Strongly Interacting Matter under Rotation

Contents
About the Editors
1 Strongly Interacting Matter Under Rotation: An Introduction
	1.1 Milestones
	1.2 Introduction
	1.3 Accessing Subatomic Vorticity
	1.4 From Signal to Physics
		1.4.1 Hydrodynamics as the Basis to Understand Hyperon Polarization
		1.4.2 Vector Meson Spin Alignment—More Complicated Physics?
		1.4.3 Future Experimental Work
	1.5 Summary and Outlook
2 Polarization in Relativistic Fluids: A Quantum Field Theoretical Derivation
	2.1 Introduction
	2.2 The Spin Density Matrix and the Definition of Mean Spin
	2.3 The Single-Particle Limit and Global Equilibrium Factorization
	2.4 The Covariant Wigner Function
		2.4.1 The Scalar Field
		2.4.2 The Dirac Field
	2.5 Fermion Polarization and the Covariant Wigner Function
	2.6 Polarization From the Angular Momentum Operator
	2.7 Local Thermodynamic Equilibrium
		2.7.1 Polarization at Local Thermodynamic Equilibrium
	2.8 Summary and Outlook
3 Thermodynamic Equilibrium of Massless Fermions with Vorticity, Chirality and Electromagnetic Field
	3.1 Introduction
	3.2 General Global Equilibrium with Electromagnetic Field
		3.2.1 Vanishing Electromagnetic Field
	3.3 Dirac Field in External Electromagnetic Field
		3.3.1 Symmetries in Constant Electromagnetic Field
	3.4 Chiral Fermions in Constant Magnetic Field
		3.4.1 Exact Thermal Solutions
		3.4.2 Thermodynamic Potential
		3.4.3 Chiral Fermion Propagator in Magnetic Field
		3.4.4 Electric Current Mean Value
		3.4.5 Axial Current Mean Value
	3.5 Constant Vorticity and Electromagnetic Field
		3.5.1 Expansion on Thermal Vorticity
		3.5.2 Currents and Chiral Anomaly
4 Exact Solutions in Quantum Field Theory Under Rotation
	4.1 Introduction
	4.2 Relativistic Kinetic Theory
		4.2.1 Rigidly Rotating Thermal Distribution
		4.2.2 Macroscopic Quantities
	4.3 Quantum Rigidly Rotating Thermal States
	4.4 Mode Solutions in Cylindrical Coordinates
	4.5 Quantum Stationary Thermal Expectation Values
		4.5.1 Fermion Condensate
		4.5.2 Charge Current
		4.5.3 Stress-Energy Tensor
	4.6 Quantum Rigidly Rotating Thermal Expectation Values
		4.6.1 Fermion Condensate
		4.6.2 Charge Current
		4.6.3 Axial Current
	4.7 Hydrodynamic Analysis of the Quantum Stress-Energy Tensor
		4.7.1 Stress-Energy Tensor Expectation Values
		4.7.2 Thermometer Frame
		4.7.3 Quantum Corrections to the SET
	4.8 Rigidly Rotating Quantum Systems in Curved Space-Time
	4.9 Summary
5 Particle Polarization, Spin Tensor, and the Wigner Distribution in Relativistic Systems
	5.1 Introduction
	5.2 Relativistic Kinetic Theory and Its Limitations
	5.3 The Relativistic Spin Tensor as a Polarization Sensitive Macroscopic Object
	5.4 Particle Polarization, the Wigner Distribution, and the Polarization Flux Pseudotensor
	5.5 Summary
6 Quantum Kinetic Description of Spin and Rotation
	6.1 Introduction
	6.2 Semi-classical Approaches
	6.3 Wigner Function Formalism
	6.4 Spin Polarization in Transport Theory
	6.5 Anomaly Induced Transport Theory
	6.6 Degenerate to Hydrodynamics
	6.7 Experiments and Numerical Simulations
	6.8 Summary
7 Global Polarization Effect and  Spin-Orbit Coupling in Strong  Interaction
	7.1 Introduction
	7.2 Orbital Angular Momenta of QGP in HIC
		7.2.1 The Reaction Plane in HIC
		7.2.2 The Global Orbital Angular Momentum
		7.2.3 The Transverse Gradient of the Momentum Distribution and the Local Orbital Angular Momentum
	7.3 Spin-Orbit Coupling in a Relativistic Quantum System
		7.3.1 Dirac Equation and Spin-Orbit Coupling
		7.3.2 Spin-Orbit Coupling in Systems Under Electromagnetic Interactions
		7.3.3 Spin-Orbit Coupling in Systems Under Strong Interactions
	7.4 Theoretical Predictions on the Global Polarization Effect of QGP in HIC
		7.4.1 Global Quark Polarization in QGP in HIC
		7.4.2 A Kinetic Approach for Quark Polarization Rate
		7.4.3 Global Hadron Polarization in HIC
		7.4.4 Comparison with Experiments
	7.5 Summary and Outlook
8 Vorticity and Polarization in Heavy-Ion Collisions: Hydrodynamic Models
	8.1 Introduction: Vorticities in a Fluid
	8.2 Polarization of Particles in the Fluid
	8.3 Hydrodynamic Modelling of Heavy-Ion Collisions
	8.4 Hydrodynamic Calculations at sqrtsNN=7…62 GeV
	8.5 Hydrodynamic Calculations at sqrtsNN=200 and 2760 GeV
	8.6 Acceleration, Grad T and Vorticity Contributions to Polarization
9 Vorticity and Spin Polarization in Heavy Ion Collisions: Transport Models
	9.1 Introduction
	9.2 Fluid Vorticity
		9.2.1 Non-relativistic Case
		9.2.2 Relativistic Case
	9.3 Spin Polarization in a Vortical Fluid
	9.4 Vorticity in Heavy Ion Collisions
		9.4.1 Setup of Computation in Transport Models
		9.4.2 Results for Kinematic Vorticity
		9.4.3 Results for Thermal Vorticity
	9.5 Λ Polarization in Heavy Ion Collisions
	9.6 Summary
10 Connecting Theory to Heavy Ion Experiment
	10.1 Introduction
	10.2 Global Polarization Transfer to the Daughter
	10.3 Spin Density Matrix for the Mother and Its Polarization
	10.4 Local Polarization Transfer to the Daughter
	10.5 Average Over the Momentum of the Mother
	10.6 Theoretical Predictions and Sign Puzzles
11 QCD Phase Structure Under Rotation
	11.1 Introduction
	11.2 Rotating Frame
	11.3 Nambu–Jona-Lasinio Model
	11.4 Rotating Fermions Without Boundary
	11.5 Boundary Conditions
	11.6 Rotating Fermions with Background Magnetic Field
	11.7 Inhomogeneity of Chiral Condensate: A BdG Treatment
	11.8 Mesonic Superfluidity
	11.9 Summary
12 Relativistic Decomposition of the Orbital and the Spin Angular Momentum in Chiral Physics and Feynman\'s Angular Momentum Paradox
	12.1 Prologue
	12.2 Basics—Angular Momenta in an Abelian Gauge Theory
	12.3 Dirac Fermions and Physical and Pure Gauge Potentials
	12.4 Potential Angular Momentum and Physical Interpretation
	12.5 Feynman\'s Angular Momentum Paradox and Possible Relevance to the Relativistic Nucleus–Nucleus Collision
	12.6 Epilogue