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Constraint analysis for variational discrete systems

Bianca Dittrich, Philipp A Hoehn

TL;DR

This work develops a general canonical formalism for variational discrete systems that may have evolving phase spaces, proving equivalence with the covariant action approach and providing a complete constraint analysis. It introduces dual (pre/post) Legendre transforms and a phase-space extension framework to handle changing numbers of degrees of freedom, and shows how constraints, symplectic structure, and observables behave under both global and local moves. A key contribution is a thorough treatment of how propagating degrees of freedom and reduced phase spaces depend on the chosen initial and final time steps, including precise characterizations of first/second class constraints and symmetry generators in the discrete setting. The results have broad relevance for discrete mechanics, lattice field theory, and quantum gravity approaches that rely on discrete spacetime structures, and they offer a robust foundation for numerical implementations and potential quantization schemes.

Abstract

A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves and naturally incorporates both constant and evolving phase spaces, the latter of which is necessary for a time varying discretization. The different roles of constraints in the discrete and the conditions under which they are first or second class and/or symmetry generators are clarified. The (non-) preservation of constraints and the symplectic structure is discussed; on evolving phase spaces the number of constraints at a fixed time step depends on the initial and final time step of evolution. Moreover, the definition of observables and a reduced phase space is provided; again, on evolving phase spaces the notion of an observable as a propagating degree of freedom requires specification of an initial and final step and crucially depends on this choice, in contrast to the continuum. However, upon restriction to translation invariant systems, one regains the usual time step independence of canonical concepts. This analysis applies, e.g., to discrete mechanics, lattice field theory, quantum gravity models and numerical analysis.

Constraint analysis for variational discrete systems

TL;DR

This work develops a general canonical formalism for variational discrete systems that may have evolving phase spaces, proving equivalence with the covariant action approach and providing a complete constraint analysis. It introduces dual (pre/post) Legendre transforms and a phase-space extension framework to handle changing numbers of degrees of freedom, and shows how constraints, symplectic structure, and observables behave under both global and local moves. A key contribution is a thorough treatment of how propagating degrees of freedom and reduced phase spaces depend on the chosen initial and final time steps, including precise characterizations of first/second class constraints and symmetry generators in the discrete setting. The results have broad relevance for discrete mechanics, lattice field theory, and quantum gravity approaches that rely on discrete spacetime structures, and they offer a robust foundation for numerical implementations and potential quantization schemes.

Abstract

A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves and naturally incorporates both constant and evolving phase spaces, the latter of which is necessary for a time varying discretization. The different roles of constraints in the discrete and the conditions under which they are first or second class and/or symmetry generators are clarified. The (non-) preservation of constraints and the symplectic structure is discussed; on evolving phase spaces the number of constraints at a fixed time step depends on the initial and final time step of evolution. Moreover, the definition of observables and a reduced phase space is provided; again, on evolving phase spaces the notion of an observable as a propagating degree of freedom requires specification of an initial and final step and crucially depends on this choice, in contrast to the continuum. However, upon restriction to translation invariant systems, one regains the usual time step independence of canonical concepts. This analysis applies, e.g., to discrete mechanics, lattice field theory, quantum gravity models and numerical analysis.

Paper Structure

This paper contains 29 sections, 9 theorems, 104 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: variational discrete systems
  3. Global dynamics of variational discrete systems
  4. General properties and definitions
  5. Lagrangian formulation
  6. Discrete Legendre transformations and canonical formulation
  7. Review: Regular discrete systems
  8. Lagrangian formulation
  9. Canonical formulation
  10. Irregular discrete systems
  11. The Lagrangian formulation
  12. The canonical formulation
  13. Evolving phase spaces and configuration and phase space extensions
  14. Bulk variables and Hamilton's principal function
  15. Local dynamics of variational discrete systems
  16. ...and 14 more sections

Key Result

Theorem 3.1

The discrete global Hamiltonian time evolution map $\mathcal{H}_n:\mathcal{C}^-_n\rightarrow\mathcal{C}^+_{n+1}$ satisfies

Figures (15)

  • Figure 1: Illustration of the fibre derivative used in the definition of the post--Legendre transform.
  • Figure 2: The implicit global discrete Hamiltonian time evolution map and pre-- and post--momenta.
  • Figure 3: A global time evolution move for a scalar field on a CDT like triangulation. Here, $Q_n=4$ and $Q_{n+1}=7$. Dashed lines represent periodic identification.
  • Figure 4: A global time evolution move for a scalar field on a triangulation with $Q_n=2$ and $Q_{n+1}=3$. Dashed lines represent periodic identification.
  • Figure 5: A fat slicing of a triangulation.
  • ...and 10 more figures

Theorems & Definitions (30)

  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Example 3.1
  • Definition 3.4
  • Theorem 3.1
  • proof
  • Example 3.2
  • Theorem 3.2
  • proof
  • ...and 20 more