This book provides the rigorous mathematical foundations of Quantum Physics, from the operational meaning of the measuring process to the most recent theories for the quantum scale of space-time geometry. Topics like relativistic invariance, quantum systems with finite and infinitely many degrees of freedom, second quantisation, scattering theory, are all presented through the formalism of Operator Algebras for a precise mathematical justification.
The book is targeted to graduate students and researchers in the area of theoretical/mathematical physics who want to learn about the mathematical foundations of quantum physics, as well as the mathematics students and researchers in the area of operator algebras/functional analysis who want to dive into some of the applications of the theory to physics.
Author(s): Gabriele Nunzio Tornetta
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 181
City: Cham, Switzerland
Tags: Quantum Physics, Operator Algebra
Preface
References
Contents
1 The Basic Postulates
1.1 The Measuring Process
1.2 The Mathematical Framework of Quantum Mechanics
1.2.1 Projections
1.2.2 State Transitions
1.2.3 Interference
1.3 The Relation with Wave Mechanics
1.4 Symmetries
1.4.1 Dynamics
1.4.2 Regular Representations
1.4.3 Superselection Rules à la Wick-Wightman-Wigner
References
2 Relativistic Invariance
2.1 The Lorentz Group
2.2 The Poincaré Group
2.3 Massive Particles
2.4 The Generalised Dirac Equation
2.4.1 Massless Particles
2.4.2 The Dirac Equation
2.4.3 Minimal Coupling
References
3 Quantum Mechanics
3.1 The Weyl Relations
3.1.1 Irreducibility of the Schrödinger Representation
3.2 The Heisenberg Group
3.2.1 Uniqueness of the Schrödinger Representation
3.3 Quantum Harmonic Oscillator
3.4 Non-commutative Functional Calculus
References
4 Quantum Field Theory
4.1 Infinite Degrees of Freedom
4.2 The Segal Field
4.3 Second Quantisation
4.3.1 Fock Space Revisited
4.3.2 Infinite Tensor Products
4.3.3 Gauge Invariance
4.4 Canonical Anticommutation Relations
4.4.1 The Fermi Oscillator
4.4.2 Representations of the CAR Algebra
4.4.3 Gauge-invariant States
4.5 Quantum Fields
4.5.1 The Scalar Field
4.5.2 Covariance
4.5.3 The Spectral Condition
4.5.4 Irreducibility
4.5.5 Complexification
4.6 Field Algebras
4.6.1 Isotony
4.6.2 Locality
4.6.3 Canonical Commutation Relations
4.6.4 Additivity
4.6.5 Duality
4.6.6 Property B
4.7 Modular Structure
4.7.1 The KMS Condition
4.7.2 Araki Duality for Wedges
4.8 The Charged Scalar Free Field
4.8.1 Superselection Sectors
4.9 The Charged Free Spinor Field
4.9.1 The Generic j2 Spin Case
4.10 The Wightman Axioms
4.10.1 Wightman Fields
4.10.2 Vacuum Expectation Values
4.10.3 The Reconstruction Theorem
4.10.4 The Borchers-Uhlmann Algebra
4.10.5 Further Properties of Wightman Fields
4.10.6 Constructive Quantum Field Theory
References
5 Further Topics
5.1 Scattering Theory
5.2 The DHR Theory
5.2.1 Superselection Sectors
5.2.2 Statistics
5.3 The DFR Models for Quantum Space-time
5.3.1 The C*-algebra of the Basic Model
5.3.2 Field Algebras on the DFR Quantum Space-time
5.3.3 The Scale-covariant Model
References
Appendix A Elements of Group Theory
A.1 Topological Groups
A.2 Group Cohomology
A.2.1 Lifting of Projective Representations
A.3 Representation Theory
A.3.1 Induced Representations
A.3.2 The Little Group Method
A.3.3 The GNS Construction
Appendix B Infinite Tensor Products
B.1 For C*-algebras
B.2 For Hilbert Spaces
B.3 For Von Neumann Algebras
B.4 Equivalences
References
Index
