Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A single-particle framework for unitary lattice gauge theory in discrete time

Pablo Arnault, Christopher Cedzich

TL;DR

The paper develops a real-time, unitary lattice gauge theory for a spin-1/2 particle in $(1+1)$ dimensions using a discrete-time quantum walk to build a DQW-based action $S_{\text{DQW}}$. It proves a lattice Noether theorem for internal symmetries, derives a conserved lattice current for the global $U(1)$ symmetry, and couples the matter field to a $U(1)$ gauge field via lattice minimal coupling. It further proposes a real-time lattice gauge-field action and derives the classical lattice Maxwell equations, including lattice Gauss and Ampère laws, with explicit expressions for link variables and plaquettes. The framework unifies quantum cellular automata concepts with lattice gauge theory, offering a strictly local, unitary description of real-time gauge dynamics that is amenable to quantum simulation and potentially sign-problem-free real-time studies, while also outlining open issues such as gauge-field gauging consistency, higher-dimensional generalizations, and fermion-doubling concerns.

Abstract

We construct a real-time lattice-gauge-theory-type action for a spin-1/2 matter field of a single particle on a (1+1)-dimensional spacetime lattice. The framework is based on a discrete-time quantum walk, and is hence inherently unitary and strictly local, i.e., transition amplitudes exactly vanish outside of a lightcone on the lattice. We then provide a lattice Noether's theorem for internal symmetries of this action. We further couple this action to an electromagnetic field by a minimal substitution on the lattice. Finally, we suggest a real-time lattice-gauge-theory-type action for the electromagnetic field in arbitrary spacetime dimensions, and derive its classical equations of motion, which are lattice versions of Maxwell's equations.

A single-particle framework for unitary lattice gauge theory in discrete time

TL;DR

The paper develops a real-time, unitary lattice gauge theory for a spin-1/2 particle in dimensions using a discrete-time quantum walk to build a DQW-based action . It proves a lattice Noether theorem for internal symmetries, derives a conserved lattice current for the global symmetry, and couples the matter field to a gauge field via lattice minimal coupling. It further proposes a real-time lattice gauge-field action and derives the classical lattice Maxwell equations, including lattice Gauss and Ampère laws, with explicit expressions for link variables and plaquettes. The framework unifies quantum cellular automata concepts with lattice gauge theory, offering a strictly local, unitary description of real-time gauge dynamics that is amenable to quantum simulation and potentially sign-problem-free real-time studies, while also outlining open issues such as gauge-field gauging consistency, higher-dimensional generalizations, and fermion-doubling concerns.

Abstract

We construct a real-time lattice-gauge-theory-type action for a spin-1/2 matter field of a single particle on a (1+1)-dimensional spacetime lattice. The framework is based on a discrete-time quantum walk, and is hence inherently unitary and strictly local, i.e., transition amplitudes exactly vanish outside of a lightcone on the lattice. We then provide a lattice Noether's theorem for internal symmetries of this action. We further couple this action to an electromagnetic field by a minimal substitution on the lattice. Finally, we suggest a real-time lattice-gauge-theory-type action for the electromagnetic field in arbitrary spacetime dimensions, and derive its classical equations of motion, which are lattice versions of Maxwell's equations.
Paper Structure (37 sections, 153 equations, 2 figures)

This paper contains 37 sections, 153 equations, 2 figures.

Table of Contents

  1. Introduction
  2. DQW-based LGT-type action for a classical matter field
  3. The continuum equation of motion
  4. The naive discretization
  5. Lattice derivatives
  6. Standard-LGT scheme: naive fermions
  7. The DQW discretization
  8. Modification of naive fermions for unitarity in discrete time
  9. The "underlying" unitary one-step scheme
  10. The lattice action of the matter field
  11. Noether's theorem in discrete spacetime for internal symmetries
  12. The framework
  13. The Noether theorem
  14. The $\mathrm{U}(1)$-gauged matter-field action and equations of motion
  15. The gauging procedure
  16. ...and 22 more sections

Figures (2)

  • Figure 1: The different links involved in Eq. \ref{['eq:gauge_one-step_scheme']} (red) and Eq. \ref{['eq:gauged_second']} (blue). Since $\mathcal{T}_0'$ and $\mathcal{T}_1'$ do not commute, the algebraic relations between the red and the blue links are different. All links, blue and red, are involved in the two-step scheme.
  • Figure 2: Terms involved in the plaquette operator $U_{10}$ from Eq. \ref{['eq:plaq_op']}.