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arXiv:math-ph/0612082 · Journal of Mathematical Physics

An approach to nonstandard quantum mechanics

A. Raab

Published Nov 19, 2004• 13 citations
arXivHTMLPDF

Abstract

We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite-dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus be treated on an equal footing. We show that the formalism extends the standard formulation of quantum mechanics. To this end we develop the Loeb-function calculus in nonstandard hulls. The idea is to perform calculations in a hyperfinite-dimensional space, but to interpret expectation values in the corresponding nonstandard hull. We further apply the framework to nonrelativistic quantum scattering theory. For time-dependent scattering theory, we identify the starting time and the finishing time of a scattering experiment, and we obtain a natural separation of time scales on which the preparation process, the interaction process, and the detection process take place. For time-independent scattering theory, we derive rigorously explicit formulas for the Mo/ller wave operators and the S-matrix.

DOI
10.1063/1.1812358

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We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite-dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus be treated on an equal footing. We show that the formalism extends the standard formulation of quantum mechanics. To this end we develop the Loeb-function calculus in nonstandard hulls. The idea is to perform calculations in a hyperfinite-dimensional space, but to interpret expectation values in the corresponding nonstandard hull. We further apply the framework to nonrelativistic quantum scattering theory. For time-dependent scattering theory, we identify the starting time and the finishing time of a scattering experiment, and we obtain a natural separation of time scales on which the preparation process, the interaction process, and the detection process take place. For time-independent scattering theory, we derive rigorously explicit formulas for the Mo/ller wave operators and the S-matrix.
Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.
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 Schrödinger equation.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.
We will extend Nonrelativistic Quantum Mechanics as a theory in $L\sp2$ to a theory in the space of distributions. We will provide 3 major theories of Nonrelativist Quantum Mechanics. First, we will extend the concept of an integral kernel for the evolution operator to a distribution kernel for the $L\sp2$ transition probability amplitude. Second, we will extend the $L\sp2$ Schrodinger's equation to a distributions Schrodinger's equation. Lastly, we will rigorously prove that; Feynman's original formulation of the real time, time-sliced path integral is well defined when formulated on the $L\sp2$ transition probability amplitude.
The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models. The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics.
The aim of this thesis is to present a reasonable account of some of the applications that functional analysis, or more specifically operator theory, has in the theory of quantum scattering. We begin by introducing the physical scattering experiment as well as establishing the standard framework for discussing quantum theory, that of a Hilbert space. This leads to a discussion of the properties of operators acting in a Hilbert space and the role played by unbounded selfadjoint operators in quantum physics. We also spend some time discussing the position, momentum and energy operators which are of most relevance here. This in turn leads logically to the problem of formulating the scattering problem mathematically using these operators. The stationary equations of scattering theory are also rigorously derived and their suitability for two-body scattering problems is proven. We show that this analysis does not extend to three-body scattering problems, and introduce the Faddeev formulation for three-body systems to supplement the theory. This thesis concludes with a brief overview of how our scattering equations can be modified to take into account effects such as particle spin and identical particles, before discussing an example of a three-body scattering problem (incorporating these features) and deriving a mathematically satisfactory set of equations which are amenable to numerical solution.