An Introduction to Covariant Quantum Mechanics

✍ Scribed by Josef Janyška, Marco Modugno

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 831
• Series: Fundamental Theories of Physics 205
• Edition: 1
• Category: Library

✦ Synopsis

This book deals with an original contribution to the hypothetical missing link unifying the two fundamental branches of physics born in the twentieth century, General Relativity and Quantum Mechanics. Namely, the book is devoted to a review of a "covariant approach" to Quantum Mechanics, along with several improvements and new results with respect to the previous related literature. The first part of the book deals with a covariant formulation of Galilean Classical Mechanics, which stands as a suitable background for covariant Quantum Mechanics. The second part deals with an introduction to covariant Quantum Mechanics. Further, in order to show how the presented covariant approach works in the framework of standard Classical Mechanics and standard Quantum Mechanics, the third part provides a detailed analysis of the standard Galilean space-time, along with three dynamical classical and quantum examples. The appendix accounts for several non-standard mathematical methods widely used in the body of the book.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 Historical Background
1.2 General Relativity and Covariance
1.2.1 General Relativity and Covariant Quantum Mechanics
1.2.2 Lorentzian and Galilean Spacetimes
1.2.3 Principle of Relativity
1.2.4 Principle of Covariance
1.2.5 Naturality
1.2.6 Intrinsic, Observed and Coordinate Languages
1.3 General Features of the Present Approach
1.3.1 Covariance
1.3.2 Minimal Axioms
1.3.3 Limits Between Different Theories
1.3.4 General Connections
1.3.5 Scales
1.4 Features of Classical Theory
1.4.1 The Role of Time
1.4.2 Galilean Metric
1.4.3 Galilean Gravitational Field
1.4.4 Galilean Electromagnetic Field
1.4.5 Example of Intrinsic, Observed and Coordinate Languages
1.4.6 Joined Spacetime Connection
1.4.7 Connection Formalism in Classical Mechanics
1.4.8 Classical Phase Space
1.4.9 Lie Algebra of Special Phase Functions
1.4.10 Classical Symmetries
1.5 Features of Quantum Theory
1.5.1 Standard Quantum Mechanics as Touchstone
1.5.2 Quantum Bundle Based on Spacetime
1.5.3 Real and Complex Quantum Bundle
1.5.4 Proper Quantum Bundle and Its Polar Real Splitting
1.5.5 η-Hermitian Quantum Metric
1.5.6 Galilean Upper Quantum Connection
1.5.7 The ``Game'' of Potentials and Distinguished Observer
1.5.8 Criterion of Projectability
1.5.9 Dynamical Quantum Objects
1.5.10 Lagrangian Formalism in Quantum Mechanics
1.5.11 Hydrodynamical Picture of Quantum Mechanics
1.5.12 Quantum Symmetries
1.5.13 Quantum Differential Operators
1.5.14 Quantum Currents
1.5.15 Quantum Expectation Forms
1.5.16 Hilbert Quantum Bundle
1.5.17 Feynman Amplitudes
1.5.18 Comparison with Geometric Quantisation
1.5.19 Open Problem: Angular Momentum
1.5.20 Examples
1.6 Algebraic and Geometric Language
1.6.1 Fibred Manifolds and Bundles
Part I Covariant Classical Mechanics
2 Spacetime
2.1 Spacetime Fibring
2.2 Tangent Space of Spacetime
2.3 Iterated Tangent Space of Spacetime
2.4 Particle and Continuum Motions
2.5 Classical Phase Space
2.6 Contact Map
2.7 Observers
3 Galilean Metric Field
3.1 Timelike Galilean Metric
3.2 Spacelike Galilean Metric
3.2.1 Definition of Spacelike Galilean Metric
3.2.2 Volumes
3.2.3 Hodge Star and Cross Product
3.2.4 Observed Kinetic Objects
3.2.5 Observed Angular Momentum
3.2.6 Fibrewise Riemannian Structure
3.2.7 Fibrewise Symplectic Structure
3.2.8 Metric Differential Operators
3.2.9 Rigid Observers
4 Galilean Gravitational Field
4.1 Special Spacetime Connections
4.1.1 Spacetime Connections
4.1.2 Curvature of Spacetime Connections
4.1.3 Torsion of Spacetime Connections
4.1.4 Bianchi Identities for Spacetime Connections
4.1.5 Special Spacetime Connections
4.2 Metric Preserving Special Spacetime Connections
4.2.1 Definition of Metric Preserving Spacetime Connection
4.2.2 Distinguished Metric Preserving Spacetime Connection
4.2.3 Observed Spacetime 2-form
4.2.4 Characterisation of Metric Preserving Connections
4.2.5 Curvature of Metric Preserving Special Connections
4.2.6 The Covariant Curvature
4.3 Galilean Spacetime Connections
4.3.1 Definition of Galilean Spacetime Connection
4.3.2 Remark on Galilean Spacetime Connections
4.3.3 Spacelike Einstein Identity
4.3.4 Postulate on the Gravitational Field
4.4 Differential Operators
4.4.1 Spacetime Connections and Volume Forms
4.4.2 Spacetime Connections and Divergence
4.4.3 Spacetime Connections and Curl
5 Galilean Electromagnetic Field
5.1 Electromagnetic Field
5.2 Magnetic Field
5.3 Observed Electric Field
5.4 Observed Splitting of the Electromagnetic Field
5.5 Transition Rule of the Electric Field
5.6 Algebraic Invariants of the Electromagnetic Field
5.7 Lorentz Force
5.8 1st Maxwell Equation
5.9 Divergence of the Electromagnetic Field
6 Joined Spacetime Connection
6.1 Coupling Scales
6.2 Electromagnetic Terms
6.3 Galilean Joined Spacetime Connection
6.4 Joined Spacetime Curvature Tensor
6.5 Joined Ricci Tensor
7 Classical Dynamics
7.1 Particle Kinematics
7.1.1 Absolute Particle Kinematics
7.1.2 Observed Particle Kinematics
7.2 Particle Dynamics
7.2.1 Absolute Particle Dynamics
7.2.2 Observed Particle Dynamics
7.3 Fluid Kinematics
7.3.1 Absolute Fluid Kinematics
7.3.2 Observed Continuum Kinematics
7.4 Fluid Dynamics
7.4.1 Absolute Fluid Dynamics
7.4.2 Observed Fluid Dynamics
8 Sources of Gravitational and Electromagnetic Fields
8.1 Galilean Version of 2nd Maxwell Equation
8.1.1 Galilei–Maxwell Equation
8.2 Galilean Version of Einstein Equation
8.2.1 Galilei–Einstein Equation
8.3 Joined Galilei–Einstein Equation
8.3.1 The Joined Galilei–Einstein Equation
9 Fundamental Fields of Phase Space
9.1 Fundamental Fields of Phase Space
9.1.1 The Fundamental Fields of Phase Space
9.1.2 Phase Volumes
9.1.3 Relations Between the Fundamental Fields of Phase Space
9.2 Spacetime Connection and Phase Fields
9.2.1 Spacetime Connections and the Phase Fields
9.2.2 Joined Phase Objects
9.2.3 Identities for Fundamental Phase Fields
10 Geometric Structures of Phase Space
10.1 Cosymplectic Structure of Phase Space
10.1.1 The Cosymplectic Pair of Phase Space
10.1.2 Upper Potential and Observed Potential
10.1.3 Dynamical Phase 1-Forms
10.1.4 Cosymplectic Versus Symplectic Structures
10.2 coPoisson Structure of Phase Space
10.2.1 The coPoisson Pair of Phase Space
11 Hamiltonian Formalism
11.1 Phase Splittings
11.2 Phase Musical Morphisms
11.2.1 Linear Phase Musical Morphisms
11.2.2 Affine Phase Musical Morphisms
11.3 Hamiltonian Phase Lift of Phase Functions
11.3.1 Scaled Hamiltonian Phase Lift of Phase Functions
11.3.2 Natural Hamiltonian Phase Lift of Phase Functions
11.4 Poisson Lie Bracket
11.5 Classical Law of Motion
11.6 Conserved Phase Functions
12 Lie Algebra of Special Phase Functions
12.1 Special Phase Functions
12.2 Tangent Lift of Special Phase Functions
12.2.1 Divergence of Special Phase Functions
12.2.2 Splittings of Special Phase Functions
