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Quantum variational calculus on a lattice

Shahn Majid, Francisco Simão

TL;DR

This paper develops a quantum-variational framework for noncommutative and lattice spacetimes by constructing a variational double complex $Ω(J^∞)$ and deriving Euler–Lagrange equations along with a partial Noether theorem. It applies the formalism to free scalar fields on lattices $\mathbb{Z}^m$, obtaining the lattice Klein–Gordon equation and, in flat backgrounds, an exact on-shell stress-energy tensor and Noether charges with modified dispersion relations. The authors extend the construction to lattices with background metrics and $U(1)$ gauge fields, obtaining covariant lattice Klein–Gordon equations and conserved currents, including a $U(1)$ charge on discrete bases. They also discuss extensions to general Abelian groups, graphs, nontrivial bundles, and gauged theories, outlining pathways toward quantum-field theory on lattices and noncommutative geometries. Overall, the work provides a rigorous variational foundation for classical field theory on discrete/noncommutative spaces and highlights how canonical conserved quantities adapt in these settings.

Abstract

We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex $Ω(J^\infty)$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a $\Bbb Z^m$ lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice and modified energy-momentum relations.

Quantum variational calculus on a lattice

TL;DR

This paper develops a quantum-variational framework for noncommutative and lattice spacetimes by constructing a variational double complex and deriving Euler–Lagrange equations along with a partial Noether theorem. It applies the formalism to free scalar fields on lattices , obtaining the lattice Klein–Gordon equation and, in flat backgrounds, an exact on-shell stress-energy tensor and Noether charges with modified dispersion relations. The authors extend the construction to lattices with background metrics and gauge fields, obtaining covariant lattice Klein–Gordon equations and conserved currents, including a charge on discrete bases. They also discuss extensions to general Abelian groups, graphs, nontrivial bundles, and gauged theories, outlining pathways toward quantum-field theory on lattices and noncommutative geometries. Overall, the work provides a rigorous variational foundation for classical field theory on discrete/noncommutative spaces and highlights how canonical conserved quantities adapt in these settings.

Abstract

We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice and modified energy-momentum relations.

Paper Structure

This paper contains 17 sections, 14 theorems, 241 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Algebraic classical variational calculus on $M$.
  3. Jet bundle and variational bicomplex
  4. Euler-Lagrange equations
  5. Symmetries and Noether's Theorem
  6. Variational calculus on the integer lattice $\mathbb{Z}$
  7. Construction of the double complex $\Omega(J^\infty)$
  8. Euler-Lagrange equations
  9. Noether current and conserved energy for classical mechanics $\mathbb{Z} \times \mathbb{R} \to \mathbb{Z}$
  10. Conserved stress-energy tensor on Abelian groups and $\mathbb{Z}^m$ lattices.
  11. EL equations and translation symmetries for $X=\mathbb{Z}^m, \mathbb{Z}^{1,m-1}$
  12. Scalar field theory in the (1+1)-dimensional case
  13. Lagrangians on lattices with background metrics and gauge fields
  14. Scalar field theory on $X$ with a generic metric
  15. Conserved charge associated to global $U(1)$ symmetry
  16. ...and 2 more sections

Key Result

Theorem 2.3

cf. Del Given a symmetry $(X_E,\sigma_X)$, there is an associated (on-shell) conserved current$j_X := \sigma_X - \iota_{X_H} (L\mathrm{Vol}) - \iota_{X_V} \Theta \in \Omega^{{\mathrm{top}}-1,0}(J^\infty)$. Then

Figures (5)

  • Figure 1: Discrete time harmonic oscillator for $m=1$ and energy $E=1$ with increasing $\omega$. For the sine modes with $\omega<2$, the dashed line is the amplitude $q_2$ such that $E={1\over 8}m \omega^2(4-\omega^2)q_2^2$, with minimum amplitude for $\omega=\sqrt{2}$. The period $2\pi/\omega'$ is significantly smaller than $2\pi/\omega$ as $\omega$ increases. As $\omega\to 2$ from below, the amplitude blows up and the period tends towards the minimum alternating case. We also see a beat frequency emerging. The $\omega>2$ case has alternating exponential growth. Plots have been smoothly interpolated for visualization.
  • Figure 2: Plots of the dispersion relation, energy and momentum densities at masses $m= 0.9,1.7, 2, 2.25,2.67$ and comparison with the continuum counterparts in the Euclidean case, shown as dashed. The $m>2$ values are the $m'=\sqrt{8-m^2}$ counterparts of the $m<2$ values but with $-0.01$ offset to separate curves in the last plot. For small masses, the lattice and continuum plots almost match. At $m=2$, waves with all $\kappa \in [-\pi,\pi]$ propagate in the lattice. As we approach $m= 2\sqrt{2}$, waves closer to $\kappa = 0$ stop propagating first, until eventually at $m=2\sqrt{2}$, only waves with $\kappa = -\pi,\pi$ propagate. The lower right plot shows the energy density against the momentum density for $A = \sqrt{2/\langle\mathcal{E}_{dens}[\phi] \rangle}$ as in equation \ref{['eq:EPrelEuc']}.
  • Figure 3: Plots for the phase and group velocity for different masses as in Figure \ref{['fig:DispEnMomEuc']}.
  • Figure 4: Dispersion relation, energy density and momentum densities in the Minkowski case for the masses $m= 0.5, \sqrt{2}, 1.9$. At lower masses, $\kappa=0$ is a local minimum of the energy density, while it is a local maximum for waves with higher masses. In the lower right, the energy density is plotted against the momentum density following equation \ref{['eq:EPrelMin']}.
  • Figure 5: Phase and group velocities in the Minkowski case for $m= 0.5, \sqrt{2}, 1.9$ and comparison with their classical counterparts shown dashed.

Theorems & Definitions (32)

  • Example 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 3.1
  • ...and 22 more