An Introduction to Covariant Quantum Mechanics

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