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Stochastic quantum mechanics and quantum spacetime : a consistent unification of relativity and quantum theory based on stochastic spaces

Eduard Prugovečki

Published Jan 1, 1984• 10 citations
Publisher

Abstract

I: Stochastic Quantum Mechanics.- 1 / Nonrelativistic Stochastic Quantum Mechanics.- 1.1 Stochastic Values and Stochastic Phase Spaces.- 1.2. Sharp Localizability and Systems of Imprimitivity.- 1.3. Stochastic Localizability and Systems of Co variance.- 1.4. Phase-Space Representations of the Galilei Group.- 1.5. Localizability in Nonrelativistic Stochastic Phase Spaces.- 1.6. Stochastic Phase-Space Systems of Covariance.- 1.7. Resolution Generators and Gauge Freedom in Stochastic Phase Spaces.- 1.8. Stochastic Probability Currents and Their Sharp-Point Limits.- 1.9. Propagators in Nonrelativistic Stochastic Phase Spaces.- 1.10. Quantum Stochastic Processes and Phase-Space Path Integrals.- 1.11. Semiclassical Propagators in Stochastic Phase Spaces.- 1.12. The Physical Nature and the Measurement of Proper Wave Functions and of Stochastic Position and Momentum.- Notes.- 2 / Relativistic Stochastic Quantum Mechanics.- 2.1. The Nonexistence of Relativistic Probability Currents at Sharp Space time Points.- 2.2. The Unfeasibility of Sharp Localization in Relativistic Quantum Mechanics.- 2.3. Phase-Space Representations of the Poincare Group.- 2.4. Resolution Generators and Relativistic Phase-Space Representations.- 2.5. Localizability in Relativistic Stochastic Phase Spaces.- 2.6. Systems of Covariance in Relativistic Stochastic Phase Spaces.- 2.7. Relativistic Canonical Commutation Relations and Gauge Freedom.- 2.8. Relativistically Covariant and Conserved Probability and Charge Currents.- 2.9. Propagators in Relativistic Stochastic Phase Spaces.- 2.10. Relativistic Extended Quantum Particles in External Electromagnetic Fields.- Notes.- 3 / Statistical Mechanics on Stochastic Phase Spaces.- 3.1. Phase-Space Representations of Quantum Statistical Mechanics.- 3.2. Informational Equivalence and Completeness of POV Measures.- 3.3. Generalized Stochastic Phase-Space Systems of Covariance.- 3.4. Classical and Quantum Mechanics in L2 (?N).- 3.5. Classical and Quantum Statistical Mechanics in Master Liouville Space ?(?N).- 3.6. Quantum Observables as Density Distributions on Stochastic Phase Spaces.- 3.7. Unified Formulations of Classical and Quantum Boltzmann Equations and Scattering Cross-Sections.- Notes.- II: Quantum Spacetime.- 4 / Reciprocity Theory and the Geometrization of Stochastic Quantum Mechanics.- 4.1. The Operational Foundations of Special and General Relativity.- 4.2. Coordinate Fluctuation Amplitudes for Coherent Flows of Quantum Test Particles.- 4.3. Quantum Lorentz Frames and Minkowski Quantum Spacetime.- 4.4. General Quantum Spacetimes and Stochastic Geometries.- 4.5. Reciprocity Theory and Born's Quantum Metric Operator.- 4.6. Unitary Internal Symmetries and Generalized Representations of Relativistic Canonical Commutation Relations.- Notes.- 5 / Field Theory on Quantum Spacetime.- 5.1. Free Scalar Fields on Minkowski Quantum Spacetime.- 5.2. LSZ Formalism and Causality on Quantum Spacetime.- 5.3. Lagrangian Field Theory on Quantum Spacetime.- 5.4. Dirac Bispinors and Free Fields on Quantum Spacetime.- 5.5. Photon Localizability and the Electromagnetic Field on Quantum Spacetime.- 5.6. Reciprocally Invariant Quantum Electrodynamics for Extended Fermions.- Notes.- Appendix A / Elements of Measure Theory.- Appendix B / Elements of Operator Theory on Hilbert Spaces.- Notes.- References.

