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arXiv:1505.08102

Functional Integral Approach to $C^*$-algebraic Quantum Mechanics I: Heisenberg and Poincaré

J. LaChapelle

Published May 27, 2015• 0 citations
arXivHTMLPDF

Abstract

The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous $C^\ast$-algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral representations of $C^\ast$-algebras. The merger achieves a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role. To illustrate the formalism, we apply the framework to non-relativistic and relativistic quantum mechanics via the Heisenberg and Poincaré groups.

DOI
10.48550/arxiv.1505.08102

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Functional Integral Approach to C∗C^*C∗-algebraic Quantum Mechanics I: Heisenberg and Poincaré
The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous C∗C^\astC∗-algebraic formulation of the theory's axioms. Conversely, the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a notion of functional integration based on topological groups that allows to construct functional integral representations of C∗C^\astC∗-algebras. The merger achieves a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role. To illustrate the formalism, we apply the framework to non-relativistic and relativistic quantum mechanics via the Heisenberg and Poincaré groups.
Functional Integral Approach to C∗C^*C∗-algebraic Quantum Mechanics
The algebraic approach to quantum mechanics has been key to the development of the theory since its inception, and the approach has evolved into a mathematically rigorous C∗C^\astC∗-algebraic formulation of the axioms. Conversely the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a recently developed notion of functional integration that allows to construct functional integral representations of C∗C^\astC∗-algebras. The merger is a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various concepts of field theory, namely stationary phase analysis, the Euclidean formulation of field theory, instanton techniques, and the role of topology in field theory, are introduced in this chapter. The path integral: general formalism Broadly speaking, there are two approaches to the formulation of quantum mechanics: the “operator approach” based on the canonical quantization of physical observables and the associated operator algebra, and the Feynman path integral.
Functional Integral Approach to C∗C^*C∗-algebraic Quantum Mechanics II: Symplectic Quantum Mechanics
We propose Sp (8,R)Sp\,(8,\R)Sp(8,R) and its Langlands dual SO(9,R)SO(9,\R)SO(9,R) as dynamical groups for closed quantum systems. Restricting here to the non-compact group Sp (8,R)Sp\,(8,\R)Sp(8,R), the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.
This paper establishes a complete and mathematically rigorous framework for path integrals in quantum mechanics through the novel concept of path integral algebraic closure. We provide comprehensive definitions, explicit combinatorial constructions, and exhaustive error analysis for path integral computations, with particular attention to measure-theoretic foundations, functional analytic properties, and combinatorial structures.We provide complete derivations of combinatorial coefficients Γ(n)m with exhaustive verification for degrees 3 through 12 and proof of optimality, comprehensive error bounds with explicit constants derived from first principles, and extensive numerical validation with rigorous error control. The framework bridges algebraic methods with functional analysis and measure theory,providing both deep theoretical insights and practical computational tools for path integrals in quantum theory.This work addresses long-standing challenges in the mathematical foundations of path integrals, including the well-definedness of iterative path integrals, the rigorous treatment of conditional Wiener measures, the explicit construction of spectral decompositions on path space, and the systematic control of all approximation errors. The proposed algebraic closure framework is shown to be a proper extension of traditional methods, achieving exponential convergence for polynomial potentials and providing a solid foundation for extensions to quantum field theory and other emerging applications.
I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.
Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.