12.3 Holonomic Phase Lift of s.p.f.
12.4 Hamiltonian Phase Lift of s.p.f.
12.5 Special Phase Lie Bracket
12.6 Lie Subalgebras of Special Phase Functions
12.6.1 Algebraic Lie Subalgebras of Special Phase Functions
12.6.2 Differential Lie Subalgebras of Special Phase Functions
13 Classical Symmetries
13.1 Symmetries of Classical Structure
13.2 Symmetries of Classical Dynamics
13.3 Classical Currents
Part II Covariant Quantum Mechanics
14 Quantum Bundle
14.1 Real Quantum Bundle
14.2 Complex Structure
14.3 Hermitian Structure
14.4 Complex Versus Real Structures
14.5 η-Hermitian Quantum Structure
14.6 Proper Quantum Bundle
14.7 Polar Splitting of the Proper Quantum Bundle
14.8 Quantum Covariance Group
14.9 Quantum Sections
14.10 Quantum Liouville Vector Field
14.11 Upper Quantum Bundle
15 Galilean Upper Quantum Connection
15.1 Quantum and Upper Quantum Connections
15.1.1 Quantum Connections
15.1.2 Upper Quantum Connections
15.1.3 Hermitian Quantum Connections
15.1.4 Hermitian Upper Quantum Connections
15.1.5 Splitting of Quantum and Upper Quantum Connection
15.1.6 Curvature of Quantum and Upper Quantum Connection
15.2 Galilean Upper Quantum Connections
15.2.1 Definition
15.2.2 Local Existence
15.2.3 Global Existence
15.2.4 Postulate on Galilean Upper Quantum Connection
15.2.5 Transition Rule for the Potential and Invariants
15.2.6 Distinguished Observer and Potential
15.3 Upper Quantum Connection Over Time
16 Quantum Differentials
16.1 1st Order Quantum Covariant Differentials
16.1.1 1st Observed Quantum Covariant Differential
16.1.2 1st Observed Quantum Covariant Differential of Quantum Bases
16.1.3 1st Observed Phase Quantum Covariant Differential
16.1.4 Polar Splitting of 1st Observed Quantum Differential
16.2 2nd Order Quantum Covariant Differentials
16.2.1 2nd Observed Quantum Covariant Differential
16.2.2 2nd Observed Phase Quantum Covariant Differential
16.2.3 Polar Splitting of the 2nd Quantum Differential
16.3 Observed Quantum Laplacian
16.3.1 Observed Phase Quantum Laplacian
16.3.2 Polar Splitting of the Observed Quantum Laplacian
16.4 Upper Quantum Covariant Differentials
16.4.1 Upper Quantum Covariant Differential
16.4.2 Phase Upper Quantum Covariant Differential
16.4.3 Polar Splitting of the Upper Quantum Differential
16.5 Remarks on Notation
17 Quantum Dynamics
17.1 Criterion of Projectability
17.2 Quantum Velocity
17.3 Kinetic Quantum Tensor
17.3.1 Definition of Kinetic Quantum Tensor
17.3.2 Kinetic Quantum Vector Field
17.4 Quantum Probability Current
17.4.1 Definition of Quantum Probability Current
17.4.2 Quantum Probability Current Form
17.5 Quantum Lagrangian
17.5.1 Definition of Quantum Lagrangian
17.5.2 Quantum Momentum Form
17.5.3 Quantum Poincaré–Cartan Form
17.6 Schrödinger Operator
17.6.1 Codifferential of the Kinetic Quantum Tensor
17.6.2 Definition of the Schrödinger Operator
17.6.3 Polar Splitting of the Schrödinger Operator
17.6.4 Schrödinger Equation
17.6.5 Polar Splitting of the Schrödinger Equation
17.6.6 Lagrangian Approach to Schrödinger Equation
17.6.7 Quantum Noether Theorem
17.7 Purely Covariant Approach
17.7.1 Covariant Operators
17.7.2 Schrödinger Operator by Covariance
17.7.3 Quantum Lagrangian by Covariance
18 Hydrodynamical Picture of QM
18.1 Kinematics of the Associated Classical Fluid
18.1.1 Associated Classical Fluid
18.1.2 Associated Mass and Charge Density Currents
18.1.3 Associated Acceleration
18.2 Dynamics of the Associated Classical Fluid
18.2.1 Law of Motion of the Associated Fluid
18.2.2 Dynamical Equations of the Associated Fluid
