Hilbert Bundle Research Articles

A New Approach to Quantising Space-Time: III. State Vestors of Functions on Arrows

In [1], a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects, Ob(Q), in a (small) category Q. The quantum states in this approach are cross-sections of a bundle A is in K[A] of Hilbert spaces over Ob(Q). The Hilbert spaces K[A], A are in Ob(Q)], depend strongly on the object A, and have to be chosen so as to get an irreducible, faithful, representation of the basic `category quantisation monoid'. In the present paper, we develop a different approach in which the state vectors are complex-valued functions on the set of arrows in Q. This throws a new light on the Hilbert bundle scheme: in particular, we recover the results of that approach in the, physically important, example when Q is a small category of finite sets.

The foliation operator in history quantum field theory

As a preliminary to discussing the quantization of the foliation in a history form of general relativity, we show how the discussion in an earlier work [J. Math. Phys. 43, 3053 (2002)] of a history version of free, scalar quantum field theory can be augmented in such a way as to include the quantization of the unit-length, timelike vector that determines a Lorentzian foliation of Minkowski space–time. We employ a Hilbert bundle construction that is motivated by (i) discussing the role of the external Lorentz group in the existing history quantum field theory [J. Math. Phys. 43, 3053 (2002)] and (ii) considering a specific representation of the extended history algebra obtained from the multi-symplectic representation of scalar field theory.

FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS V: INTERPRETATION, SUMMARY AND DISCUSSION

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.

FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS IV: MIXED STATES AND EVOLUTION TRANSPORT'S CURVATURE

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. The present fourth part of this series is devoted mainly to the fibre bundle description of mixed quantum states. We show that to the conventional density operator there corresponds a unique density lifting of paths for which the corresponding equations of motion are derived. It is also investigated the bundle description of mixed quantum states in the different pictures of motion. We calculate the curvature of the evolution transport and prove that it is curvature free iff the values of the Hamiltonian operator at different moments commute.

Pseudoforces in quantum mechanics

Dynamical evolution is described as a parallel section on an infinite-dimensional Hilbert bundle over the base manifold of all frames of reference. The parallel section is defined by an operator-valued connection whose components are the generators of the relativity group acting on the base manifold. In the case of Galilean transformations we show that the property that the curvature for the fundamental connection must be zero is just the Heisenberg equations of motion and the canonical commutation relation in geometric language. We then consider linear and circular accelerating frames and show that pseudoforces must appear naturally in the Hamiltonian.

Fibre bundle formulation of nonrelativistic quantum mechanics: II. Equations of motion and observables

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced by an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper.

Fibre bundle formulation of nonrelativistic quantum mechanics: III. Pictures and integrals of motion

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrodinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. In the third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, we find the state liftings and observable liftings corresponding to state vectors and observables respectively in the different pictures of motion. The equations of motion for these quantities are derived. Using the results obtained, problems concerning the integrals of motion are considered from the bundle viewpoint. Necessary and sufficient invariant bundle conditions for a dynamical variable to be an integral of motion are found.

Classical and quantum mechanics with time-dependent parameters

We show that composite bundles Q→Σ→R, where Σ→R is the parameter bundle, provide the adequate mathematical description of mechanical systems with time-dependent parameters both in classical and quantum mechanics. In particular, the Berry phase phenomenon is described in terms of connections on composite Hilbert space bundles.

Crossed products of Hilbert C*-bimodules by bundles

AbstractWe present the definition of crossed products of Hilbert C*-bimodules by Hilbert bundles with commuting finite group actions and finite dimensional fibers. This is a general construction containing the bundle construction and crossed products of Hilbert C*-bimodule by finite groups. We also study the structure of endomorphism algebras of the tensor products of bimodules. We also define the multiple crossed products using three bimodules containing more than 2 bundles and show the associativity law. Moreover, we present some examples of crossed product bimodules easily computed by our method.