DOI
10.1007/978-94-009-4492-3

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By taking into account the quantum aspects intrinsic to any operational definition of spatio-temporal relationships, a stochastic concept of spacetime emerges. In relation to its classical counterpart is realized as a stochastic mean around which quantum fluctuations become negligible only in the limit of macroscopic spacetime intervals. The test-particle propagators used in the proposed quantum concept of spacetime are derived by solving in a consistent manner the localizability problem for relativistic particles. This is achieved in the framework of the stochastic phase space formulation of quantum mechanics, which in the nonrelativistic context is shown to result from systems of imprimitivity related to phase space conserved probability currents derivable from bona fide convariant probability densities in stochastic phase spaces of one particle systems, which can be interpreted as due to measurements performed with extended rather than pointlike test particles. The associated particle propagators can be therefore consistently related to coordinate probability densities measurable by the exchange of photons in between test particles from a chosen standard. Quantum spacetime is defined as the family of propagators corresponding to all conceivable coherent flows of test particles. This family of free-fall propagators has to satisfy certain self-consistency conditions as well as consistent laws ofmore » motion which inplicitly determine the stochastic geometro-dynamics of quantum space-time. Field theory on quantum spacetime retains many of the formal features of conventional quantum field theory. On a fundamental epistemological level stochastic geometries emerge as essential prerequisites in the construction of spacetime models that would be operationally based and yet consistent with the relativity principle as well as with the uncertinty principle.« less

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Total Cross Sections in Non-Relativistic Scattering Theory.- Classical-Quantum Correspondence in Non-Linear Systems.- Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels.- Statistical Inference in Quantum Mechanics.- Schrodinger Operators with Oscillating Potentials.- Time Delay and Resonance in Simple Scattering.- A Study of the Helmholtz Operator.- Invariant Manifolds and Bifurcations in the Taylor Problem.- On the Dimension of a Finite Difference Approximation to Divergence-Free Vectors.- Intramolecular Dynamics in the Quasiperiodic and Stochastic Regimes.- Problems in the Semiclassical Quantization of Integrable and Nonintegrable Classical Dynamical Systems.- Nonselfadjoint Operators in Diffraction and Scattering.- One-Dimensional Crystals in an External Field.- Dynamics of Forced Coupled Oscillators: Classical Phenomenology of Infrared Multiphoton Absorption.- Irreversibility and Stochasticity of Chemical Processes.- On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I).- Regular and Chaotic Regimes in Quantum Mechanics.- Barrier Penetration and Exponential Decay in the Stark Effect.- A Nonlinear Schrodinger Equation Yielding the Shape of Molecules by Spontaneous Symmetry Breaking.- Classical Limit of the Number of Quantum States.- Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report.- On a Generalized Weyl-Von Neumann Converse Theorem.- Scattering Theory in Many-Body Quantum Systems. Analyticity of the Scattered Matrix.- Existence, Uniqueness, Stability and Calculation of the Stationary Regimes in Some Nonlinear Systems.- Weighted Trigonometric Approximations in L2(Rn).- The Phenomenon of Magnetic Pairing and Exactly Soluble Models of Magnetic Resonances.- Dyons Solutions to Yang-Mills Equations.- Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model.- Self-Consistent Field Methods for Metastable Electronic States: A Promising Extension of the Complex Coordinate Technique.- Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems.- Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces.- Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients.- Orthogonal Laurent Polynomials and Gaussian Quadrature.- Measurement Theory with Instruments Treated Partially Quantum Mechanically.- Transition Maps and Locality.- The Feynman-Kac Formula for Boson Wiener Processes.- On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes.

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Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].

