[Paper Review] Lattice p-Form Electromagnetism and Chain Field Theory

Derek K. Wise|ArXiv.org|Oct 8, 2005

Noncommutative and Quantum Gravity Theories25 references3 citations

TL;DR

This paper introduces a discrete formulation of p-form electromagnetism using chain complexes of abelian groups, enabling a generalization of lattice gauge theory to arbitrary discrete spacetimes beyond regular lattices. The key contribution is the development of 'chain field theory'—a discrete analog of topological quantum field theory that supports time evolution across spacelike slices and allows for quantization via the Euclidean path integral, with convergence ensured by Hodge theory and cohomological structure.

ABSTRACT

Since Wilson's work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as p-form electromagnetism, including the Kalb-Ramond field in string theory, and its nonabelian generalizations. It is desirable to discretize such `higher gauge theories' in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of p-form electromagnetism. We show that discrete p-form electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for p-form electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of p-form electromagnetism as a `chain field theory' -- a theory analogous to topological quantum field theory, but with chain complexes replacing manifolds. This, in particular, gives a notion of time evolution from one `spacelike slice' of discrete spacetime to another.

Motivation & Objective

• To develop a discrete, gauge-invariant formulation of p-form electromagnetism that generalizes lattice gauge theory beyond regular cubical lattices.
• To construct a discrete spacetime model using n-complexes that supports arbitrary geometric structures, not just regular lattices.
• To provide a quantization procedure for p-form electromagnetism using the Euclidean path integral formalism on discrete spacetime.
• To establish a framework—'chain field theory'—that generalizes topological quantum field theory by replacing manifolds with chain complexes.
• To demonstrate discretization independence and convergence of path integrals in p+1 dimensions using Hodge theory and cohomological criteria.

Proposed method

• Models discrete spacetime as an n-complex, where p-cells represent p-dimensional geometric elements, generalizing the notion of a lattice.
• Represents p-form fields as cochains in a cochain complex, with the exterior derivative d realized as a boundary operator between chain groups.
• Defines the action for discrete p-form electromagnetism using a quadratic form on cochains, analogous to the continuum Maxwell action.
• Applies the Gaussian path integral formalism on tori to compute expectation values, leveraging the structure of symmetric positive-definite matrices and their inverses.
• Uses Hodge's theorem and cohomological decomposition to analyze convergence of path integrals and ensure topological invariance.
• Derives the discrete p-form Maxwell equations from the action principle, showing consistency with the continuum limit.

Experimental results

Research questions

• RQ1 How can p-form electromagnetism be discretized in a way that preserves gauge invariance and allows for general spacetime geometry?
• RQ2 Can a discrete spacetime model be constructed that avoids the symmetry-breaking of regular lattices and supports diffeomorphism-like invariance?
• RQ3 What is the appropriate discrete analog of topological quantum field theory for p-form fields?
• RQ4 How can the Euclidean path integral be defined and shown to converge for discrete p-form electromagnetism?
• RQ5 What role does cohomology play in ensuring discretization independence and topological invariance in the quantum theory?

Key findings

• The paper establishes that discrete p-form electromagnetism can be formulated algebraically using chain complexes of abelian groups, providing a rigorous discrete analog of continuum field theory.
• The path integral for U(1) p-form electromagnetism converges when the quadratic form in the action is positive definite, ensuring well-defined quantum amplitudes.
• The model allows for arbitrary discrete spacetimes, not restricted to regular cubical lattices, thus supporting a broader class of geometric discretizations.
• The theory exhibits discretization independence in p+1 dimensions, meaning physical observables do not depend on the specific choice of discrete spacetime structure.
• The framework of chain field theory provides a notion of time evolution from one spacelike slice to another, generalizing the concept of dynamics in discrete field theories.
• The p-form Bohm-Aharonov effect is recovered in the discrete setting, confirming the topological nature of the theory through cohomological invariants.