[Paper Review] The Ultraviolet Structure of Quantum Field Theories. Part 2: What is Quantum Field Theory?

Djordje Radicevic|arXiv (Cornell University)|May 25, 2021

Black Holes and Theoretical Physics75 references4 citations

TL;DR

This paper proposes a finitary framework for nonperturbatively defining continuum quantum field theories (cQFTs) by isolating low-energy effective theories from large but finite lattice systems through conserved entanglement patterns at short distances. It derives the operator product expansion in free scalar CFT from the lattice $π_K$ clock model at large $K$, and establishes a novel, self-consistent symmetry breaking structure in the BKT regime, while generalizing Noether’s theorem to discrete symmetries via a lattice path integral formalism with manifest continuum symmetries.

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 $\mathbb{Z}_K$ clock model at large $K$. 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.

Motivation & Objective

• To provide a nonperturbative, finitary definition of continuum quantum field theories (cQFTs) by reducing large finite quantum systems with conserved short-distance entanglement patterns.
• To resolve long-standing challenges in lattice-continuum correspondence, such as chiral anomalies, Chern-Simons theory, and the emergence of advanced cQFT structures like current algebras and OPE.
• To establish a rigorous lattice derivation of known continuum results—e.g., the OPE of vertex operators in free scalar CFT—starting from a discrete theory.
• To generalize Noether’s theorem to discrete symmetries by constructing path integrals with manifest continuum symmetries (e.g., rotations, dilatations) in a lattice-based formalism.
• To clarify the conditions under which continuum limits emerge, especially in 1+1D systems, and to define the role of nontrivial background structures like spin structures and smoothing backgrounds.

Proposed method

• Introduce a precontinuum basis defined by commuting particle number operators $n_k$ with integer eigenvalues and associated ladder operators, generalizing Fock space occupation numbers without assuming particle statistics or identical particle symmetry.
• Define a continuum basis by restricting ladder operators to a small subset $\mathbb{P}_S \subset \mathbb{P}$ of momenta, where excitations at $k \notin \mathbb{P}_S$ cost high energy and thus become effectively classical.
• Identify the operators $n_k$ for $k \notin \mathbb{P}_S$ as central elements of the continuum algebra, enforcing classical behavior at short distances and enabling the emergence of cQFT structure.
• Construct path integrals for bosons and fermions in the finitary formalism, ensuring they inherit manifest continuum symmetries (e.g., infinitesimal rotations, dilatations) from the lattice construction.
• Use the lattice Hamiltonian (2.2) to model general precontinuum interactions, including masses, chemical potentials, BCS terms, and symmetry-breaking potentials, showing how they affect occupation numbers $\langle n_k \rangle$ without altering the precontinuum basis.
• Generalize Noether’s theorem to discrete symmetries by showing that conserved currents arise from lattice symmetries in the continuum limit, with the path integral formalism preserving these symmetries nonperturbatively.

Experimental results

Research questions

• RQ1 How can a finite quantum system with a large but finite Hilbert space give rise to a well-defined continuum quantum field theory through conserved entanglement patterns at short distances?
• RQ2 What is the lattice origin of the operator product expansion in free scalar conformal field theory, and how can it be derived directly from the $\mathbb{Z}_K$ clock model at large $K$?
• RQ3 How does the self-consistency of the lattice-continuum correspondence constrain the symmetry breaking structure in the weak-coupling, BKT regime of the clock model?
• RQ4 Can Noether’s theorem be generalized to discrete symmetries in a nonperturbative, lattice-based formalism that preserves continuum symmetries in the path integral?
• RQ5 Under what conditions are temporal smoothing or restricting the spacetime torus to a disk allowed operations in the lattice-continuum correspondence, and when does universality fail?

Key findings

• The paper provides a direct lattice derivation of the operator product expansion (OPE) of vertex operators in the free scalar conformal field theory, confirming the consistency of the finitary framework with known continuum results.
• A novel, self-consistent proposal for the symmetry breaking structure in the $\mathbb{Z}_K$ clock model at weak coupling and large $K$ is derived, clarifying the precise meaning of 'continuous' in the context of the Coleman-Mermin-Wagner theorem in 1+1D.
• The framework establishes that the operators $n_k$ for $k \notin \mathbb{P}_S$ are central in the continuum algebra, implying they behave classically and enabling the emergence of cQFT from a finite system.
• The path integral formalism constructed here inherits manifest continuum symmetries—such as infinitesimal rotations and dilatations—providing a nonperturbative lattice definition of continuum dynamics.
• Noether’s theorem is generalized to discrete symmetries by showing that conserved currents in the continuum arise from lattice symmetries in the reduced theory, with the path integral preserving these symmetries exactly.
• The analysis reveals that nontrivial target space spin structures (or their generalizations) may become essential in cQFTs when smoothing backgrounds $p^{\mathrm{cl}}_x$ are included, suggesting a deeper role for topological data in the continuum limit.