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Consistent formulation of relativistic dynamics for massive spin-zero particles in external fields

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It is shown that in the context of stochastic-phase-space formulations of quantum mechanics a nonlocal type of nonrelativistic as well as relativistic dynamics for a massive spin-zero particle moving in an external electromagnetic field can be formulated in ${L}^{2}(\ensuremath{\Gamma})$ space. In a sharp-point limit $s\ensuremath{\rightarrow}+0$ the nonrelativistic dynamics merges into the conventional local model. The relativistic dynamics is gauge invariant, it can be formulated covariantly in the relativistic phase space ${M}_{m}$, it possesses a covariant propagator, it leaves the space of positive-energy solutions of the Klein-Gordon equation invariant, and it gives rise to a covariant and conserved probability current on stochastic phase space. Furthermore, for states describing a particle moving at nonrelativistic speeds in relation to a given inertial frame, the relativistic propagation contracts into its nonrelativistic counterpart. The sharp-point limit $s\ensuremath{\rightarrow}+0$ of the relativistic $S$ matrix also exists in each such inertial frame. The necessity of interactions acting at nonsharp configuration points for a consistent formulation of covariant relativistic one-particle dynamics is traced to the question of the nonexistence of covariant position operators in relativistic theories. The mandatory character of this feature is shown to be firmly rooted in the theory of quantum measurement.

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Stochastic processes in quantum physics