19 Quantum Symmetries
19.1 Symmetries of the Hermitian Quantum Metric
19.1.1 Quantum Lifts of Special Phase Functions
19.1.2 Classification of Hermitian Quantum Vector Fields
19.2 Symmetries of Quantum Structure
19.3 Symmetries of Quantum Dynamics
20 Quantum Differential Operators
20.1 Quantum Differential Operators and s.p.f
20.1.1 Quantum Differential Operators
20.1.2 η-Hermitian Quantum Vector Fields as Operators
20.1.3 Special Quantum Differential Operators
20.1.4 Polar Splitting of Quantum Differential Operators
20.1.5 Commutator of Special Quantum Differential Operators
21 Quantum Currents and Expectation Forms
21.1 Quantum Currents
21.1.1 Quantum Currents
21.1.2 Vertical Quantum Currents
21.2 Quantum Current Forms
21.2.1 Quantum Current Forms
21.2.2 Vertical Quantum Current Forms
21.3 Quantum Expectation Forms
22 Sectional Quantum Bundle
22.1 Concise Introduction to F-smooth Spaces
22.2 The F-smooth Sectional Quantum Space
22.3 The F-smooth Sectional Quantum Bundle
22.4 The Pre-Hilbert Sectional Quantum Bundle
22.5 Quantum Operators
22.6 Schrödinger Connection
23 Feynman Path Integral
23.1 Upper Quantum Covariant Differential Over Time
23.2 Feynman Amplitudes
Part III Examples
24 Flat Newtonian Spacetime
24.1 Flat Newtonian Spacetime
24.2 Inertial Observers
24.3 Inertial Observers Versus Affine Spacetime
24.4 Uniformly Accelerated Observer
24.5 Uniformly Rotating Observer
25 Dynamical Example 1: No Electromagnetic Field
25.1 Classical Objects
25.1.1 Starting Hypothesis of the Classical Theory
25.1.2 Inertial Observer
25.1.3 Uniformly Accelerated Observer
25.1.4 Uniformly Rotating Observer
25.2 Quantum Objects
25.2.1 Starting Hypothesis of the Quantum Theory
25.2.2 Discussion on the Chosen Distinguished Gauge
25.2.3 Inertial Observer
25.2.4 Uniformly Accelerated Observer
25.2.5 Uniformly Rotating Observer
26 Dynamical Example 2: Radial Electric Field
26.1 Classical Objects
26.1.1 Starting Hypothesis of the Classical Theory
26.1.2 Inertial Observer
26.1.3 Uniformly Accelerated Observer
26.2 Quantum Objects
26.2.1 Starting Hypothesis of the Quantum Theory
26.2.2 Inertial Observer
26.2.3 Uniformly Accelerated Observer
27 Dynamical Example 3: Constant Magnetic Field
27.1 Classical Objects
27.1.1 Starting Hypothesis of the Classical Theory
27.1.2 Inertial Observer
27.1.3 Uniformly Rotating Observer
27.2 Quantum Objects
27.2.1 Inertial Observer
27.2.2 Uniformly Rotating Observer
28 Curved Newtonian Spacetime
28.1 Curved Newtonian Spacetime
28.2 Gravitational Connection
28.3 Gravitational Curvature
28.4 Newton Law of Gravitation
28.5 Further Properties
Part IV Conclusions and Further Developments
29 Conclusions
29.1 Main Features of Our Approach
29.2 Open Problems
30 Developments in Galilean Spin Particle
30.1 Classical Spinning Particle
30.1.1 Classical Sphere Bundle
30.1.2 Lie Algebra of Spin Special Phase Functions
30.1.3 Classical Spin Bundle
30.1.4 Spin Connection
30.1.5 Pauli Map
30.2 Quantum Spin
30.2.1 Quantum Spin Bundle
30.2.2 Quantum Spin Connection
30.2.3 Quantum Spin Lagrangian
30.2.4 Pauli Equation on the Curved Galilean Spacetime
30.2.5 Quantum Spin Operators
31 Developments in Einsteinian General Relativity
31.1 Einsteinian Spacetime
31.1.1 The Einsteinian Spacetime
31.1.2 Gravitational Connection
31.1.3 Motions
31.2 Einsteinian Phase Space
31.2.1 The Einsteinian Phase Space
31.2.2 Contact Map and Contact Form
31.2.3 Orthogonal Projection
31.2.4 Vertical Space of the Phase Space
31.2.5 Observers
31.2.6 Observed Spacelike Volume
31.3 Phase Objects
31.3.1 Dynamical Phase Objects
