Classical and Quantum Physics: 60 Years Alberto Ibort Fest Geometry, Dynamics, and Control (Springer Proceedings in Physics)

Preface
Alberto Ibort
	Ph.D. Students of Alberto Ibort Latre
	Master Students
	List of Publications of Alberto Ibort
Contents
Contributors
1 On a New Asymptotic Behaviour of Toeplitz Determinants
	1.1 Introduction
	1.2 The Two Szegő\'s Theorems
	1.3 The Fisher-Hartwig Conjecture Revisited
	1.4 Generalization of the Fisher-Hartwig Conjecture
	1.5 Application to a Principal Submatrix
	1.6 Conclusions
	References
2 Bulk-Edge Dualities in Topological Matter
	2.1 Introduction
	2.2 Hall Effect in Planar Systems
	2.3 Boundary Effects and Atiyah-Patodi-Singer Theorem
	2.4 Quantization of the Hall Conductivity
	2.5 Conclusions
	References
3 Near-Horizon Modes and Self-adjoint Extensions of the Schrödinger Operator
	3.1 Introduction
	3.2 Self-adjoint Extensions of the Effective Schrödinger Operator
		3.2.1 Statistical Mechanics and Thermal Equilibrium
		3.2.2 A Prescription for the Extension Selection
	3.3 Final Remarks
	References
4 The Gauss Law: A Tale
	4.1 Introduction
	4.2 The Structure of the Gauge Group: The Gauss Law
	4.3 The Group mathcalG0infty
	4.4 The Group mathcalG0: the Emergence of Global Groups
	4.5 The Sky Group 0
	4.6 Winding Number Gauge Transformations
	4.7 On Local Observables and Gauge Invariance
	4.8 On Superselection Groups
		4.8.1 Charge and Colour
		4.8.2 QCD θ-vacua
		4.8.3 The Sky Group
	4.9 Global Symmetries: Lorentz and Flavour Groups
		4.9.1 Axial U(1) Anomaly
		4.9.2 The Axial Flavour Anomaly
		4.9.3 How QED Breaks Lorentz Invariance
		4.9.4 The Higgs Field
	4.10 Non-linear Models and Edge Excitations
	References
5 Quantum Control at the Boundary
	5.1 Introduction
	5.2 Control of Quantum Systems
	5.3 Magnetic Laplacian
	5.4 Existence of Dynamics in Boundary Control Systems
	5.5 Approximate Controllability of Boundary Control Systems
	5.6 Conclusions
	References
6 Application of Lie Systems to Quantum Mechanics: Superposition Rules
	6.1 Introduction
	6.2 Fundamentals
		6.2.1 The Superposition Rule
		6.2.2 Additional Structures in Lie Systems
	6.3 The Geometrical Description of Quantum Mechanics
		6.3.1 The Linear, Complex, and Hermitian Structure
		6.3.2 Observables: Hamiltonian Dynamics and Killing Vector Fields
		6.3.3 Projective Hilbert Spaces as Kähler Manifolds
	6.4 Lie-Kähler Systems in Quantum Mechanics
		6.4.1 2-Level Lie Systems
		6.4.2 Schrödinger Equations and Lie-Kähler Systems
	6.5 Lie Systems on the Quantum Quotient Manifolds
		6.5.1 Lie Systems on the Manifold Q
		6.5.2 Lie Systems on the Manifold R
		6.5.3 Lie-Kähler System on the Projective Manifold P
	6.6 Superposition Rules for Schrödinger Equations
		6.6.1 Particular Solutions of the Schrödinger Equation
		6.6.2 Constants of Motion and Superposition Rules
	6.7 Superposition Rules for 2-Level Systems
		6.7.1 Superposition Rule for a 2-Level System on MQ and Q
		6.7.2 Superposition Rules for the 2-Level System on R and P
	6.8 Conclusions and Outlook
	References
7 Killing Vector Fields and Quantisation of Natural Hamiltonians
	7.1 Introduction
	7.2 Hamiltonian Dynamical Systems
	7.3 Dynamical Systems of Mechanical Type
	7.4 Geometric Approach to Quantum Mechanics
	7.5 How to Find a Quantum Model for a Classical One?
	7.6 Position Dependent Mass Systems
	7.7 Classical Motion on a Cycloid: A Case Study
	7.8 Quantisation of Motions on Curves
	7.9 Quantisation of Position Dependent Mass Systems
	7.10 Constant Curvature Surfaces
	7.11 Quantisation of Noether Momenta
	7.12 Conclusions and Outlook
	References
8 Conditions for Legitimate Memory Kernel Master Equation
	8.1 Introduction
	8.2 Preliminaries
	8.3 Quantum Jump Representation of the Markovian Semigroup
	8.4 A Class of Legitimate Memory Kernels
	8.5 Properties of Legitimate Pairs
	8.6 Non-homogeneous Memory Kernel Master Equation
	8.7 Semi-Markov Evolution
	8.8 Conclusions
	References
9 From Classical Trajectories to Quantum Commutation Relations
	9.1 Introduction
	9.2 Differential Equations from Experimental Data
