Quantum mechanics on Riemannian manifold in Schwinger's quantization approach I

Schwinger's quantization scheme is extended in order to solve the problem of the formulation of quantum mechanics on a space with a group structure. The importance of Killing vectors in the quantization scheme is shown. Usage of these vectors makes the algebraic properties of the operators consistent with the geometrical structure of the manifold. The procedure of the definition of the quantum Lagrangian of a free particle and the norm of the velocity (momentum) operators is given. These constructions are invariant under a general coordinate transformation. The unified procedure for constructing the quantum theory on a space with a group structure is developed. Using this, quantum mechanics on a Riemannian manifold with a simply transitive group acting on it is investigated. Received: 27 June 2000 / Revised version: 10 May 2001 / Published online: 19 July 2001     

Geometric algebra and star products on the phase space

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover, it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincaré and Lie-Poisson reduction can be formulated in this formalism.     

Deformation Quantization for Systems with Fermions

Deformation quantization, which achieves the passage from classical mechanics to quantum mechanics by the replacement of the pointwise multiplication of functions on phase space by the star product, is a powerful tool for treating systems involving bosonic degrees of freedom, both in quantum mechanics and in quantum field theory. In the present paper we show how these methods may be naturally extended to systems involving fermions. In particular we show how supersymmetric quantum mechanics can be formulated in this approach and consider examples involving both non-relativistic and relativistic systems.     

Algebraic model of a classical non-relativistic electron

An algebraic structure is constructed which serves as an algebraic analog of a phase space for a model of a non-relativistic classical electron. The structure consists of a type of Poisson bracket defined on the tensor product of a commutative algebra and a Grassmann algebra. The equivalent of Hamiltonian dynamics is defined and applied to specific models of an electron. A quantization procedure is introduced which leads to the usual quantum equivalents of the classical models.     

On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the Weyl algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of aq-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e., if one works in position-momentum realization, can be mapped on aq-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not possess a proper group of global transformations.     

The Aharonov–Casher effect for spin-1 particles in non-commutative quantum mechanics

By using a generalized Bopp’s shift formulation, instead of the star product method, we investigate the Aharonov–Casher (AC) effect for a spin-1 neutral particle in non-commutative (NC) quantum mechanics. After solving the Kemmer equations both on a non-commutative space and a non-commutative phase space, we obtain the corrections to the topological phase of the AC effect for a spin-1 neutral particle both on a NC space and a NC phase space. PACS 02.40.Gh, 11.10.Nx, 03.65.-w     

Quantization of contact manifolds and thermodynamics

The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this ‘quantization’ describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.     

Euler-Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system

This Letter focuses on studying generalized Euler-Lagrange equation and Hamiltonian framework from nonlocal-in-time kinetic energy of nonconservative system. According to Suykens' approach, we extend his results and formulate some work related to the nonconservative system. With the Lagrangian and nonconservative force in nonlocal-in-time form, we obtain the higher order generalized Euler-Lagrange equation which leads to an extension of Newton's second law of motion. The Hamiltonian is studied in relation to the Lagrangian in the canonical phase space. Finally, the particle with nonconservative force case is studied and compared with quantum mechanical results. The extended equation gives a possible approach for understanding the connection between classical and quantum mechanics.     

Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.     

Spectrum of Schrödinger field in a noncommutative magnetic monopole

The energy spectrum of a nonrelativistic particle on a noncommutative sphere in the presence of a magnetic monopole field is calculated. The system is treated in the field theory language, in which the one-particle sector of a charged Schrödinger field coupled to a noncommutative U(1) gauge field is identified. It is shown that the Hamiltonian is essentially the angular momentum squared of the particle, but with a nontrivial scaling factor appearing, in agreement with the first-quantized canonical treatment of the problem. Monopole quantization is recovered and identified as the quantization of a commutative Seiberg–Witten mapped monopole field.     