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I Markov Processes.- 1.1 Classical Mechanics.- 1.2 Movement of a Particle with Noise.- 1.3 Transition Probability and the Markov Property.- 1.4 Diffusion Equations.- 1.5 Brownian Motions.- 1.6 The Ito formula.- Appendix. Monotone Lemmas.- II Time Reversal and Duality.- 2.1 Time Reversal of Markov Processes and Duality.- 2.2 Space-Time Markov Processes and Space-Time Duality.- 2.3 Time Reversal and Schroedinger's Representation.- III Non-Relativistic Quantum Theory.- 3.1 Non-Relativistic Equation of Motion.- 3.2 Stationary States and Eigenvalue Problem.- 3.3 Time Reversal of Diffusion Processes.- 3.4 Duality Relation of Diffusion Processes.- 3.5 Equation of Motion in General Cases.- 3.6 Principle of Superposition of Markov Processes.- 3.7 Non-Relativistic Schroedinger Equation.- 3.8 State Preparations and Measurements.- 3.9 Diffusion or Schroedinger Equations ?.- 3.10 The First Technical Convention.- IV Stationary Schroedinger Processes.- 4.1 Stationary States.- 4.2 One-Dimensional Harmonic Oscillator.- 4.3 An Example in Two-Dimension.- 4.4 Superposition of Eigenfunctions.- 4.5 Further Excited States.- 4.6 Hydrogen Atom.- V Construction of the Schroedinger Processes.- 5.1 The Feynman-Kac Formula.- 5.2 Solving the Equation of Motion.- 5.3 Transformation by Multiplicative Functionals.- 5.4 Renormalization.- 5.5 A Variational Method.- 5.6 The Maruyama-Girsanov Formula.- 5.7 A Lagrangian Formulation.- 5.8 The Second Technical Convention.- VI Markov Processes with Jumps.- 6.1 Poisson and Compound Poisson Processes.- 6.2 Poisson Random measures and Point Processes.- 6.3 Stochastic Integrals with Poisson Point Processes.- 6.4 Levy Processes.- 6.5 Stable Processes.- 6.6 Bochner's Subordination.- 6.7 Duality of Subordinate Semi-Groups.- 6.8 Harmonic Transformation of Subordinate Semi-Groups.- 6.9 Duality of Fractional Powers of Time-Dependent Operators.- VII Relativistic Quantum Particles.- 7.1 A Relativistic Schroedinger Equation for a Spinless Paticle.- 7.2 Equation of Motion for Relativistic Quantum Particles.- 7.3 Stationary States of the Relativistic Schroedinger Equation.- 7.4 Stochastic Processes for Relativistic Spinless Particles.- 7.5 Non-Relativistic Limit.- 7.6 A Diffusion Approximation.- VIII Stochastic Differential Equations of Pure-Jumps.- 8.1 Markov Processes with the Generators of Fractional Power.- 8.2 Stochastic Differential Equations of Pure-Jumps.- 8.3 The Case with no Potential Term.- 8.4 To Solve the Stochastic Differential Equations of Pure-Jumps.- 8.5 To Construct Pure-Jump Markov Processes.- 8.6 A Remark on the Integrability Condition.- IX Variational Principle for Relativistic Quantum Particles.- 9.1 Absolute Continuity.- 9.2 Pure-Jump Markov Processes.- 9.3 A Multiplicative Functional.- 9.4 Renormalization and Variational Principle.- X Time Dependent Subordination and Markov Processes with Jumps.- 10.1 Time-Inhomogeneous Subordination.- 10.2 Lemmas.- 10.3 Stochastic Differential Equation with Jumps.- 10.4 A Formula of Feynman-Kac Type.- 10.5 Markov Processes with Jumps.- Appendix. Integration by Parts Formulae.- XI Concave Majorants of Levy Processes and the Light Cone.- 14.1 The Vertex Process of a Levy Process.- 14.2 Propositions on Random walks.- 14.3 Proof of Propositions on Random Walks.- 14.4 Proof of the main Theorems.- 14.5 Examples.- 14.6 The light Cone.- XII The Locality in Quantum Physics.- 12.1 Historical Overview.- 12.2 Hidden-Variable Theories.- 12.3 Locality of Hidden-Variable Theories.- 12.4 Spin-Correlation of Three Particles.- 12.5 Gudder's Hidden-Variable Theory.- 12.6 Spin-Correlations in Gudder's Theory.- 12.7 Some Remarks.- XIII Micro Statistical Theory.- 13.1 The Source of the Noise.- 13.2 Large Deviations of the Renormalized Processes.- 13.3 The Propagation of Chaos.- 13.4 Micro Statistical Mechanics.- 13.5 Propagation of Chaos of Pure-Jump Processes.- 13.6 Superposition of Movements.- 12.7 A Remark on the Gibbs Distribution.- XIV Processes on Open Time Intervals.- 14.1 Diffusion Processes on Open Time Intervals.- 14.2 Time-Reversed Schroedinger Processes.- 14.3 A Theorem of Jeulin-Yor.- 14.4 Reflecting Brownian Motion.- 14.5 Two-Sided Skorokhod Type Problem.- 14.6 Skorokhod Problem with Singular Drift.- 14.7 The Minimum and Maximum Solutions.- 14.8 The Uniqueness and Non-Uniqueness of Solutions.- 14.9 An Application: The Origin of Universes.- XV Creation and Killing of Particles.- 15.1 Non-Linear Differential Equations.- 15.2 Branching Markov Processes.- 15.3 The Expected Number of Particles.- 15.4 Killing.- XVI The lto Calculus.- 16.1 The Ito Integral.- 16.2 Martingales.- 16.3 The Ito Integral with Local Martingales.- 16.4 Ito's formula.- 16.5 Stochastic Differential Equations.- 16.6 Stochastic Differential Calculus.- References.

Birkhäuser-Verlag eBooks•Jan 1, 2000•2 citations

Quantum mechanics : a stochastic approach

Ramabhadra Vasudevan, Kanniyappan Parthasarathy, Ravishankar Ramanathan

Preface / Forward / Acknowledgements / Introduction / Invitation to Stochastic Calculus / Hydrodynamical Formulation of Quantum Mechanics / Stochastic Diffusion Processes / Selected Problems in Stochastic Mechanics / Stochastic Variational Principles / Euclidean Quantum Mechanics / Stochastic Path Space Picture / Relativistic Stochastic Mechanics / Dirac Equation - Stochastic Viewpoint / Stochastic Field Theory / Stochastic Mechanics on Manifolds / Stochastic Route to Quantum Mechanics - Quest Continues / References / Author Index / Subject Index.

