Abstract
In this excursion, which is inspired by Albeverio et al. [3, Sect. 6.6] and the classical article by Nelson [58], we give another demonstration of the usefulness of radically elementary mathematics in mathematical physics, by providing a rigorous, radically elementary definition of Feynman path integrals in Minimal Internal Set Theory. A summary of these ideas—combined with a brief introduction to radically elementary mathematics for mathematical physicists and some references to previous attempts at formalizing the Feynman path integral by means of nonstandard analysis—can be found in [35].
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An analysis of the proof of the Lie–Trotter product formula (cf. e.g. Nelson [58, Appendix B, proof of Theorem 9]) shows that the condition which n has to meet is
$$\frac{\exp \left (-\frac{\mathrm{i}} {\hslash } \frac{t} {n}{H}_{0}\right )\exp \left (-\frac{\mathrm{i}} {\hslash } \frac{t} {n}V \right )\psi - \psi + \frac{\mathrm{i}} {\hslash } \frac{t} {n}H\psi } {t/n} \simeq 0.$$At least when we impose limited bounds on m, V and ψ, this holds for all nonstandard n.
References
Albeverio, S., Høegh-Krohn, R., Fenstad, J., Lindstrøm, T.: Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, vol. 122. Academic, Orlando, FL (1986)
Herzberg, F.: Radically elementary mathematics and the Feynman path integral. Manuscript (2012)
Nelson, E.: Feynman integrals and the Schrödinger equation. J. Math. Phys. 5(3), 332–343 (1964)
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Herzberg, F.S. (2013). Excursion to Mathematical Physics: A Radically Elementary Definition of Feynman Path Integrals. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_8
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DOI: https://doi.org/10.1007/978-3-642-33149-7_8
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