Quantum Mechanics and Quantum Field Theory from Algebraic and Geometric Viewpoints

Preface
Contents
1 Quantum Theory in Algebraic and Geometric Approaches
	1.1 Traditional Quantum Mechanics
	1.2 Geometric and Algebraic Approaches to Quantum Theory
	1.3 Gelfand-Naimark-Segal (GNS) Construction
	1.4 Algebraic Approach to Classical Mechanics
	1.5 Quantum Mechanics as a Deformation of Classical Mechanics. Weyl Algebra
	1.6 Quadratic Hamiltonians
	1.7 Stationary States
	1.8 Fock Space
	1.9 Hamiltonians Preserving the Number of Particles
	1.10 Representations of Weyl Algebra
	1.11 Grassmann Algebra
	1.12 Clifford Algebras and Their Representation
	1.13 Adiabatic Approximation. Decoherence
	1.14 Statistical Physics
2 Scattering
	2.1 Solitons as Classical Analogs of Quantum Particles
	2.2 Particles and Quasiparticles
		2.2.1 Excitations. Elementary Excitations
		2.2.2 Particles in Non-relativistic Quantum Mechanics
	2.3 Scattering; in- and out-states
		2.3.1 in- and out-states
		2.3.2 Møller Matrices
		2.3.3 Scattering Matrix. in- and out-oparators
		2.3.4 Inclusive Scattering Matrix
		2.3.5 Generalization
	2.4 Correlation Functions. Cluster Property
	2.5 Green\'s Functions. Generalized Green\'s Functions
		2.5.1 Green\'s Functions
		2.5.2 Generalized Green\'s Functions
	2.6 Scattering Amplitudes in Terms of Green\'s Functions. LSZ Formula
		2.6.1 Introduction
		2.6.2 LSZ Formula
		2.6.3 Proof of LSZ Formula
	2.7 Inclusive Scattering Matrix
	2.8 Perturbation Theory. Feynman Diagrams
		2.8.1 Perturbation Theory for Evolution Operator
		2.8.2 Perturbation Theory for Green\'s Functions
		2.8.3 Scattering Matrix from Adiabatic Scattering Matrix
		2.8.4 Perturbation Theory for Generalized Green\'s Functions
3 Poincaré Group. Relativistic Theories
	3.1 Introduction
	3.2 Lorentz Group. Poincaré Group
	3.3 Particles in Relativistic Theories
	3.4 Examples
		3.4.1 Scalar Field
		3.4.2 Particles with Spin 1/2
		3.4.3 Electromagnetic Field
	3.5 Free Theories
	3.6 Interacting Field Theories
4 Deterministic Physical Theories from Geometric Viewpoint
	4.1 Introduction
	4.2 Classical Theory with a Restricted Set of Observables
	4.3 Geometric Theories
	4.4 Decoherence. Probabilities
	4.5 Particles as Elementary Excitations
	4.6 Scattering. Inclusive Scattering Matrix
	4.7 L-functionals
	4.8 Diagram Techniques in Formalism of L-functionals
		4.8.1 Inclusive Scattering Matrix from Adiabatic Scattering Matrix in the Formalism of L-functionals
	4.9 Infrared Problem in Quantum Electrodynamics
Appendix  Background
A.1  Notations, Conventions, Definitions
A.2  Convex Sets
A.3  Asymptotic Behavior of Solutions of Linear Equations
References