Relativistic formulation of quantum-state diffusion

A relativistic generalization of the quantum-state diffusion model is developed. The model describes a Dirac electron which is coupled to an external electromagnetic field and a dissipative environment. A relativistically covariant stochastic Dirac equation is obtained by regarding the state vector as a functional on a certain set of spacelike hypersurfaces in Minkowski space and by the definition of an appropriate Hilbert bundle on this set. The integrability condition of the stochastic process and the corresponding covariant density matrix equation are derived. Further, the relativistic equations governing the dynamical state-vector localization are deduced.

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincare symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.

Geometro-stochastically quantized fields with internal spin variables

The use of internal variables for the description of relativistic particles with arbitrary mass and spin in terms of scalar functions is reviewed and applied to the stochastic phase space formulation of quantum mechanics. Following Bacry and Kihlberg a four-dimensional internal spin space S̄ is chosen possessing an invariant measure and being able to represent integer as well as half integer spins. S̄ is a homogeneous space of the group SL(2,C) parametrized in terms of spinors α∈C2 and their complex conjugates ᾱ. The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin [m,s], with m⩾0 and s=0,12,1,32,… , with spin described in terms of the usual (2s+1)-component fields. Viewed from the internal space description of spin this reduction amounts to a restriction of the variable α to a compact subspace of S̄, i.e., a “spin shell” Sr=2s2 of radius r=2s in C2. This formulation of single particles or single antiparticles of type [m,s] is then used to study the geometro-stochastic (i.e., quantum) propagation of amplitudes for arbitrary spin on a curved background space–time possessing a metric and axial vector torsion treated as external fields. A Poincaré gauge covariant path integral-like representation for the probability amplitude (generalized wave function) of a particle with arbitrary spin is derived satisfying a second order wave equation on the Hilbert bundle constructed over curved space–time. The implications for the stochastic nature of polarization effects in the presence of gravitation are pointed out and the extension to Fock bundles of bosonic and fermionic type is briefly mentioned.

WAVELETS IN HIGH-RESOLUTION RADAR IMAGING AND CLINICAL MAGNETIC RESONANCE TOMOGRAPHY

Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high-resolution synthetic aperture radar SAR imaging and clinical magnetic resonance imaging MRI . The unitary filter bank construction is performed by the Kepplerian temporospatial Hilbert bundle strategy of physical astronomy, which allows stroboscopic and synchronous cross-sectional quadrature filtering of phase histories in contiguous local frequency encoding subband multichannels with respect to the rotating coordinate frame of quadrature reference. The Kepplerian temporospatial quadrature strategy and the associated unitary filter bank construction take place in symplectic affine planes and are implemented in the quadrature format of a phase-splitting network by Fourier analysis of the Heisenberg nilpotent Lie group G. The tomographic slices, frequency selected by the MRI scanner, are identified with the nontrivial coadjoint orbits of G. Thus the pathway of this paper leads from Keppler to Heisenberg and to the fascinating aspects of electronic engineering concerned with the implementation of the symmetries inherent to the semiclassical approach to clinical magnetic resonance tomography by large-scale integrated LSI circuit technology.

On quantum-geometric connections and propagators in curved spacetime

The basic properties of Poincaré gauge invariant Hilbert bundles over Lorentzian manifolds are derived. Quantum connections are introduced in such bundles, which govern a parallel transport that is shown to satisfy the strong equivalence principle in the quantum regime. Path-integral expressions are presented for boson propagators in Hilbert bundles over globally hyperbolic curved spacetimes. Their Poincaré gauge covariance is proven, and their special relativistic limit is examined. A method for explicitly computing such propagators is presented for the case of cosmological models with Robertson - Walker metric.

On quantum and parallel transport in a Hilbert bundle over spacetime

We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugovecki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugovecki. Furthermore, we show that the quantum-geometric propagator in curved spacetime proposed by Prugovecki, yielding a Feynman path integral-like formula involving integrations over intermediate phase-space variables, is Poincaré gauge-covariant (i.e. is gauge-invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a `contact point measure' in the soldered stochastic phase-space bundle raised over curved spacetime.

