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Quantum optical phase, rigged Hilbert spaces and complementarity.

Khai Bordon¹, Joan A Vaccaro

¹School of Science, Centre for Quantum Dynamics, Griffith University, Nathan, Australia.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|December 24, 2024

Summary

This summary is machine-generated.

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Loudon's quantum phase theory inspired new formalisms. Pegg and Barnett developed a quantum phase formalism using extended Hilbert spaces, resolving ambiguities in quantum-classical correspondence.

Area of Science:

Quantum optics
Quantum mechanics
Mathematical physics

Background:

Loudon's quantum treatment of optical phase.
The quantum theory of light.
Historical context of quantum phase.

Purpose of the Study:

To place Loudon's quantum phase theory in historical context.
To outline research inspired by Loudon's work.
To explain the development of the Pegg-Barnett quantum phase formalism.

Main Methods:

Historical analysis of quantum phase theory.
Examination of the Pegg-Barnett quantum phase formalism.
Investigation of extended rigged Hilbert spaces.
Identification of complementarity structures.

Keywords:

Pegg–Barnett phase formalism complementarity quantum phase operator rigged hilbert spaces

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Last Updated: May 8, 2025

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Main Results:

Loudon's work inspired the Pegg-Barnett formalism.
The formalism rigorously defines phase operators and eigenstates.
An extended rigged Hilbert space supports limits of phase operators.
A complementarity structure provides quantum-classical correspondence free of ambiguity.

Conclusions:

Pegg and Barnett's formalism offers a rigorous approach to quantum phase.
The formalism resolves ambiguities in quantum-classical correspondence.
The underlying complementarity structure is key to this advancement.