The geometric phase and ray space isometries

2007

In this paper we approach the question of the existence of a (x, p) phase space in a new way. Rather than abandoning all hope of constructing such a phase-space for quantum phenomena, we take aspects from both the Wigner-Moyal and Bohm approaches and show that although there is no unique phase space, we can form `shadow' phase spaces. We then argue that this is a consequence of the non-commutative geometry defined by the operator algebra.

arXiv preprint math/0408233, 2004

Abstract: For simple Lie groups, the only homogeneous manifolds $ G/K $, where $ K $ is maximal compact subgroup, for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric ...

Journal of Physics A: Mathematical and General, 1999

The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.

2013

Wigner's quasi-probability distribution function in phase-space is a special (Weyl-Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics, nuclear physics, quantum computing, decoherence, and chaos. It is also of importance in signal processing, and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory: The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but are composed together in novel algebraic ways. This treatise provides an introductory overview and includes an extensive bibliography. Still, the bibliography makes no pretense to exhaustiveness. The overview collects often-used practical formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. As a concise treatise, it provides supplementary material which may be used for an advanced undergraduate or a beginning graduate course in quantum mechanics. It represents an expansion of a previous overview with selected papers collected by the authors, and includes a historical narrative account due the subject. This Historical Survey is presented first, in Section 1, but it might be skipped by students more anxious to get to the mathematical details beginning with the Introduction in Section 2. Alternatively, Section 1 may be read alone by anyone interested only in the history of the subject. Peter Littlewood and Harry Weerts are thanked for allotting time to make the treatise better.

Contemporary Problems in Mathematical Physics - Proceedings of the Second International Workshop, 2002

Phase space is the state space of classical mechanics, and this manifold is normally endowed only with a symplectic form. The geometry of quantum mechanics is necessarily more complicated. Arguments will be given to show that augmenting the symplectic manifold of classical phase space with a Riemannian metric is sufficient for describing quantum mechanics. In particular, using such spaces, a fully satisfactory geometric version of quantization will be developed and described.

Linear Algebra and its Applications, 2021

Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + f (y) , f (x) − f (y) } = { x + y , x − y }, x, y ∈ X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = σU , where U : X → Y is a surjective linear isometry and σ : X → {−1, 1}. This is a Wigner's type result for real normed spaces.

Journal of Geometry and Physics, 2007

The geometry of Grassmann manifolds Gr K (H), of orthogonal projection manifolds P K (H) and of Stiefel bundles St(K , H) is reviewed for infinite dimensional Hilbert spaces K and H. Given a loop of projections, we study Hamiltonians whose evolution generates a geometric phase, i.e. the holonomy of the loop. The simple case of geodesic loops is considered and the consistence of the geodesic holonomy group is discussed. This group agrees with the entire U (K) if H is finite dimensional or if dim(K) ≤ dim(K ⊥). In the remaining case we show that the holonomy group is contained in the unitary Fredholm group U ∞ (K) and that the geodesic holonomy group is dense in U ∞ (K).

Physical Review A, 2003

We present a theory of the geometric phase based logically on the Bargmann invariant of quantum mechanics, and null phase curves in ray space, as the fundamental ingredients. Null phase curves are themselves defined entirely in terms of the ͑third order͒ Bargmann invariant, and it is shown that these are the curves natural to geometric phase theory, rather than geodesics used in earlier treatments. The natural symplectic structure in ray space is seen to play a crucial role in the definition of the geometric phase. Logical consistency of the formulation is explicitly shown, and the principal properties of geometric phases are deduced as systematic consequences.

Wigner's quasi-probability distribution function in phase space is a special (Weyl-Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics; nuclear physics; and quantum computing, decoherence, and chaos. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter century: It furnishes a third, alternative formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate and momentum space. It works in full phase-space while accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but which compose together in novel algebraic ways. This volume is a selection of 23 classic and/or useful papers about the phase-space formulation, with an introductory overview that provides a trail-map to these papers, and with an extensive bibliography. The overview collects often-used formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. It thereby provides supplementary material that may be used for a beginning graduate course in quantum mechanics.

Foundations of Physics, 1981

We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a Jbundation for understanding their relationship. We use the Wigner-Moyal trans]brmation as a change of representation in phase space, and we avoM the problem of"negative probabilities" by regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the tlueory to cover thermodynamic processes invoh, ing entropy changes, as well as the usual reversible processes.

Physical Review Letters, 2003

The geometric phase for a pure quantal state undergoing an arbitrary evolution is a "memory" of the geometry of the path in the projective Hilbert space of the system. We find that Uhlmann's geometric phase for a mixed quantal state undergoing unitary evolution not only depends on the geometry of the path of the system alone but also on a constrained bi-local unitary evolution of the purified entangled state. We analyze this in general, illustrate it for the qubit case, and propose an experiment to test this effect. We also show that the mixed state geometric phase proposed recently in the context of interferometry requires uni-local transformations and is therefore essentially a property of the system alone. PACS numbers: 03.65.Vf, 42.50.Dv Pancharatnam [1] was first to introduce the concept of geometric phase in his study of interference of light in distinct states of polarization. Its quantal counterpart was discovered by Berry [2], who proved the existence of geometric phases in cyclic adiabatic evolutions. This was generalized to the case of nonadiabatic [3] and noncyclic [4] evolutions. The geometric phase was also derived on the basis of purely kinematic considerations [5].

Brazilian Journal of Physics

The decomplexification procedure allows one to show mathematically (stricto sensu) the equivalence (isomorphism) between the quantum dynamics of a system with a finite number of basis states and a classical dynamics system. This unique way of connecting different dynamics was used in the past to analyze the relationship between the well-known geometric phase present in the quantum evolution discovered by Berry and its generalizations, with their analogs, the Hannay phases, in the classical domain. In here, this analysis is carried out for several quantum hermitian and non-hermitian PT-symmetric Hamiltonians and compared with the Hannay phase analysis in their classical isomorphic equivalent systems. As the equivalence ends in the classical domain with oscillator dynamics, we exploit the analogy to propose resonant electric circuits coupled with a gyrator, to reproduce the geometric phase coming from the theoretical solutions, in simulated laboratory experiments.

We develop a general 'classicalization' procedure that links Hilbert-space and phasespace operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in anh-Taylor series and fix parameters with the condition that coefficients ofh 0 , −ih 1 ∂/∂Q and ih 1 ∂/∂P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact. In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S(Q, P , t) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.

Journal of Mathematical Physics, 2013

Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.

2005

Drawing inspiration from Dirac's work on functions of non commuting observables, we develop a fresh approach to phase space descriptions of operators and the Wigner distribution in quantum mechanics. The construction presented here is marked by its economy, naturalness and more importantly, by its potential for extensions and generalisations to situations where the underlying configuration space is non Cartesian.