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On Hilbert's Sixth Problem

Journal article published in 2013 by B. R. F. Jefferies
This paper is available in a repository.
This paper is available in a repository.

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Abstract

Feynman path integrals are now a standard tool in quantum physics and their use in di↵erential geometry leads to new mathematical insights. A logical treatment of quantum phenomena seems to require a proper analysis of path integrals. The subject is complicated by the fact that their application in flat space-time is quite di↵erent from how path integrals are used in, say, Topological Quantum Field Theory, where there is no natural notion of time translation. An historical background is given in this paper and a few approaches to Feynman path integrals in the context of nonrelativistic quantum mechanics and scalar quantum fields with polynomial self-interactions are outlined. David Hilbert presented his sixth problem at the Paris conference of the International Congress of Mathematicians, speaking on 8 August, 1900 in the Sorbonne. It roughly calls for the axiomatization of physics. The explicit statement reads: 6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. Things were going swimmingly at the time. War was already on the horizon. Great Britain had close ties with Germany, but Newtonian mechanics and Maxwell's equations were solving problems in the real world from building skyscrapers to transmitting voice over radio. The Navier-Stokes equations were fine for artillery ballistics too. There was not much work to be done writing down the axioms of classical mechanics and fluid mechanics. Much had already been achieved by Newton, d'Alembert, Lagrange, Poisson, Hamilton, Stokes, Gibbs and many others. There were rumblings in physics already. Maxwell's equations are invariant under Poincaré transformations, as noticed by Minkowski in 1907 [45, 46]. The Michelson-Morley experiment seemed to indicate strange viscous behaviour of the ether, governed by Lorentz transformations. But even Einstein's special theory of relativity produced in his annus mirabilus 1905 1 can be reduced to a few simple principles: Einstein himself wrote them down. Later, in 1915, during actual war, the General Theory of Relativity came in the form of simply formulated mathematical principles — as long as you had di↵erential geometry under your belt. Meanwhile, the French, with their usual devotion to higher ab-stractions like patriotism and valor, sent many of their promising young mathematicians to be slaughtered in the trenches 2 . Einstein balked at the new quantum theory, but the quantum mechanics of nonrela-tivistic particles is also subject to mathematical axioms, essentially formulated by J. von Neumann [57] in 1932 using the then recent techniques of functional analysis and operator theory. Heisenberg's oracular 'matrix mechanics' made sense after all. The combination Date: July 12, 2013. 2000 Mathematics Subject Classification. Primary 47A60; Secondary 47A13 47N50 60H25.