The Ultraviolet Structure of Quantum Field Theories. Part 2: What is Quantum Field Theory?
arXiv:2105.12147
Abstract
This paper proposes a general framework for nonperturbatively defining continuum quantum field theories. Unlike most such frameworks, the one offered here is finitary: continuum theories are defined by reducing large but finite quantum systems to subsystems with conserved entanglement patterns at short distances. This makes it possible to start from a lattice theory and use rather elementary mathematics to isolate the entire algebraic structure of the corresponding low-energy continuum theory. The first half of this paper illustrates this approach through a persnickety study of (1 + 1)D continuum theories that emerge from the clock model at large . This leads to a direct lattice derivation of many known continuum results, such as the operator product expansion of vertex operators in the free scalar CFT. Many new results are obtained too. For example, self-consistency of the lattice-continuum correspondence leads to a rich, novel proposal for the symmetry breaking structure of the clock model at weak coupling, deep in the BKT regime. This also makes precise what one means by "continuous" when saying that continuous symmetries cannot be broken in (1+1)D. The second half of this paper is devoted to path integrals for continuum theories of bosons and fermions defined in this finitary formalism. The path integrals constructed here have both nonperturbative lattice definitions and manifest continuum properties, such as symmetries under infinitesimal rotations or dilatations. Remarkably, this setup also makes it possible to generalize Noether's theorem to discrete symmetries.
arXiv comment92 pages, 2 figures. v2: Minor improvements. Comments from readers are welcome!
References in corpus (30)
- Symmetry protected topological orders and the group cohomology of their symmetry group
- Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation
- Analytic Smearing of SU(3) Link Variables in Lattice QCD
- Theta, Time Reversal, and Temperature
- A Duality Web in 2+1 Dimensions and Condensed Matter Physics
- Chiral Fermions on the Lattice
- SU(N) gauge theories in 2+1 dimensions
- Free parafermions
- Lattice supersymmetry and order-disorder coexistence in the tricritical Ising model
- Restoration of Rotational Symmetry in the Continuum Limit of Lattice Field Theories
- Parafermionic conformal field theory on the lattice
- Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
- Locally smeared operator product expansions in scalar field theory
- Conformal fields and operator product expansion in critical quantum spin chains
- Conformal invariance of lattice models
- Emergence of conformal symmetry in quantum spin chains: anti-periodic boundary conditions and supersymmetry
- Discrete Holomorphicity at Two-Dimensional Critical Points
- Quantum Yang-Mills theory: an overview of a programme
- Chiral and Geometric Anomalies in Finite Systems
- Spin Structures and Exact Dualities in Low Dimensions
- Lattices and Their Continuum Limits
- Continuum Limits of Quantum Lattice Systems
- Canonical quantization of 1+1-dimensional Yang-Mills theory: An operator-algebraic approach
- Yang-Mills for probabilists
- Abelian Bosonization, OPEs, and the "String Scale" of Fermion Fields
- The Lattice-Continuum Correspondence in the Ising Model
- The Ultraviolet Structure of Quantum Field Theories. Part 3: Gauge Theories
- The Ultraviolet Structure of Quantum Field Theories. Part 1: Quantum Mechanics
- Conservation Laws and Non-Lie Symmetries
- Discrete Noether Currents