Theory of filters

We consider the following statistical problem: suppose we have a light beam and a collection of semi-transparent windows which can be placed in the way of the beam. Assume that we are colour blind and we do not possess any colour sensitive detector. The question is, whether by only measurements of the decrease in the beam intensity in various sequences of windows we can recognize which among our windows are light beam filters absorbing photons according to certain definite rules? To answer this question a definition of physical systems is formulated independent of “quantum logic” and lattice theory, and a new idea of quantization is proposed. An operational definition of filters is given: in the framework of this definition certain nonorthodox classes of filters are admissible with a geometry incompatible to that assumed in orthodox quantum mechanics. This leads to an extension of the existing quantum mechanical structure generalizing the schemes proposed by Ludwig [10] and the present author [13]. In the resulting theory, the quantum world of orthodox quantum mechanics is not the only possible but is a special member of a vast family of “quantum worlds” mathematically admissible. An approximate classification of these worlds is given, and their possible relation to the quantization of non-linear fields is discussed. It turns out to be obvious that the convex set theory has a similar significance for quantum physics as the Riemannian geometry for space-time physics.     

Trajectories and causal phase-space approach to relativistic quantum mechanics

We analyze phase-space approaches to relativistic quantum mechanics from the viewpoint of the causal interpretation. In particular, we discuss the canonical phase space associated with stochastic quantization, its relation to Hilbert space, and the Wigner-Moyal formalism. We then consider the nature of Feynman paths, and the problem of nonlocality, and conclude that a perfectly consistent relativistically covariant interpretation of quantum mechanics which retains the notion of particle trajectory is possible.     

Contact geometry in Lagrangian mechanics

We present a picture of Lagrangian mechanics, free of some unnatural features (such as complete divergences). As a byproduct, a completely natural U (1)-bundle over the phase space appears. The correspondence between classical and quantum mechanics is very clear, e.g. no topological ambiguities remain. Contact geometry is the basic tool.     

Schwinger’s action principle,supersymmetry and path integrals

We use Schwinger’s action principle in quantum mechanics to obtain the quantisation from Lagrangian for the fermionic variables, as well as when it contains auxiliary coordinates. We illustrate this with a supersymmetric Lagrangian which naturally includes auxiliary variables. We further show that the action principle also leads to Feynman’s path integral quantisation, which is aesthetically pleasing.     

An alternative approach to canonical quantizationof the radiation damping

Inspired by some works on quantization of dissipative systems, in particular the ones treating the damped harmonic oscillator and by a paper due to Lukierski, we consider the dissipative system of a charge interacting with its own radiation, which is the origin of radiation damping. Using the indirect Lagrangian representation we obtained a Lagrangian formalism with a Chern-Simons-like term. A Hamiltonian analysis is also done in commutative and non-commutative scenarios, which leads to the quantization of the system. Received: 23 August 2005, Published online: 21 October 2005 PACS: 41.60.-m, 04.60.Ds     

An Enlarged Canonical Quantization Scheme and Quantization of a Free Particle on Two-Dimensional Sphere

For a non-relativistic particle that freely moves on a curved surface, the fundamental commutation relations between positions and momenta are insufficient to uniquely determine the operator form of the momenta. With introduction of more commutation relations between positions and Hamiltonian and those between momenta and Hamiltonian,our recent sequential studies imply that the Cartesian system of coordinates is physically preferable, consistent with Dirac's observation. In present paper, we study quantization problem of the motion constrained on the two-dimensional sphere and develop a discriminant that can be used to show how the quantization within the intrinsic geometry is improper. Two kinds of parameterization of the spherical surface are explicitly invoked to investigate the quantization problem within the intrinsic geometry.     

Relation Between Relativistic and Non-Relativistic Quantum Mechanics as Integral Transformation

A formulation of quantum mechanics (QM) in the relativistic configurational space (RCS) is considered. A transformation connecting the non-relativistic QM and relativistic QM (RQM) has been found in an explicit form. This transformation is a direct generalization of the Kontorovich–Lebedev transformation. It is shown also that RCS gives an example of non-commutative geometry over the commutative algebra of functions.     

Dirac Field on Moyal-Minkowski Spacetime and Non-commutative Potential Scattering

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov’s formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.