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  • Cited by 5
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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Hörmann, Günther 2018. Limits of Regularizations for Generalized Function Solutions to the Schrödinger Equation with ‘Square Root of Delta’ Initial Value. Journal of Fourier Analysis and Applications, Vol. 24, Issue. 4, p. 1160.

    Perlis, Don 2016. Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Vol. 10, Issue. , p. 243.

    Albeverio, S. Cianci, R. and Khrennikov, A. Yu. 1997. Representation of a quantum field Hamiltonian inp-adic Hilbert space. Theoretical and Mathematical Physics, Vol. 112, Issue. 3, p. 1081.

    Альбеверио, Сержио А Albeverio, Sergio A Хренников, Андрей Юрьевич Khrennikov, Andrei Yur'evich Чанчи, Р and Cianci, R 1997. Представление гамильтониана квантового поля в $p$-адическом гильбертовом пространстве. Теоретическая и математическая физика, Vol. 112, Issue. 3, p. 355.

    Albeverio, Sergio and Wu, Jiang-Lun 1996. A mathematical flat integral realization and a large deviation result for the free Euclidean field. Acta Applicandae Mathematicae, Vol. 45, Issue. 3, p. 317.

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  • Print publication year: 1988
  • Online publication date: June 2012

APPLICATIONS OF NONSTANDARD ANALYSIS IN MATHEMATICAL PHYSICS

Summary

Abstract. The aim of this article is to give a short introduction to applications of nonstandard analysis in mathematical physics. Two basic techniques, the hyperfinite and the hypercontinuous are presented, together with illustrations mainly from quantum mechanics, polymer physics and quantum field theory.

INTRODUCTION

Nonstandard analysis is a specific mathematical technique as well as a way of thinking: both aspects are also represented in the interaction between nonstandard analysis and mathematical physics, which is the subject of this paper. As in other domains of application of nonstandard analysis, mathematical physics (or the mathematical study of problems of physics) has particular aspects that make some of the nonstandard methods most natural to use. Often in mathematical physics one has to study systems with many interacting components, idealized as systems with infinitely many degrees of freedom. To look upon a fluid or a gas as a composed of infinitely many particles might seem at first sight to be a very rough abstraction, but is a useful one for mathematical purposes, being in some sense easier to handle than the more realistic case of finitely many particles. On the other hand, in quantum field theory, for example, the abstraction itself creates its own problems, like the famous ones connected with divergences, about which we will say a few more words below; sometimes also it is only in a limit, like that of infinitely many degrees of freedom, that one “sees” some specific phenomenon, raising challenging problems, like phase transitions in thermodynamic systems (only perceived in the so called ”infinite volume“ or “thermodynamic limit”),or exact invariance properties (under a continuous group of symmetries), in systems idealized as “continua” (as in field theories).

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Nonstandard Analysis and its Applications
  • Online ISBN: 9781139172110
  • Book DOI: https://doi.org/10.1017/CBO9781139172110
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