics, he found that a quantum particle has not just one path available to it but an infinite number, and they are – loosely speaking – centred around the path predicted by classical mechanics. He also found a way of taking into account all the paths available to the particle to calculate the probability that the quantum particle moves from one place to another...

“Normally, he would submit a paper like this to a British journal, such as the Proceedings of the Royal Society, but this time he chose to demonstrate his support for Soviet physics by sending the paper to a new Soviet journal... Dirac was quietly pleased with his ‘little paper’ and wrote in early November to one of his colleagues in Russia: ‘It appears that all the important things in the classical [...] treatment can be taken over, perhaps in a rather disguised form, into the quantum theory’” (Farmelo, pp. 215-6).

G. Farmelo, *The strangest man*, 1988. Feynman’s discovery of Dirac’s paper, and his derivation of the method of path-integrals from it, is described in his own words in his Nobel Prize address (nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html). Laurie M. Brown, Feynman’s thesis. A new approach to quantum theory, 2005 (this is the first publication of Feynman’s complete Ph.D. thesis; it reprints the present paper of Dirac as an appendix). The journal *Physikalische Zeitschrift der Sowjetunion* is uncommon even in institutional collections (COPAC lists only eight UK holdings).

8vo (229 x 152 mm), offprint from Physikalische Zeitschrift der Sowjetunion, Vol. 3 (1933), pp. 64-72, original printed wrappers, light vertical crease from having been folded for postage, else fine. Very rare.

Item #2920

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Price:
$8,500.00
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