# Dirac's algorithm in the presence of boundaries: a practical guide to a geometric approach

@article{JFernandoBarbero2019DiracsAI, title={Dirac's algorithm in the presence of boundaries: a practical guide to a geometric approach}, author={G. J.FernandoBarbero and Bogar D'iaz and Juan Margalef-Bentabol and Eduardo J S Villase{\~n}or}, journal={arXiv: General Relativity and Quantum Cosmology}, year={2019} }

The goal of this paper is to propose and discuss a practical way to implement the Dirac algorithm for constrained field models defined on spatial regions with boundaries. Our method is inspired in the geometric viewpoint developed by Gotay, Nester, and Hinds (GNH) to deal with singular Hamiltonian systems. We pay special attention to the specific issues raised by the presence of boundaries and provide a number of significant examples -among them field theories related to general relativity- to… Expand

#### 12 Citations

Three Roads to the Geometric Constraint Formulation of Gravitational Theories with Boundaries

- Computer Science, Physics
- Symmetry
- 2021

The Hamiltonian description of mechanical or field models defined by singular Lagrangians plays a central role in physics. A number of methods are known for this purpose, the most popular of them… Expand

Generalizations of the Pontryagin and Husain-Kuchař actions to manifolds with boundary

- Physics, Mathematics
- Journal of High Energy Physics
- 2019

Abstract
In this paper we study a family of generalizations of the Pontryagin and Husain-Kuchǎr actions on manifolds with boundary. In some cases, they describe well- known models — either at the… Expand

Constructing the theory at the boundary, its dynamics and degrees of freedom

- Physics, Mathematics
- 2020

Given a theory in a region of space-time with boundaries, defined by a Lagrangian, we propose a method to find the corresponding theory at the boundary (either space-like or time-like), that allows… Expand

Constrained systems, generalized Hamilton-Jacobi actions, and quantization

- Physics, Mathematics
- 2020

Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this… Expand

Hamiltonian GNH analysis of the parametrized unimodular extension of the Holst action

- Physics, Mathematics
- 2021

We give a detailed account of the Hamiltonian GNH analysis of the parametrized unimodular extension of the Holst action. The purpose of the paper is to derive, through the clear geometric picture… Expand

Hamiltonian Gotay-Nester-Hinds analysis of the parametrized unimodular extension of the Holst action

- Physics
- 2021

We give a detailed account of the Hamiltonian Gotay-Nester-Hinds (GNH) analysis of the parametrized unimodular extension of the Holst action. The purpose of the paper is to derive, through the clear… Expand

Quantum field theory with dynamical boundary conditions and the Casimir effect

- Physics, Mathematics
- 2020

We study a coupled system that describes the interacting dynamics between a bulk field, confined to a finite region with timelike boundary, and a boundary observable. In our system the dynamics of… Expand

A Non-Local Action for Electrodynamics: Duality Symmetry and the Aharonov-Bohm Effect, Revisited

- Physics, Computer Science
- Symmetry
- 2019

It is shown how the electric-magnetic duality symmetry of the electromagnetic field and the Aharonov–Bohm effect can be derived from the simple harmonic oscillator character of vacuum electrodynamics, while also demonstrating how the magnetic version of the latter naturally arises in an explicitly non-local manner. Expand

Quantum Chern-Simons theories on cylinders: BV-BFV partition functions

- Physics, Mathematics
- 2020

We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I×Σ, with I an interval and dim Σ = 4l+2, in the BVBFV formalism (a refinement of the Batalin–Vilkovisky… Expand

Concise symplectic formulation for tetrad gravity

- Physics, Mathematics
- 2021

We discuss a simple symplectic formulation for tetrad gravity that leads to the real Ashtekar variables in a direct and transparent way. It also sheds light on the role of the Immirzi parameter and… Expand

#### References

SHOWING 1-10 OF 31 REFERENCES

Hamiltonian treatment of linear field theories in the presence of boundaries: a geometric approach

- Physics, Mathematics
- 2013

The purpose of this paper is to study in detail the constraint structure of the Hamiltonian and symplectic-Lagrangian descriptions for the scalar and electromagnetic fields in the presence of spatial… Expand

Generalizations of the Pontryagin and Husain-Kuchař actions to manifolds with boundary

- Physics, Mathematics
- Journal of High Energy Physics
- 2019

Abstract
In this paper we study a family of generalizations of the Pontryagin and Husain-Kuchǎr actions on manifolds with boundary. In some cases, they describe well- known models — either at the… Expand

Dirac's Canonical Quantization Programme

- Mathematics, Physics
- 1996

This is a collection of lectures given at the University of Heidelberg, especially but not exclusively for people who want to learn something about the canonical approach to quantum gravity, which is… Expand

On Dirac's incomplete analysis of gauge transformations

- Physics, Mathematics
- 2004

Dirac’s approach to gauge symmetries is discussed. We follow closely the steps that led him from his conjecture concerning the generators of gauge transformations at a given time —to be contrasted… Expand

Presymplectic manifolds and the Dirac-Bergmann theory of constraints

- Mathematics
- 1978

We present an algorithm which enables us to state necessary and sufficient conditions for the solvability of generalized Hamilton‐type equations of the form ι (X) ω=α on a presymplectic manifold… Expand

Role of surface integrals in the Hamiltonian formulation of general relativity

- Physics
- 1974

Abstract It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces,… Expand

Boundary conditions as Dirac constraints

- Physics
- 2001

Abstract. In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an… Expand

Bering’s proposal for boundary contribution to the Poisson bracket

- Mathematics, Physics
- 2000

It is shown that the Poisson bracket with boundary terms proposed by Bering can be deduced from the Poisson bracket proposed by the present author if one omits terms free of Euler–Lagrange… Expand

Momentum maps and classical relativistic fields. Part 1: Covariant Field Theory

- Mathematics, Physics
- 1997

This is the first paper of a five part work in which we study the Lagrangian and Hamiltonian structure of classical field theories with constraints. Our goal is to explore some of the connections… Expand

Hamiltonian description of the parametrized scalar field in bounded spatial regions

- Physics, Mathematics
- 2016

We study the Hamiltonian formulation for a parametrized scalar field in a regular bounded spatial region subject to Dirichlet, Neumann and Robin boundary conditions. We generalize the work carried… Expand