David F. The Formalisms of Quantum Mechanics: An Introduction

Springer International Publishing, 2015. — 157 p. — (Lecture Notes in Physics 893). — ISBN: 978-3-319-10539-0, 978-3-319-10538-3.These lecture notes present a concise and introductory, yet as far as possible coherent, view of the main formalizations of quantum mechanics and of quantum field theories, their interrelations and their theoretical foundations.
The “standard” formulation of quantum mechanics (involving the Hilbert space of pure states, self-adjoint operators as physical observables, and the probabilistic interpretation given by the Born rule) on one hand, and the path integral and functional integral representations of probabilities amplitudes on the other, are the standard tools used in most applications of quantum theory in physics and chemistry. Yet, other mathematical representations of quantum mechanics sometimes allow better comprehension and justification of quantum theory. This text focuses on two of such representations: the algebraic formulation of quantum mechanics and the “quantum logic” approach. Last but not least, some emphasis will also be put on understanding the relation between quantum physics and special relativity through their common roots - causality, locality and reversibility, as well as on the relation between quantum theory, information theory, correlations and measurements, and quantum gravity.
Quantum mechanics is probably the most successful physical theory ever proposed and despite huge experimental and technical progresses in over almost a century, it has never been seriously challenged by experiments. In addition, quantum information science ha
s become an important and very active field in recent decades, further enriching the many facets of quantum physics. Yet, there is a strong revival of the discussions about the principles of quantum mechanics and its seemingly paradoxical aspects: sometimes the theory is portrayed as the unchallenged and dominant paradigm of modern physical sciences and technologies while sometimes it is considered a still mysterious and poorly understood theory, waiting for a revolution. This volume, addressing graduate students and seasoned researchers alike, aims to contribute to the reconciliation of these two facets of quantum mechanics.Motivation
Organization
What This Course is Not!
The Standard Formulations of Classical and Quantum Mechanics
Classical Mechanics
The Lagrangian Formulation
The Hamiltonian Formulation
The Algebra of Classical Observables
Probabilities
The Frequentist Point of View
The Bayesian Point of View
Conditional Probabilities
Quantum Mechanics: The “Canonical Formulation”
Principles
Representations of Quantum Mechanics
Quantum Statistics and the Density Matrix
Path and Functional Integrals Formulations
Path Integrals
Field Theories, Functional Integrals
Quantum Probabilities and Reversibility
Is Quantum Mechanics Reversible or Irreversible?
Reversibility of Quantum Probabilities
Causal Reversibility
The Algebraic Quantum FormalismObservables as Operators
Operator Algebras
The Algebraic Approach
The Algebra of Observables
The Mathematical Principles
Physical Discussion
Physical Observables and Pure States
The C^* -Algebra of Observables
The Norm on Observables, A is a Banach Algebra
The Observables form a Real G*-AIgebra
Spectrum of Observables and Results of Measurements
Complex C[sup]* [sup]-Algebras
The GNS Construction, Operators and Hilbert Spaces
Finite Dimensional Algebra of Observables
Infinite Dimensional Real Algebra of Observables
The Complex Case, the GNS Construction
Why Complex Algebras?
Dynamics
Locality and Separability
Quaternionic Hilbert Spaces
Superselection Sectors
Definition
A Simple Example: The Particle on a Circle
General Discussion
von Neumann Algebras
Definitions
Classification of Factors
The Tomita-Takesaki Theory
3.8 Locality and Algebraic Quantum Field Theory
Algebraic Quantum Field Theory in a Dash
Axiomatic QFT
3.9 Discussion
The Quantum Logic FormalismWhy an Algebraic Structure?
Measurements as “Logical Propositions”
The Quantum Logic Approach
A Presentation of the Principles
Projective Measurements as Propositions
Causality, POSET’s and the Lattice of Propositions
Reversibility and Orthocomplementation
Subsystems of Propositions and Orthomodularity
Pure States and AC Properties
The Geometry of Orthomodular AC Lattices
Prelude: The Fundamental Theorem of Projective Geometry
The Projective Geometry of Orthomodular AC Lattices
Towards Hilbert Spaces
Gleason’s Theorem and the Born Rule
States and Probabilities
Gleason’s Theorem
Principle of the Proof
The Born Rule
Physical Observables
Discussion
Information, Correlations, and More
Quantum Information Formulations
Quantum Correlations
Entropic Inequalities
Bipartite Correlations
Hidden Variables, Contextuality and Local Realism
Hidden Variables and “Elements of Reality”
Context-Free Hidden Variables
Gleason’s Theorem and Contextuality
The Kochcn-Specker Theorem
The Bell-CHSH Inequalities and Local Realism
Contextual Models
Summary Discussion on Quantum Correlations
Locality and Realism
Chance and Correlations
Measurements
What are the Questions?
The von Neumann Paradigm
Decoherence, Ergodicity and Mixing
Discussion
Formalisms, Interpretations and Alternatives to Quantum Mechanics
What About Interpretations?
Formalisms
Interpretations
Alternatives
What About Gravity?
Bibliography
Index