•Jan 1, 2008•4 citations
I: Stochastic Quantum Mechanics.- 1 / Nonrelativistic Stochastic Quantum Mechanics.- 1.1 Stochastic Values and Stochastic Phase Spaces.- 1.2. Sharp Localizability and Systems of Imprimitivity.- 1.3. Stochastic Localizability and Systems of Co variance.- 1.4. Phase-Space Representations of the Galilei Group.- 1.5. Localizability in Nonrelativistic Stochastic Phase Spaces.- 1.6. Stochastic Phase-Space Systems of Covariance.- 1.7. Resolution Generators and Gauge Freedom in Stochastic Phase Spaces.- 1.8. Stochastic Probability Currents and Their Sharp-Point Limits.- 1.9. Propagators in Nonrelativistic Stochastic Phase Spaces.- 1.10. Quantum Stochastic Processes and Phase-Space Path Integrals.- 1.11. Semiclassical Propagators in Stochastic Phase Spaces.- 1.12. The Physical Nature and the Measurement of Proper Wave Functions and of Stochastic Position and Momentum.- Notes.- 2 / Relativistic Stochastic Quantum Mechanics.- 2.1. The Nonexistence of Relativistic Probability Currents at Sharp Space time Points.- 2.2. The Unfeasibility of Sharp Localization in Relativistic Quantum Mechanics.- 2.3. Phase-Space Representations of the Poincare Group.- 2.4. Resolution Generators and Relativistic Phase-Space Representations.- 2.5. Localizability in Relativistic Stochastic Phase Spaces.- 2.6. Systems of Covariance in Relativistic Stochastic Phase Spaces.- 2.7. Relativistic Canonical Commutation Relations and Gauge Freedom.- 2.8. Relativistically Covariant and Conserved Probability and Charge Currents.- 2.9. Propagators in Relativistic Stochastic Phase Spaces.- 2.10. Relativistic Extended Quantum Particles in External Electromagnetic Fields.- Notes.- 3 / Statistical Mechanics on Stochastic Phase Spaces.- 3.1. Phase-Space Representations of Quantum Statistical Mechanics.- 3.2. Informational Equivalence and Completeness of POV Measures.- 3.3. Generalized Stochastic Phase-Space Systems of Covariance.- 3.4. Classical and Quantum Mechanics in L2 (?N).- 3.5. Classical and Quantum Statistical Mechanics in Master Liouville Space ?(?N).- 3.6. Quantum Observables as Density Distributions on Stochastic Phase Spaces.- 3.7. Unified Formulations of Classical and Quantum Boltzmann Equations and Scattering Cross-Sections.- Notes.- II: Quantum Spacetime.- 4 / Reciprocity Theory and the Geometrization of Stochastic Quantum Mechanics.- 4.1. The Operational Foundations of Special and General Relativity.- 4.2. Coordinate Fluctuation Amplitudes for Coherent Flows of Quantum Test Particles.- 4.3. Quantum Lorentz Frames and Minkowski Quantum Spacetime.- 4.4. General Quantum Spacetimes and Stochastic Geometries.- 4.5. Reciprocity Theory and Born's Quantum Metric Operator.- 4.6. Unitary Internal Symmetries and Generalized Representations of Relativistic Canonical Commutation Relations.- Notes.- 5 / Field Theory on Quantum Spacetime.- 5.1. Free Scalar Fields on Minkowski Quantum Spacetime.- 5.2. LSZ Formalism and Causality on Quantum Spacetime.- 5.3. Lagrangian Field Theory on Quantum Spacetime.- 5.4. Dirac Bispinors and Free Fields on Quantum Spacetime.- 5.5. Photon Localizability and the Electromagnetic Field on Quantum Spacetime.- 5.6. Reciprocally Invariant Quantum Electrodynamics for Extended Fermions.- Notes.- Appendix A / Elements of Measure Theory.- Appendix B / Elements of Operator Theory on Hilbert Spaces.- Notes.- References.
