The space of state vectors: A hyperfinite approach

• Type: Article in International Scientific Journal
• Date: 2004

Authors

Almeida, J (Author) Other; Teixeira, J (Author) Other

Journal

Title: Journal of Mathematical Physics

Vol. 45

Pages: 1-20

ISSN: 0022-2488

Publisher: American Institute of Physics

Indexing

ISI Web of Knowledge - 6 Citations

Scopus - 3 Citations

Other information

Authenticus ID: P-000-D31

DOI: 10.1063/1.1631394

Abstract (EN): We present a version of the formalism of Dirac based on nonstandard analysis, allowing us to deal with state vectors and operators using the resources of finite-dimensional linear algebra. The space of state vectors is a nonstandard Hilbert space with hyperfinite dimension, which includes all square-integrable functions, together with vectors representing states of definite position or momentum. Every vector is normalizable, even when its norm is infinite. Observables are represented by Hermitian operators, which are always (hyper)bounded and defined on the whole space. The connection with the standard theory is established by postulating the existence of "hyper-observables" and nonstandard states. Each observable in the usual sense appears as a kind of standard-scale approximation of some hyper-observable. We show that the probabilistic predictions are consistent with those of the standard theory. Consistency extends to time evolution, in the sense that if an initial nonstandard state is "near-standard," then the state after a finite time shall be infinitely near the standard state obtained through the Schrodinger equation. (C) 2004 American Institute of Physics.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 20