Geometro-differential conception of extended particles and their quantum theory in de Sitter space

A geometro-differential quantum theory of extended particles is presented. The geometrical selling is that of Hilbert fiber bundles whose base manifolds are pseudo-Riemannian space-times of points χ which are interpreted as partial aspects of physical reality (the extended particle). The fibers are carrier spaces of induced (internal configuration and momentum) representations of the structural group (the de Sitter group here). Sections of these bundles are seen as physical representations of the particle, and their values in the fibers are interpreted as states of internally localized modes composing the particle. This geometrical structure is “analogous” to that of a geometro-stochastic quantization developed in recent years. The physical interpretation is a combination of those of an old functional quantum theory and of an induced representation scheme based on an interpretation of intertwining, between configuration and momentum representations, as localization, materialization, and propagation of particles. Our model is applied in two cases: (1) Induced representation is applied in both space-time and internal space: both have de Sitter symmetry whose connection is ignored. Intertwining is considered in both spaces in a composed fashion. (2) For generic spacetimes and when the connection is taken into account, intertwining is applied only in the internal space. Parallel transport is combined with intertwining for a redefinition of localization, materialization, and propagation (the latter in a path integral context inspired from geometro-stochastic quantization).

An Infinite-Dimensional Geometric Structure on the Space of all the Probability Measures Equivalent to a Given One

Let $\mathscr{M}_\mu$ be the set of all probability densities equivalent to a given reference probability measure $\mu$. This set is thought of as the maximal regular (i.e., with strictly positive densities) $\mu$-dominated statistical model. For each $f \in \mathscr{M}_\mu$ we define (1) a Banach space $L_f$ with unit ball $\mathscr{V}_f$ and (2) a mapping $s_f$ from a subset $\mathscr{U}_f$ of $\mathscr{M}_\mu$ onto $\mathscr{V}_f$, in such a way that the system $(s_f, \mathscr{U}_f, f \in \mathscr{M}_\mu)$ is an affine atlas on $\mathscr{M}_\mu$. Moreover each parametric exponential model dominated by $\mu$ is a finite-dimensional affine submanifold and each parametric statistical model dominated by $\mu$ with a suitable regularity is a submanifold. The global geometric framework given by the manifold structure adds some insight to the so-called geometric theory of statistical models. In particular, the present paper gives some of the developments connected with the Fisher information metrics (Rao) and the Hilbert bundle introduced by Amari.

Spectral geometry: two exactly solvable models

Two exactly solvable models illustrating the links between spectral properties of Hamiltonians, connections on the induced Hilbert bundles and topological characteristics of the basis spaces are considered. The first of them is based on the extension theory for symmetric operators and the second on the one-dimensional Laplace operator with parametrical boundary conditions.

Spectral geometry: two exactly solvable models

Two exactly solvable models illustrating the links between spectral properties of Hamiltonians, connections on the induced Hilbert bundles and topological characteristics of the basis spaces are considered. The first of them is based on the extension theory for symmetric operators and the second on the one-dimensional Laplace operator with parametrical boundary conditions.

Delicacies of mass perturbation in the Schwinger model on a circle.

The Hilbert bundle for the massless fermions of the Schwinger model on a circle, over the space of gauge field configurations, is topologically non-trivial (twisted). The corresponding bundle for massive fermions is topologically trivial (periodic). Since the structure of the fermionic Hilbert bundle changes discontinuously the possibility of perturbing in the mass is thrown into doubt. In this article, we show that a direct application of the anti-adiabatic theorem of Low, allows the structure of the massless theory to be dynamically preserved in the strong coupling limit, ${e\over m}>>1$. This justifies the use of perturbation theory in the bosonized version of the model, in this limit.