By taking into account the quantum aspects intrinsic to any operational definition of spatio-temporal relationships, a stochastic concept of spacetime emerges. In relation to its classical counterpart is realized as a stochastic mean around which quantum fluctuations become negligible only in the limit of macroscopic spacetime intervals. The test-particle propagators used in the proposed quantum concept of spacetime are derived by solving in a consistent manner the localizability problem for relativistic particles. This is achieved in the framework of the stochastic phase space formulation of quantum mechanics, which in the nonrelativistic context is shown to result from systems of imprimitivity related to phase space conserved probability currents derivable from bona fide convariant probability densities in stochastic phase spaces of one particle systems, which can be interpreted as due to measurements performed with extended rather than pointlike test particles. The associated particle propagators can be therefore consistently related to coordinate probability densities measurable by the exchange of photons in between test particles from a chosen standard. Quantum spacetime is defined as the family of propagators corresponding to all conceivable coherent flows of test particles. This family of free-fall propagators has to satisfy certain self-consistency conditions as well as consistent laws ofmore » motion which inplicitly determine the stochastic geometro-dynamics of quantum space-time. Field theory on quantum spacetime retains many of the formal features of conventional quantum field theory. On a fundamental epistemological level stochastic geometries emerge as essential prerequisites in the construction of spacetime models that would be operationally based and yet consistent with the relativity principle as well as with the uncertinty principle.« less
Total Cross Sections in Non-Relativistic Scattering Theory.- Classical-Quantum Correspondence in Non-Linear Systems.- Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels.- Statistical Inference in Quantum Mechanics.- Schrodinger Operators with Oscillating Potentials.- Time Delay and Resonance in Simple Scattering.- A Study of the Helmholtz Operator.- Invariant Manifolds and Bifurcations in the Taylor Problem.- On the Dimension of a Finite Difference Approximation to Divergence-Free Vectors.- Intramolecular Dynamics in the Quasiperiodic and Stochastic Regimes.- Problems in the Semiclassical Quantization of Integrable and Nonintegrable Classical Dynamical Systems.- Nonselfadjoint Operators in Diffraction and Scattering.- One-Dimensional Crystals in an External Field.- Dynamics of Forced Coupled Oscillators: Classical Phenomenology of Infrared Multiphoton Absorption.- Irreversibility and Stochasticity of Chemical Processes.- On the Representations of the Local Current Algebra and the Group of Diffeomorphisms (I).- Regular and Chaotic Regimes in Quantum Mechanics.- Barrier Penetration and Exponential Decay in the Stark Effect.- A Nonlinear Schrodinger Equation Yielding the Shape of Molecules by Spontaneous Symmetry Breaking.- Classical Limit of the Number of Quantum States.- Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report.- On a Generalized Weyl-Von Neumann Converse Theorem.- Scattering Theory in Many-Body Quantum Systems. Analyticity of the Scattered Matrix.- Existence, Uniqueness, Stability and Calculation of the Stationary Regimes in Some Nonlinear Systems.- Weighted Trigonometric Approximations in L2(Rn).- The Phenomenon of Magnetic Pairing and Exactly Soluble Models of Magnetic Resonances.- Dyons Solutions to Yang-Mills Equations.- Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model.- Self-Consistent Field Methods for Metastable Electronic States: A Promising Extension of the Complex Coordinate Technique.- Timestep Control for the Numerical Solutions of Initial-Boundary-Value Problems.- Quantum Mechanical Angular Distributions and Group Representations on Banach Spaces.- Tensor Product Decomposition of Holomorphically Induced Representations and Clebsch-Gordan Coefficients.- Orthogonal Laurent Polynomials and Gaussian Quadrature.- Measurement Theory with Instruments Treated Partially Quantum Mechanically.- Transition Maps and Locality.- The Feynman-Kac Formula for Boson Wiener Processes.- On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