	9.3 Dynamical Systems and Geometrical Structures: Lagrangian Picture
	9.4 Dynamical Systems and Geometrical Structures: Hamiltonian Picture
	9.5 Dynamical Systems and Geometrical Structures: Quantum Systems
	9.6 Conclusions
	References
10 On the Thermodynamics of Supersymmetric Haldane–Shastry Spin Chains
	10.1 Introduction
	10.2 Preliminaries
	10.3 Partition Function and Spectrum
	10.4 The Free Energy
	10.5 The su(1|1) Supersymmetric Spin Chain
	10.6 Conclusions
	References
11 Towards a Quantum Sampling Theory: The Case of Finite Groups
	11.1 Introduction
	11.2 The Mathematical Setting
	11.3 The Sampling Result
	11.4 Some Simple Examples
	11.5 Discussion and Conclusions
	References
12 On the Kinematics of the Last Wigner Particle
	12.1 Introduction
	12.2 The Schwinger Decomposition of the Pauli–Lubański Operator
		12.2.1 The Wigner Rotation, Tamed
	12.3 The Invariant Formalism for the WP
		12.3.1 Equations of Motion
		12.3.2 Invariant Wavefunctions
	12.4 The Propagator
	12.5 Connecting with the Standard Formalism
	References
13 Dimensional Deception  for the Noncommutative Torus
	13.1 Dimensions à la Weyl
	13.2 Matrix Approximations to the Noncommutative Torus
	13.3 The Truncation Map
	13.4 Derivations
	13.5 Weyl Dimension at Different Scales
	13.6 Discussion and Conclusions
	References
14 Notions of Infinity in Quantum Physics
	14.1 Introduction
	14.2 Operators and Operator Algebras in Hilbert Spaces
	14.3 Følner C*-Algebras
	14.4 Quantum Physics
		14.4.1 Type I Algebras and Quantum Mechanics
		14.4.2 The CAR-Algebra
		14.4.3 Local Quantum Physics
		14.4.4 The Theory of Superselection Sectors
	References
15 Poisson-Nijenhuis Manifolds,  Classical Yang-Baxter Equations,  and Frobenius Algebras
	15.1 Introduction
	15.2 Linear Poisson-Nijenhuis Manifolds
	15.3 Classical Yang-Baxter Equations
	15.4 Frobenius Algebras
	15.5 Concluding Remarks
	References
16 Hermite Polynomial Representation of Qubit States in Quantum Suprematism Picture
	16.1 Introduction
	16.2 Jordan–Schwinger Map
	16.3 The Spin States in Terms of Oscillator\'s Wave Functions
	16.4 H-Representation
	16.5 Qubit State
	16.6 Quantum Suprematism Representation of Qubit States
	16.7 Generic Qudit State in the H-Representation
	16.8 Conclusions
	References
17 On Sympletic Lifts of Actions  for Complete Lagrangian Fibrations
	17.1 Introduction
	17.2 Symplectic Cotangent Lifts of Actions on a Manifold
	17.3 Symplectic Lifts of Actions on a Complete G-Lagrangian Fibration
	References
18 Some Properties of Multisymplectic Manifolds
	18.1 Introduction
	18.2 Multisymplectic Manifolds
	18.3 Hamiltonian Structures in Multisymplectic Manifolds
	18.4 Characteristic Submanifolds of Multisymplectic Manifolds
	18.5 Canonical Models for Multisymplectic Manifolds. Darboux-Type Coordinates
	18.6 Other Kinds of Multisymplectic Manifolds
	18.7 Invariance Theorems
	18.8 Conclusions and Discussion
	References
19 A Simple Model of Double Dynamics on Lie Groups
	19.1 Introduction
	19.2 The Isotropic Rigid Rotator
	19.3 Poisson-Lie Groups and the Double Lie Algebra mathfraksl(2,mathbbC)
	19.4 A Model over the Dual Group SB(2,mathbbC)
	19.5 The Generalized Action
		19.5.1 The Hamiltonian Formalism
		19.5.2 Poisson Brackets
	19.6 Conclusions
	References
20 Loops of Legendrians in Contact 3-Manifolds
	20.1 Introduction
	20.2 Preliminaries on Contact 3-Manifolds
		20.2.1 Contact 3-Manifolds
	20.3 Legendrian Submanifolds
		20.3.1 Legendrian Submanifolds
		20.3.2 Classical Invariants
		20.3.3 Invariants for Loops of Legendrian Embeddings
	20.4 The Formal Viewpoint
		20.4.1 Formal Legendrian Embeddings
	20.5 The Action of Cont(mathbbS3,ξstd) on the Space mathfrakLeg\"0362mathfrakLeg(mathbbS3,ξstd)
		20.5.1 The Action of the Contactomorphism Group on the Space of Legendrians
		20.5.2 Homotopy Injection of Cont(mathbbS3,ξstd) in mathfrakLeg\"0362mathfrakLeg(mathbbS3,ξstd)
		20.5.3 Non Injectivity of the Map π1 (mathfrakLeg\"0362mathfrakLeg(mathbbS3,ξstd),L)rightarrowπ1(mathcalK\"0362mathcalK(mathbbS3),L)
	References
Author Index