Geometric Formulation of Classical and Quantum Mechanics

The following sections are included:

• Preface

• Contents

We address classical and quantum mechanics in a general setting of arbitrary time-dependent coordinate and reference frame transformations.

The technique of symplectic manifolds is well known to provide the adequate Hamiltonian formulation of autonomous mechanics [1; 104; 157]…

The following sections are included:

• Preliminary. Fibre bundles over ℝ

• Autonomous dynamic equations

• Dynamic equations

• Dynamic connections

• Non-relativistic geodesic equations

• Reference frames

• Free motion equations

• Relative acceleration

• Newtonian systems

• Integrals of motion

The following sections are included:

• Lagrangian formalism on Q → ℝ

• Cartan and Hamilton–De Donder equations

• Quadratic Lagrangians

• Lagrangian and Newtonian systems

• Lagrangian conservation laws
  • Generalized vector fields
  • First Noether theorem
  • Noether conservation laws
  • Energy conservation laws

• Gauge symmetries

The following sections are included:

• Geometry of Poisson manifolds
  • Symplectic manifolds
  • Presymplectic manifolds
  • Poisson manifolds
  • Lichnerowicz–Poisson cohomology
  • Symplectic foliations
  • Group action on Poisson manifolds

• Autonomous Hamiltonian systems
  • Poisson Hamiltonian systems
  • Symplectic Hamiltonian systems
  • Presymplectic Hamiltonian systems

• Hamiltonian formalism on Q → ℝ

• Homogeneous Hamiltonian formalism

• Lagrangian form of Hamiltonian formalism

• Associated Lagrangian and Hamiltonian systems

• Quadratic Lagrangian and Hamiltonian systems

• Hamiltonian conservation laws

• Time-reparametrized mechanics

The following sections are included:

• GNS construction
  • Involutive algebras
  • Hilbert spaces
  • Operators in Hilbert spaces
  • Representations of involutive algebras
  • GNS representation
  • Unbounded operators

• Automorphisms of quantum systems

• Banach and Hilbert manifolds
  • Real Banach spaces
  • Banach manifolds
  • Banach vector bundles
  • Hilbert manifolds
  • Projective Hilbert space

• Hilbert and C*-algebra bundles

• Connections on Hilbert and C*-algebra bundles

• Instantwise quantization

The following sections are included:

• Geometric quantization of symplectic manifolds

• Geometric quantization of a cotangent bundle

• Leafwise geometric quantization
  • Prequantization
  • Polarization
  • Quantization

• Quantization of non-relativistic mechanics
  • Prequantization of T*Q and V*Q
  • Quantization of T*Q and V*Q
  • Instantwise quantization of V*Q
  • Quantization of the evolution equation

• Quantization with respect to different reference frames

The following sections are included:

• Autonomous Hamiltonian systems with constraints

• Dirac constraints

• Time-dependent constraints

• Lagrangian constraints

• Geometric quantization of constraint systems

The following sections are included:

• Partially integrable systems with non-compact invariant submanifolds
  • Partially integrable systems on a Poisson manifold
  • Bi-Hamiltonian partially integrable systems
  • Partial action-angle coordinates
  • Partially integrable system on a symplectic manifold
  • Global partially integrable systems

• KAM theorem for partially integrable systems

• Superintegrable systems with non-compact invariant submanifolds

• Globally superintegrable systems

• Superintegrable Hamiltonian systems

• Example. Global Kepler system

• Non-autonomous integrable systems

• Quantization of superintegrable systems

The following sections are included:

• The vertical extension of Lagrangian mechanics

• The vertical extension of Hamiltonian mechanics

• Jacobi fields of completely integrable systems

The following sections are included:

• Lagrangian mechanics with parameters

• Hamiltonian mechanics with parameters

• Quantum mechanics with classical parameters

• Berry geometric factor

• Non-adiabatic holonomy operator

The following sections are included:

• Jets of submanifolds

• Lagrangian relativistic mechanics

• Relativistic geodesic equations

• Hamiltonian relativistic mechanics

• Geometric quantization of relativistic mechanics

The following sections are included:

• Commutative algebra

• Geometry of fibre bundles
  • Fibred manifolds
  • Fibre bundles
  • Vector bundles
  • Affine bundles
  • Vector fields
  • Multivector fields
  • Differential forms
  • Distributions and foliations
  • Differential geometry of Lie groups

• Jet manifolds
  • First order jet manifolds
  • Second order jet manifolds
  • Higher order jet manifolds
  • Differential operators and differential equations

• Connections on fibre bundles
  • Connections
  • Flat connections
  • Linear connections
  • Composite connections

• Differential operators and connections on modules

• Differential calculus over a commutative ring

• Infinite-dimensional topological vector spaces

The following sections are included:

• Bibliography

• Index