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Galerkin Formulation of Path Integrals in Lattice Field Theory

Brian K. Tran, Ben S. Southworth

TL;DR

The paper develops a Galerkin formulation of path integrals for lattice field theories by identifying the FE subspace DOFs as the fundamental lattice variables, and by expressing the partition function and expectation values within a finite-dimensional $X_h$ via $S_h=S\circ i_h$ and $\mathfrak{S}_h=\mathbb{S}_h\circ \mathbb{E}_h^{-1}$. It constructs an $n$-point correlation basis on DOFs and shows how spatial correlators for any $x,y\in D$ can be recovered when using a $C^0$ FE space, with a generating functional $Z_h[\vec{J}]$ enabling computation of polynomial observables. A key result is that for a free massive scalar field, the two-point function in the Galerkin discretization is $\langle \phi_h(x)\phi_h(y)\rangle=\mathbf{v}_i(x)(K+M)^{-1}_{ij}\mathbf{v}_j(y)$ and satisfies a weak propagator identity, with convergence to the continuum propagator established via FE error estimates; higher-order FE spaces improve accuracy, including propagator behavior near the Brillouin zone boundary. The framework accommodates curved geometries and symmetry-preserving spaces, and opens routes to higher-order lattice gauge theories and efficient FE-based solvers, with future work including MCMC integration, multigrid strategies, and rigorous convergence analysis.

Abstract

We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the degrees of freedom. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, that this satisfies a weak propagator (or Green's function) identity, in analogy to the continuum case. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.

Galerkin Formulation of Path Integrals in Lattice Field Theory

TL;DR

The paper develops a Galerkin formulation of path integrals for lattice field theories by identifying the FE subspace DOFs as the fundamental lattice variables, and by expressing the partition function and expectation values within a finite-dimensional via and . It constructs an -point correlation basis on DOFs and shows how spatial correlators for any can be recovered when using a FE space, with a generating functional enabling computation of polynomial observables. A key result is that for a free massive scalar field, the two-point function in the Galerkin discretization is and satisfies a weak propagator identity, with convergence to the continuum propagator established via FE error estimates; higher-order FE spaces improve accuracy, including propagator behavior near the Brillouin zone boundary. The framework accommodates curved geometries and symmetry-preserving spaces, and opens routes to higher-order lattice gauge theories and efficient FE-based solvers, with future work including MCMC integration, multigrid strategies, and rigorous convergence analysis.

Abstract

We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the degrees of freedom. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, that this satisfies a weak propagator (or Green's function) identity, in analogy to the continuum case. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.

Paper Structure

This paper contains 8 sections, 2 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Galerkin Formulation of the Partition Function
  3. The Variational Principle on Degrees of Freedom
  4. Galerkin Formulation of Expectation Values
  5. Analytical Example: Free Massive Scalar Field
  6. Convergence Test
  7. A remark on more general finite element spaces
  8. Conclusion

Key Result

Proposition 2.1

The variational principle applied to the discrete action on $X_h$, $\mathbb{S}_h$, and applied to the discrete action on degrees of freedom, $\mathfrak{S}_h$, are equivalent. That is, the following diagram commutes: Here, DEL denotes the discrete Euler--Lagrange equations arising from the variational principle and $i_h^*$ denotes pullback by $i_h$, i.e., $i_h^*S = S \circ i_h$.

Figures (4)

  • Figure 1: Comparison of momentum space propagator with piecewise linear, piecewise linear lumped, piecewise quadratic and piecewise cubic elements for the same $h$ with $h = 1, m = 0.7$. Dashed lines indicate the Brillouin zone $[-\pi/h,\pi/h]$.
  • Figure 2: Comparison of momentum space propagator with piecewise linear, piecewise linear lumped, piecewise quadratic and piecewise cubic elements for the same number of degrees of freedom per unit length with $a = 1, m = 0.7$. Dashed lines indicate the Brillouin zone $[-\pi/a,\pi/a]$.
  • Figure 3: Triangulation with (top left) $N=128$, (top right) $N=512$, (bottom left) $N=2048$, (bottom right) $N=8192$.
  • Figure 4: Convergence test for piecewise linear, quadratic, and cubic elements used to obtain the two-point correlation function for a scalar field on the conformal two-sphere.

Theorems & Definitions (9)

  • Remark 2.1
  • Proposition 2.1: Variational Principle on Degrees of Freedom
  • proof
  • Remark 2.2
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5