Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime

Abstract

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed `wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance scaling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.

latex2e, 41 pages

References in corpus (5)

• A general worldline quantum inequality
• Passivity and microlocal spectrum condition
• The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson-Walker Spacetimes
• The Reeh-Schlieder Property for Quantum Fields on Stationary Spacetimes
• Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime

Cited by in corpus (81)

• The generally covariant locality principle -- A new paradigm for local quantum physics
• Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction
• A Quantum Weak Energy Inequality for Spin-One Fields in Curved Spacetime
• Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems
• The Necessity of the Hadamard Condition
• The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor
• Quantum Field Theory on Curved Backgrounds -- A Primer
• Quantum out-states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property
• Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law
• A smooth introduction to the wavefront set
• On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario
• Construction of Hadamard states by pseudo-differential calculus
• The locally covariant Dirac field
• Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property
• Classical and Quantum Fields on Lorentzian Manifolds
• Nuclearity, Local Quasiequivalence and Split Property for Dirac Quantum Fields in Curved Spacetime
• Approximate KMS states for scalar and spinor fields in Friedmann-Robertson-Walker spacetimes
• Construction of Hadamard states by characteristic Cauchy problem
• Global Existence of Solutions of the Semiclassical Einstein Equation for Cosmological Spacetimes
• The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States
• The generalised principle of perturbative agreement and the thermal mass
• Temperature and entropy-area relation of quantum matter near spherically symmetric outer trapping horizons
• The renormalized locally covariant Dirac field
• On the Backreaction of Scalar and Spinor Quantum Fields in Curved Spacetimes - From the Basic Foundations to Cosmological Applications
• Renormalised fermion vacuum expectation values on anti-de Sitter space-time
• Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states
• Quantum field theory on affine bundles
• Remarks on the Reeh-Schlieder property for higher spin free fields on curved spacetimes
• Trace anomaly for chiral fermions via Hadamard subtraction
• On the issue of imposing boundary conditions on quantum fields
• Hadamard property of the in and out states for Klein-Gordon fields on asymptotically static spacetimes
• Quantization of the Maxwell field in curved spacetimes of arbitrary dimension
• An explicit quantum weak energy inequality for Dirac fields in curved spacetimes
• The Fermionic Signature Operator and Quantum States in Rindler Space-Time
• Quantum Virasoro algebra with central charge c=1 on the horizon of a 2D-Rindler spacetime
• A rigorous geometric derivation of the chiral anomaly in curved backgrounds
• The holographic Hadamard condition on asymptotically Anti-de Sitter spacetimes
• The Fermionic Signature Operator in De Sitter Spacetime
• The Regularized Hadamard Expansion
• Local Thermal Equilibrium in Quantum Field Theory on Flat and Curved Spacetimes
• Hadamard state in Schwarzschild-de Sitter spacetime
• On cosmic hair and "de Sitter breaking" in linearized quantum gravity
• Supersymmetric Field-Theoretic Models on a Supermanifold
• The current density in quantum electrodynamics in external potentials
• Hadamard states for quantized Dirac fields on Lorentzian manifolds of bounded geometry
• An absolute quantum energy inequality for the Dirac field in curved spacetime
• On Wick polynomials of boson fields in locally covariant algebraic QFT
• A Mechanism of Baryogenesis for Causal Fermion Systems
• Constructing Hadamard states via an extended Møller operator
• Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime
• The Lambda CDM-model in quantum field theory on curved spacetime and Dark Radiation
• Local causal structures, Hadamard states and the principle of local covariance in quantum field theory
• Pure quasifree states of the Dirac field from the fermionic projector
• A new class of Fermionic Projectors: Møller operators and mass oscillation properties
• Scale-Invariant Curvature Fluctuations from an Extended Semiclassical Gravity
• Snowmass White Paper: The Quest to Define QFT
• Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds
• Green's functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges
• Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect
• The Semiclassical Einstein Equation on Cosmological Spacetimes
• Holography and $SL(2,\bR)$ symmetry in 2D Rindler spacetime
• Dynamical Casimir Effect with Semi-Transparent Mirrors, and Cosmology
• Hadamard states for quantum Abelian duality
• Dynamical Gravitational Coupling as a Modified Theory of General Relativity
• The Chaplygin Gas Equation of State for the Quantized Free Scalar Field on Cosmological Spacetimes
• The quantization of Proca fields on globally hyperbolic spacetimes: Hadamard states and Møller operators
• Local Thermal Equilibrium States in Relativistic Quantum Field Theory
• The Fermionic Signature Operator and Hadamard States in the Presence of a Plane Electromagnetic Wave
• Local incompatibility of the microlocal spectrum condition with the KMS property along spacelike directions in quantum field theory on curved spacetime
• A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT
• On the local structure of the Klein-Gordon field on curved spacetimes
• Unique fermionic vacuum in de Sitter spacetime from hybrid quantum cosmology
• Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes
• On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods
• Hadamard property of the in and out states for Dirac fields on asymptotically static spacetimes
• Stable, Renormalizable, Scalar Tachyonic Quantum Field Theory with Chronology Protection
• Back-action from inertial and non-inertial Unruh-DeWitt detectors revisited in covariant perturbation theory
• Continuity of KMS States for Quantum Fields on Manifolds
• Explicit construction of Hadamard states for Quantum Field Theory in curved spacetimes
• The microlocal spectrum condition, initial value formulations and background independence
• A Quantum Energy Inequality for a Non-commutative QFT