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arXiv:hep-th/9304036 · Physica Scripta

Quantum phase and quantum phase operators: some physics and some history

Michael Martin Nieto

Published Jan 1, 1993• 61 citations
arXivHTMLPDF

Abstract

After reviewing the role of phase in quantum mechanics, I discuss, with the aid of a number of unpublished documents, the development of quantum phase operators in the 1960's. Interwoven in the discussion are the critical physics questions of the field: are there (unique) quantum phase operators and are there quantum systems which can determine their nature? I conclude with a critique of recent proposals which have shed new light on the problem.

DOI
10.1088/0031-8949/1993/t48/001

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After reviewing the role of phase in quantum mechanics, I discuss, with the aid of a number of unpublished documents, the development of quantum phase operators in the 1960's. Interwoven in the discussion are the critical physics questions of the field: are there (unique) quantum phase operators and are there quantum systems which can determine their nature? I conclude with a critique of recent proposals which have shed new light on the problem.
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.
Quantum mechanics, one of the most successful theories in the history of science, was created to account for physical systems not describable by classical physics. Though it is consistent with all experiments conducted thus far, many of its core concepts (amplitudes, global phases, etc.) can not be directly accessed and its interpretation is still the subject of intense debate, more than 100 years since it was introduced. So, a fundamental question is why this particular mathematical model is the one that nature chooses, if indeed it is the correct model. In the past two decades there has been a renewed effort to determine what physical or informational principles define quantum mechanics. In this paper, recent attempts at establishing reasonable physical principles are reviewed and their degree of success is tabulated. An alternative approach using joint quasi-probability distributions is shown to provide a common basis of representing most of the proposed principles. It is argued that having a common representation of the principles can provide intuition and guidance to relate current principles or advance new principles. The current state of affairs, along with some alternative views are discussed.
This is a brief review of the history and development of quantum theories. Starting from the experimental findings and theoretical results which marked the crisis of the classical framework, I overview the rise of axiomatic quantum mechanics through matrix and wave mechanics. I discuss conceptual problems such as the measurement problem that led scientific realists to explore other, more satisfactory, quantum theories, as well as Bell’s theorem and quantum nonlocality, concluding with a short review of relativistic theories.
This is the story of David Bohm's hitherto unpublished notes, uncovered from a series of lectures given in Israel in 1957 to a group of Marxist physicists from the Hashomer Hatsair (young guard) movement. The notes explore quantum theory and dialectical materialism, as well as shedding light on the intellectual and emotional resources of his persistent rebellion against orthodoxies in science. The discussion places his lectures notes in a wider historical, scientific, and philosophical context and traces Bohm's intellectual development from an early attempt to better understand the Copenhagen interpretation of quantum mechanics to his elaboration of an alternative causal interpretation based on the notion of "hidden variables." As the lecture notes strikingly illustrate, dialectical materialism played a fundamental role for the further development of his understanding of quantum mechanics. Bohm placed the notions of infinity of nature and the unity of opposites in the center of his natural philosophy, drawing from the works of Hegel, Marx, Engels, and Lenin. The authors also examine Bohm's legacy in the history of quantum mechanics and his contribution to preparing the "second quantum revolution." Bohm's engagement with dialectical materialism and his role as a dissident to the orthodox interpretation of quantum mechanics were pivotal in bringing about the novel insights and technologies associated with this second quantum revolution. During the 1960s, Bohm developed a growing interest in the nature of consciousness and engaged with the philosopher Jiddu Krishnamurti. In spite of a seemingly fundamental break with earlier philosophical positions, this new phase in Bohm's intellectual odyssey was not completely unrelated to his fascination with and deep understanding of Hegel's philosophy, which the lecture notes amply document for the first time.
It is generally thought that, in the same way that position and momentum are canonical variables in quantum mechanics, the number operator should have a canonical operator, i.e., there should be a possibility of defining a well-defined phase operator associated with it. From the point of view of the phase space, it becomes clearer as position and momentum uncertainties are well reflected. Indeed, the same occurs in the phase space for phase and intensity. In this contribution, we do a small survey on the possibilities of defining a phase operator via discrete Fourier operators and introduce some possible applications.