31.3.2 Gravitational Phase Objects
31.4 Electromagnetic Field
31.5 Joined Phase Objects
31.6 Dynamical 1-Forms
31.7 Hamiltonian Lift
31.8 Phase Lie Brackets
31.8.1 Poisson Lie Bracket
31.8.2 Special Phase Lie Bracket
31.9 Classical Symmetries
31.10 Quantum Stuff
31.10.1 Quantum Bundle
31.10.2 Hermitian Vector Fields
31.10.3 Quantum Dynamics
31.11 Further Hints
Appendix Appendix on Geometric Methods
Appendix A Fibred Manifolds and Bundles
A.1 Fibred Manifolds
A.2 Bundles
A.3 Structured Bundles
A.3.1 Vector Bundles
A.3.2 Affine Bundles
A.3.3 Lie Group Bundles
A.3.4 Lie Affine Bundles
A.3.5 Principal Bundles
Appendix B Tangent Bundle
B.1 Tangent Prolongation of Manifolds
B.2 Tangent Prolongation of Fibred Manifolds
B.3 Tangent Prolongation of Structured Bundles
B.3.1 Tangent Prolongation of Vector Bundles
B.3.2 Tangent Prolongation of Affine Bundles
B.3.3 Tangent Prolongation of Lie Group Bundles
B.3.4 Tangent Prolongation of Lie Affine Bundles
B.4 Iterated Tangent Bundle
Appendix C Tangent Valued Forms
C.1 Conventions on Exterior Forms
C.2 Tangent Valued Forms on a Manifold
C.2.1 Tangent Valued Forms on a Fibred Manifold
C.2.2 Vector Valued Forms on a Vector Bundle
Appendix D Lie Derivatives
D.1 Lie Derivatives of Sections
D.2 Lie Derivatives of Vertical Covariant Tensors
D.3 Infinitesimal Symmetries of Tensors
Appendix E The Frölicher–Nijenhuis Bracket
E.1 The FN-Bracket on a Manifold
E.2 The FN-Bracket on a Fibred Manifold
E.3 The FN-Bracket on a Structured Bundle
E.3.1 The FN-Bracket on a Vector Bundle
E.3.2 The FN-Bracket on an Affine Bundle
E.4 The FN-Bracket of Vector Valued Forms
Appendix F Connections
F.1 General Connections
F.1.1 Connections as Tangent Valued Forms
F.1.2 Covariant Differential of Tangent Valued Forms
F.1.3 Curvature
F.1.4 Identities for Curvature
F.1.5 Lie Derivatives of the Connection
F.1.6 Torsion
F.1.7 Identities for Torsion
F.2 Linear Connections of Vector Bundles
F.2.1 Covariant Differential of Vector Calued Forms
F.2.2 Curvature
F.2.3 Torsion
F.3 Affine Connections of Affine Bundles
F.3.1 Curvature
F.3.2 Torsion
F.4 Linear Connections of a Manifold
Appendix G Jets
G.1 Jet Spaces of Fibred Manifolds
G.1.1 Multi-indices
G.1.2 Jet Spaces
G.1.3 Vertical Bundle of Jet Spaces
G.2 Jet Spaces of Double Fibred Manifolds
G.3 Contact Structure
G.3.1 Contact Maps
G.3.2 Complementary Contact Maps
G.3.3 Contact Splitting of the Tangent Space
G.4 Jet Functor
G.5 The Exchange Map
G.6 Holonomic Prolongation of Vector Fields
Appendix H Lagrangian Formalism
H.1 Momentum and Poincaré–Cartan Form
H.2 Euler–Lagrange Operator
H.3 Currents
Appendix I Geometric Structures
I.1 Schouten Bracket
I.1.1 Regular Pairs
I.1.2 Dual Regular Pairs
I.1.3 Cosymplectic and coPoisson Structures
Appendix J Covariance
J.1 Categories and Functors
J.1.1 Categories
J.1.2 Functors
J.2 Natural Bundle Functors and Operators
J.2.1 Natural Bundle Functors
J.2.2 Natural Differential Operators
J.3 Gauge Natural Bundles and Operators
J.3.1 Gauge Natural Bundle Functors
J.3.2 Natural Operators of Gauge Natural Bundles
J.3.3 Generalised Lie Derivatives
J.4 Naturality and Covariance
J.4.1 Equivariant Sections and Morphisms
J.4.2 Covariant Sections and Morphisms
J.4.3 Gauge Covariant Sections and Morphisms
Appendix K Scales
K.1 Positive Spaces
K.1.1 Definition of Positive Spaces
K.1.2 Tensor Product of Positive Spaces
K.1.3 Rational Maps Between Positive Spaces
K.1.4 Rational Powers of a Positive Space
K.2 Physical Scales
K.2.1 Units and Scales
K.2.2 Scaled Objects
Appendix References
Index