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Journal of Mathematical Physics

Transition amplitude spaces

Stanley Gudder, Sylvia Pulmannová

Published Feb 1, 1987• 10 citations
Publisher

Abstract

Motivated by physically plausible axioms, the concept of a transition amplitude space (tas) is defined. Various connections between this framework and other axiomatic approaches to the foundations of quantum mechanics are derived. In particular, transition probability spaces, the algebraic approach, the operational statistics and quantum logic approaches, and traditional Hilbert space quantum mechanics are considered. It is shown that a tas always admits a Hilbert space representation. Results are obtained concerning isomorphisms and automorphisms for tas’s. The concept of an A-form is introduced and the relationship between certain A-forms and bounded linear operators on a Hilbert space are studied. Finally, it is shown that sums and tensor products of tas’s can be formulated in a natural way.

DOI
10.1063/1.527669

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The investigation of transition amplitude spaces (tas’s) introduced by Gudder and Pulmannová [J. Math. Phys. 28, 376 (1987)] is continued. In particular, ordered structures related to tas’s are considered. Under some conditions, which are analogous to the conditions obtained for transition probability spaces by Pulmannová [J. Math. Phys. 27, 1791 (1986)], the ordered structure related to a tas can be represented by the orthocomplemented lattice of all f-closed subspaces of a generalized Hilbert space (𝒱,𝒟,θ, f ). It is shown that, provided that the above representation takes place for a total tas, the division ring 𝒟 must be isomorphic with a subfield C1 of the field of complex numbers C. Sufficient conditions are also given under which the ordered structure of a tas can be represented by the lattice of all closed subspaces of a complex Hilbert space.

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Motivated by physically plausible axioms, the concept of a transition amplitude space (tas) is defined. Various connections between this framework and other axiomatic approaches to the foundations of quantum mechanics are derived. In particular, transition probability spaces, the algebraic approach, the operational statistics and quantum logic approaches, and traditional Hilbert space quantum mechanics are considered. It is shown that a tas always admits a Hilbert space representation. Results are obtained concerning isomorphisms and automorphisms for tas’s. The concept of an A-form is introduced and the relationship between certain A-forms and bounded linear operators on a Hilbert space are studied. Finally, it is shown that sums and tensor products of tas’s can be formulated in a natural way.
The investigation of transition amplitude spaces (tas’s) introduced by Gudder and Pulmannová [J. Math. Phys. 28, 376 (1987)] is continued. In particular, ordered structures related to tas’s are considered. Under some conditions, which are analogous to the conditions obtained for transition probability spaces by Pulmannová [J. Math. Phys. 27, 1791 (1986)], the ordered structure related to a tas can be represented by the orthocomplemented lattice of all f-closed subspaces of a generalized Hilbert space (𝒱,𝒟,θ, f ). It is shown that, provided that the above representation takes place for a total tas, the division ring 𝒟 must be isomorphic with a subfield C1 of the field of complex numbers C. Sufficient conditions are also given under which the ordered structure of a tas can be represented by the lattice of all closed subspaces of a complex Hilbert space.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states and to transitions between quantum effects. We also present their dependence on a measured operation or instrument. We begin by defining our concepts on a general quantum effect algebra. These concepts are illustrated using Holevo operations and instruments. We then present transition probabilities in the special case of the Hilbert space formulation of quantum mechanics. We show that for pure states and particular types of operations the transition probabilities reduce to their usual form. We give examples in terms of Lüders operations and instruments.