It is shown that in the context of stochastic-phase-space formulations of quantum mechanics a nonlocal type of nonrelativistic as well as relativistic dynamics for a massive spin-zero particle moving in an external electromagnetic field can be formulated in ${L}^{2}(\ensuremath{\Gamma})$ space. In a sharp-point limit $s\ensuremath{\rightarrow}+0$ the nonrelativistic dynamics merges into the conventional local model. The relativistic dynamics is gauge invariant, it can be formulated covariantly in the relativistic phase space Mm{M}_{m}Mm​, it possesses a covariant propagator, it leaves the space of positive-energy solutions of the Klein-Gordon equation invariant, and it gives rise to a covariant and conserved probability current on stochastic phase space. Furthermore, for states describing a particle moving at nonrelativistic speeds in relation to a given inertial frame, the relativistic propagation contracts into its nonrelativistic counterpart. The sharp-point limit $s\ensuremath{\rightarrow}+0$ of the relativistic SSS matrix also exists in each such inertial frame. The necessity of interactions acting at nonsharp configuration points for a consistent formulation of covariant relativistic one-particle dynamics is traced to the question of the nonexistence of covariant position operators in relativistic theories. The mandatory character of this feature is shown to be firmly rooted in the theory of quantum measurement.
I Markov Processes.- 1.1 Classical Mechanics.- 1.2 Movement of a Particle with Noise.- 1.3 Transition Probability and the Markov Property.- 1.4 Diffusion Equations.- 1.5 Brownian Motions.- 1.6 The Ito formula.- Appendix. Monotone Lemmas.- II Time Reversal and Duality.- 2.1 Time Reversal of Markov Processes and Duality.- 2.2 Space-Time Markov Processes and Space-Time Duality.- 2.3 Time Reversal and Schroedinger's Representation.- III Non-Relativistic Quantum Theory.- 3.1 Non-Relativistic Equation of Motion.- 3.2 Stationary States and Eigenvalue Problem.- 3.3 Time Reversal of Diffusion Processes.- 3.4 Duality Relation of Diffusion Processes.- 3.5 Equation of Motion in General Cases.- 3.6 Principle of Superposition of Markov Processes.- 3.7 Non-Relativistic Schroedinger Equation.- 3.8 State Preparations and Measurements.- 3.9 Diffusion or Schroedinger Equations ?.- 3.10 The First Technical Convention.- IV Stationary Schroedinger Processes.- 4.1 Stationary States.- 4.2 One-Dimensional Harmonic Oscillator.- 4.3 An Example in Two-Dimension.- 4.4 Superposition of Eigenfunctions.- 4.5 Further Excited States.- 4.6 Hydrogen Atom.- V Construction of the Schroedinger Processes.- 5.1 The Feynman-Kac Formula.- 5.2 Solving the Equation of Motion.- 5.3 Transformation by Multiplicative Functionals.- 5.4 Renormalization.- 5.5 A Variational Method.- 5.6 The Maruyama-Girsanov Formula.- 5.7 A Lagrangian Formulation.- 5.8 The Second Technical Convention.- VI Markov Processes with Jumps.- 6.1 Poisson and Compound Poisson Processes.- 6.2 Poisson Random measures and Point Processes.- 6.3 Stochastic Integrals with Poisson Point Processes.- 6.4 Levy Processes.- 6.5 Stable Processes.