Canonical quantization of constrained theories on discrete space-time lattices

We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.Comment: 18 pages, RevTex, one figur

Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories

Symplectic integrators offer many advantages for the numerical solution of Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two of the central Hamiltonian systems encountered in plasma physics --- the flow of magnetic field lines and the guiding center motion of magnetized charged particles --- resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates, i.e., coordinates lacking the familiar $(q, p)$ partitioning. Recent efforts made progress toward non-canonical symplectic integration of these systems by appealing to the variational integration framework; however, those integrators were multistep methods and later found to be numerically unstable due to parasitic mode instabilities. This work eliminates the multistep character and, therefore, the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem ``degenerate variational integration''. Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that their resultant Euler-Lagrange equations are systems of first-order ODEs. We show that retaining the same degree of degeneracy when constructing a discrete Lagrangian yields one-step variational integrators preserving a non-canonical symplectic structure on the original Hamiltonian phase space. The advantages of the new algorithms are demonstrated via numerical examples, demonstrating superior stability compared to existing variational integrators for these systems and superior qualitative behavior compared to non-conservative algorithms

Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

The Midpoint Rule as a Variational--Symplectic Integrator. I. Hamiltonian Systems

Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the foundation for such applications. The midpoint rule for Hamilton's equations is examined from the perspectives of variational and symplectic integrators. It is shown that the midpoint rule preserves the symplectic form, conserves Noether charges, and exhibits excellent long--term energy behavior. The energy behavior is explained by the result, shown here, that the midpoint rule exactly conserves a phase space function that is close to the Hamiltonian. The presentation includes several examples.Comment: 11 pages, 8 figures, REVTe

Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces

A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While the present manuscript focuses on global evolution moves and, for simplicity, restricts to Euclidean configuration spaces, a companion article discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a `creation from nothing'. Subtleties arising when applying such a formalism to quantum gravity models are discussed.Comment: 45 pages, 1 appendix, 6 figures (additional explanations, now matches published version