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International Journal of Modern Physics B

MATHEMATICAL ASPECTS OF QUANTUM PHASE

Daniel A. Dubin, Mark A. Hennings, T.B. SMITH

Published Sep 15, 1995• 16 citations
Publisher

Abstract

We consider the current state of the quest for a quantum phase operator, which started in the earliest days of quantum mechanics.4–7 Particular emphasis has been placed on analysis of the structure of the several distinct theories, both physical and mathematical, which has led us from classical function theory to the quantum theory of measurement.

DOI
10.1142/s0217979295000987

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We consider the current state of the quest for a quantum phase operator, which started in the earliest days of quantum mechanics.4–7 Particular emphasis has been placed on analysis of the structure of the several distinct theories, both physical and mathematical, which has led us from classical function theory to the quantum theory of measurement.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
This manuscript presents a conservative mathematical framework investigating whether observable multiplicity and distinguishability in physical theories can emerge from a more primitive, undifferentiated starting point. A singular foundational state SSS is defined as a minimal-ontology entity lacking internal partitions, while interpretation operators III are introduced as structural maps that induce differentiated state spaces M=I(S)M = I(S)M=I(S). The paper develops the formal properties of these operators using moderate operator algebra, category theory, and structural analysis, and situates the framework alongside established programs in quantum reconstruction, operator-algebraic foundations, information-theoretic approaches, and relational quantum mechanics. Connections are drawn between interpretive non-commutativity and contextuality, and between induced multiplicity and the emergence of classical state identity in measurement theory. This is an exploratory theoretical contribution intended to motivate further mathematical development. It does not propose empirical modifications to quantum mechanics, but provides a structural model for understanding how distinctions, state-space geometry, and observable outcomes may be generated rather than postulated. The framework may have relevance for foundational studies in quantum information, measurement, generalized probabilistic theories, and the structure of operator families underlying physical state differentiation.