- 6.6 Bochner's Subordination.- 6.7 Duality of Subordinate Semi-Groups.- 6.8 Harmonic Transformation of Subordinate Semi-Groups.- 6.9 Duality of Fractional Powers of Time-Dependent Operators.- VII Relativistic Quantum Particles.- 7.1 A Relativistic Schroedinger Equation for a Spinless Paticle.- 7.2 Equation of Motion for Relativistic Quantum Particles.- 7.3 Stationary States of the Relativistic Schroedinger Equation.- 7.4 Stochastic Processes for Relativistic Spinless Particles.- 7.5 Non-Relativistic Limit.- 7.6 A Diffusion Approximation.- VIII Stochastic Differential Equations of Pure-Jumps.- 8.1 Markov Processes with the Generators of Fractional Power.- 8.2 Stochastic Differential Equations of Pure-Jumps.- 8.3 The Case with no Potential Term.- 8.4 To Solve the Stochastic Differential Equations of Pure-Jumps.- 8.5 To Construct Pure-Jump Markov Processes.- 8.6 A Remark on the Integrability Condition.- IX Variational Principle for Relativistic Quantum Particles.- 9.1 Absolute Continuity.- 9.2 Pure-Jump Markov Processes.- 9.3 A Multiplicative Functional.- 9.4 Renormalization and Variational Principle.- X Time Dependent Subordination and Markov Processes with Jumps.- 10.1 Time-Inhomogeneous Subordination.- 10.2 Lemmas.- 10.3 Stochastic Differential Equation with Jumps.- 10.4 A Formula of Feynman-Kac Type.- 10.5 Markov Processes with Jumps.- Appendix. Integration by Parts Formulae.- XI Concave Majorants of Levy Processes and the Light Cone.- 14.1 The Vertex Process of a Levy Process.- 14.2 Propositions on Random walks.- 14.3 Proof of Propositions on Random Walks.- 14.4 Proof of the main Theorems.- 14.5 Examples.- 14.6 The light Cone.- XII The Locality in Quantum Physics.- 12.1 Historical Overview.- 12.2 Hidden-Variable Theories.- 12.3 Locality of Hidden-Variable Theories.- 12.4 Spin-Correlation of Three Particles.- 12.5 Gudder's Hidden-Variable Theory.- 12.6 Spin-Correlations in Gudder's Theory.- 12.7 Some Remarks.- XIII Micro Statistical Theory.- 13.1 The Source of the Noise.- 13.2 Large Deviations of the Renormalized Processes.- 13.3 The Propagation of Chaos.- 13.4 Micro Statistical Mechanics.- 13.5 Propagation of Chaos of Pure-Jump Processes.- 13.6 Superposition of Movements.- 12.7 A Remark on the Gibbs Distribution.- XIV Processes on Open Time Intervals.- 14.1 Diffusion Processes on Open Time Intervals.- 14.2 Time-Reversed Schroedinger Processes.- 14.3 A Theorem of Jeulin-Yor.- 14.4 Reflecting Brownian Motion.- 14.5 Two-Sided Skorokhod Type Problem.- 14.6 Skorokhod Problem with Singular Drift.- 14.7 The Minimum and Maximum Solutions.- 14.8 The Uniqueness and Non-Uniqueness of Solutions.- 14.9 An Application: The Origin of Universes.- XV Creation and Killing of Particles.- 15.1 Non-Linear Differential Equations.- 15.2 Branching Markov Processes.- 15.3 The Expected Number of Particles.- 15.4 Killing.- XVI The lto Calculus.- 16.1 The Ito Integral.- 16.2 Martingales.- 16.3 The Ito Integral with Local Martingales.- 16.4 Ito's formula.- 16.5 Stochastic Differential Equations.- 16.6 Stochastic Differential Calculus.- References.
Preface / Forward / Acknowledgements / Introduction / Invitation to Stochastic Calculus / Hydrodynamical Formulation of Quantum Mechanics / Stochastic Diffusion Processes / Selected Problems in Stochastic Mechanics / Stochastic Variational Principles / Euclidean Quantum Mechanics / Stochastic Path Space Picture / Relativistic Stochastic Mechanics / Dirac Equation - Stochastic Viewpoint / Stochastic Field Theory / Stochastic Mechanics on Manifolds / Stochastic Route to Quantum Mechanics - Quest Continues / References / Author Index / Subject Index.