[Paper Review] The Ultraviolet Structure of Quantum Field Theories. Part 1: Quantum Mechanics

Djordje Radicevic|arXiv (Cornell University)|May 24, 2021

Noncommutative and Quantum Gravity Theories45 references4 citations

TL;DR

This paper introduces a systematic framework—taming—for deriving continuum quantum mechanics from finite-dimensional lattice theories by enforcing smoothness and compactness on wavefunctions. It demonstrates that canonical commutation relations, contact terms, spacetime symmetries, and supersymmetry algebras emerge naturally in the tame subspace, and proves that any lattice supersymmetric theory must have a vanishing Witten index.

ABSTRACT

This paper fires the opening salvo in the systematic construction of the lattice-continuum correspondence, a precise dictionary that describes the emergence of continuum quantum theories from finite, nonperturbatively defined models ("lattice theories"). Here the focus will be on quantum field theory in (0+1)D, i.e. quantum mechanics. The main conceptual achievement is an explicit and systematic procedure for reducing a theory with a large but finite Hilbert space to a subtheory in which wavefunctions satisfy prescribed smoothness and compactness constraints. This reduction, here named taming, in effect defines quantum mechanics on a continuum target space. When appropriate lattice theories are tamed, many familiar continuum notions explicitly emerge, e.g. canonical commutation relations, contact terms in correlation functions, continuous spacetime symmetries, and supersymmetry algebras. All of these are thus "put on the lattice" using the present framework. This analysis also leads to further insights into old subjects: for example, it is proven that any supersymmetric lattice theory must have a vanishing Witten index.

Motivation & Objective

• To establish a rigorous, systematic lattice-continuum correspondence in quantum field theory by focusing on subsectors of finite Hilbert spaces.
• To resolve foundational issues in QFT by constructing continuum-like behavior from nonperturbative, finite models without relying on functional analysis.
• To demonstrate that key continuum structures—such as commutation relations, contact terms, and spacetime symmetries—emerge from lattice systems via controlled reductions.
• To analyze supersymmetry in finite Hilbert spaces and prove that the Witten index must vanish in any lattice realization of exact SUSY.
• To provide a blueprint for extending the framework to higher-dimensional QFTs through target space taming and spatial smoothing.

Proposed method

• Introduce a procedure called 'taming' to project a large but finite Hilbert space onto a subspace where wavefunctions satisfy smoothness and compactness constraints.
• Use controlled reductions (smoothings and tamings) of the clock algebra to approximate continuum quantum mechanics models with arbitrary precision.
• Express the path integral formulation by restricting the set of states inserted at each time step, leading to effective actions with familiar quadratic forms.
• Apply uncontrolled temporal smoothing in path integrals, supplemented by counterterms, to recover continuum-like behavior and correlation functions.
• Construct a non-Lagrangian, minimal supersymmetric model on a finite Hilbert space and show that the SUSY algebra only emerges after taming.
• Leverage the lattice-based definition of saddle point sums and distinguish compact vs. noncompact QM via the taming procedure.

Experimental results

Research questions

• RQ1 How can continuum quantum mechanics be systematically derived from finite, nonperturbatively defined lattice theories?
• RQ2 Under what conditions do canonical commutation relations and contact terms in correlation functions emerge from lattice models?
• RQ3 Can spacetime symmetries and supersymmetry algebras be realized on a finite Hilbert space, and if so, how?
• RQ4 What constraints does exact supersymmetry impose on lattice theories, particularly regarding the Witten index?
• RQ5 How does the taming procedure enable the emergence of continuum-like structures in a finitary framework?

Key findings

• The taming procedure successfully recovers the free particle on ℝ and S¹, the harmonic oscillator, and the supersymmetric harmonic oscillator from a finite clock algebra with arbitrary precision.
• Path integral formulations on the lattice yield effective actions with standard quadratic forms when the taming parameters are chosen appropriately, and the resulting path integrals approximate exact results as per equation (5.27).
• Counterterms and contact terms in correlation functions arise naturally from the lattice-based smoothing and taming procedures, providing a nonperturbative origin for these continuum features.
• Spacetime symmetries, including reparameterization invariance, emerge from the lattice after appropriate taming and smoothing, even in the absence of explicit continuum structure.
• The Witten index of any lattice theory with exact supersymmetry must be zero, implying all states are at least doubly degenerate.
• Supersymmetry algebra and the symmetry between bosons and fermions only become manifest after taming, demonstrating that the symmetry is emergent in the tame subspace.