Functional Evolution of Free Quantum Fields

Abstract

We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger, and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.

21 pages, RevTeX, minor improvements in exposition, to appear in Classical and Quantum Gravity

References in corpus (1)

• Loop Quantum Gravity

Cited by in corpus (70)

• Quantum Gravity: a Progress Report
• The generally covariant locality principle -- A new paradigm for local quantum physics
• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• A Master Equation for Gravitational Decoherence: Probing the Textures of Spacetime
• The Rest-Frame Instant Form of Metric Gravity
• Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate
• The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field
• Quantum Gowdy $T^3$ model: A unitary description
• Quantization of Midisuperspace Models
• Minkowski vacuum in background independent quantum gravity
• Unitarity and ultraviolet regularity in cosmology
• Unitary evolution in Gowdy cosmology
• Quantum Dynamics of the Polarized Gowdy Model
• The Chrono-geometrical Structure of Special and General Relativity: a Re-Visitation of Canonical Geometrodynamics
• Quantum Gowdy $T^3$ Model: Schrodinger Representation with Unitary Dynamics
• Gravitational effects in macroscopic quantum systems: a first-principles analysis
• On Unitary Time Evolution in Gowdy $T^3$ Cosmologies
• Observations in Quantum Cosmology
• Tall tales from de Sitter space II: Field theory dualities
• Quantum Field Theory based Quantum Information: Measurements and Correlations
• On the Schroedinger Representation for a Scalar Field on Curved Spacetime
• From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect
• Dirac Quantization of Parametrized Field Theory
• Spin-rotation couplings: spinning test particles and Dirac field
• On Unitary Evolution of a Massless Scalar Field In A Schwarzschild Background: Hawking Radiation and the Information Paradox
• Complex Structures for Klein-Gordon Theory on Globally Hyperbolic Spacetimes
• Generalized Schrodinger equation for free field
• Generalized Schroedinger equation in Euclidean field theory
• Quantum evolution of the Hawking state for black holes
• Schrödinger evolution of the Hawking state
• Lessons for Loop Quantum Gravity from Parametrised Field Theory
• Perturbative quantum evolution of the gravitational state and dressing in general backgrounds
• Relational Reality in Relativistic Quantum Mechanics
• Note on Canonical Quantization and Unitary Equivalence in Field Theory
• Nonlinear quantum gravity on the constant mean curvature foliation
• Hybrid Geometrodynamics: A Hamiltonian description of classical gravity coupled to quantum matter
• Functional evolution of quantum cylindrical waves
• Another Proof of Born's Rule on Arbitrary Cauchy Surfaces
• Towards conformal loop quantum gravity
• Quantum probing of null-singularities
• Unitary evolution of free massless fields in de Sitter space-time
• Towards general relativity through parametrized theories
• Maslov's complex germ and the Weyl--Moyal algebra in quantum mechanics and in quantum field theory
• Gaussian transform of the Weil representation
• The Functional Schrodinger Equation in the Semiclassical Limit of Quantum Gravity with a Gaussian Clock Field
• On the Coherent State Path Integral for Linear Systems
• Schrödinger from Wheeler-DeWitt: The Issues of Time and Inner Product in Canonical Quantum Gravity
• An action principle for the quantization of parametric theories and nonlinear quantum cosmology
• On Relativistic Entanglement and Localization of Particles and on their Comparison with the Non-Relativistic Theory
• Post-Minkowskian Gravity: Dark Matter as a Relativistic Inertial Effect?
• Path integral quantization of parametrised field theory
• Bogoliubov Transformation and Schrodinger Representation on Curved Space
• A new mathematical problem related to quantization of fields
• No-Counterterm approach to quantum field theory
• Functional Time Evolution, Anomaly Potentials, and the Geometric Phase
• Bogoliubov Transformations Beyond Shale-Stinespring: Generic $v^{\ast }v$ for bosons
• The background as a quantum observable: Einstein's hole argument in a quasiclassical context
• Mathematical renormalization of Hamiltonian field theories
• Tetrad Gravity and Dirac's Observables
• Relativistic Entanglement from Relativistic Quantum Mechanics in the Rest-Frame Instant Form of Dynamics
• Symmetry Reduction of Quasi-Free States
• Functional evolution of scalar fields in bounded one-dimensional regions
• Relational Description of the Measurement Process in Quantum Field Theory
• Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition
• Relativistic Atomic Physics: from Atomic Clock Synchronization towards Relativistic Entanglement
• On the Work of Henry P. Stapp
• Unitarity in the Schroedinger Formalism of QFT in Curved Space-Time
• Einstein causality of quantum measurements in the Tomonaga-Schwinger picture
• Solution to the problem of time
• Quantization of multidimensional variational principles