Snowmass White Paper: The Quest to Define QFT

Abstract

This article provides a review of the literature on rigorous definitions and constructions in Quantum Field Theory, spanning the period of seven decades. Comparing with the ideas and constructions found in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians.

arXiv comment12 pages + references; Please send your comments! v2: added more references, improved and expanded several statements, improved the discussion of CFT

References in corpus (105)

• Quantum Gravity at a Lifshitz Point
• Notes on Some Entanglement Properties of Quantum Field Theory
• The generally covariant locality principle -- A new paradigm for local quantum physics
• Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds
• Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime
• The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes
• Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime
• Quantum fields in curved spacetime
• Modular Structure and Duality in Conformal Quantum Field Theory
• Exotic $U(1)$ Symmetries, Duality, and Fractons in 3+1-Dimensional Quantum Field Theory
• Sphere Partition Functions and the Zamolodchikov Metric
• Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes
• Field Theories With a Vector Global Symmetry
• On the Renormalization Group in Curved Spacetime
• Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime
• Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory
• A local (perturbative) construction of observables in gauge theories: the example of QED
• Marginal triviality of the scaling limits of critical 4D Ising and ${\varphi }_4^4$ models
• Axiomatic quantum field theory in curved spacetime
• Renormalized Quantum Yang-Mills Fields in Curved Spacetime
• Perturbative quantum gauge theories on manifolds with boundary
• The low-energy limit of some exotic lattice theories and UV/IR mixing
• Conformal Subnets and Intermediate Subfactors
• Entanglement entropy and superselection sectors I. Global symmetries
• Entanglement measures and their properties in quantum field theory
• Classical BV theories on manifolds with boundary
• Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems
• QFT on curved spacetimes: axiomatic framework and examples
• Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity
• The Necessity of the Hadamard Condition
• Lifshitz-type Quantum Field Theories in Particle Physics
• Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion
• Batalin-Vilkovisky formalism in the functional approach to classical field theory
• Modular Groups of Quantum Fields in Thermal States
• The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory
• Classification of Two-dimensional Local Conformal Nets with c<1 and 2-cohomology Vanishing for Tensor Categories
• Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law
• An Algebraic Spin and Statistics Theorem
• A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework
• More Exotic Field Theories in 3+1 Dimensions
• Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime
• Entropic order parameters for the phases of QFT
• Quantization of Maxwell's equations on curved backgrounds and general local covariance
• Dynamical locality and covariance: What makes a physical theory the same in all spacetimes?
• Functional Integral Construction of the Thirring model: axioms verification and massless limit
• Removal of violations of the Master Ward Identity in perturbative QFT
• On completeness and generalized symmetries in quantum field theory
• Charged sectors, spin and statistics in quantum field theory on curved spacetimes
• "Vacuum-like" Hadamard states for quantum fields on curved spacetimes
• Quantized Abelian principal connections on Lorentzian manifolds
• Massless Sine-Gordon and Massive Thirring Models: proof of the Coleman's equivalence
• Distributions in CFT II. Minkowski Space
• Generalized Symmetries of the Graviton
• On the Reeh-Schlieder Property in Curved Spacetime
• Linear bosonic and fermionic quantum gauge theories on curved spacetimes
• Star products and perturbative quantum field theory
• Fermionic fields in the functional approach to classical field theory
• Quantum Field Theory on Curved Backgrounds, I. The Euclidean Functional Integral
• Distributions in CFT I. Cross-Ratio Space
• Endomorphisms and automorphisms of locally covariant quantum field theories
• Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization
• Superselection Sectors and General Covariance.I
• Quantum Field Theory on Curved Backgrounds
• Homotopy theory of algebraic quantum field theories
• A C*-algebraic approach to interacting quantum field theories
• The Renormalization Group According to Balaban - I. Small fields
• KMS, etc
• Markov quantum fields on a manifold
• Dynamical locality of the free scalar field
• AQFT from n-functorial QFT
• Homotopy colimits and global observables in Abelian gauge theory
• Operads for algebraic quantum field theory
• Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT
• Differential cohomology and locally covariant quantum field theory
• The stack of Yang-Mills fields on Lorentzian manifolds
• Representations of Conformal Nets, Universal C*-Algebras and K-Theory
• The Renormalization Group According to Balaban - III. Convergence
• The Renormalization Group According to Balaban - II. Large fields
• The split property for locally covariant quantum field theories in curved spacetime
• Quantization of the Maxwell field in curved spacetimes of arbitrary dimension
• Linear Yang-Mills theory as a homotopy AQFT
• N=2 superconformal nets
• Higher Structures in Algebraic Quantum Field Theory
• Constructive Field Theory and Applications: Perspectives and Open Problems
• A cellular topological field theory
• Dynamical locality of the free Maxwell field
• Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds
• Dynamical locality of the nonminimally coupled scalar field and enlarged algebra of Wick polynomials
• Quantum field theories on categories fibered in groupoids
• Locally covariant quantum field theory with external sources
• Spin, statistics, orientations, unitarity
• The Virasoro vertex algebra and factorization algebras on Riemann surfaces
• Renormalized Perturbation Theory: A Missing Chapter
• Categorification of algebraic quantum field theories
• Regularization in quantum field theory from the causal point of view
• Asymptotic Freedom in the BV Formalism
• Pure quasifree states of the Dirac field from the fermionic projector
• Lorentzian 2d CFT from the pAQFT perspective
• Dynamical C*-algebras and kinetic perturbations
• Algebraic Approach to the 1/N Expansion in Quantum Field Theory
• Universal factorization spaces and algebras
• Conformal nets IV: The 3-category
• From path integrals to dynamical algebras: a macroscopic view of quantum physics
• Operator algebras and vertex operator algebras
• Coloring Operads for Algebraic Field Theory

Cited by in corpus (4)

• QCD Running Couplings and Effective Charges
• von Neumann Subfactors and Non-invertible Symmetries
• Self-adjointness and domain of generalized spin-boson models with mild ultraviolet divergences
• Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition