Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories (original) (raw)

Canonical analysis of field theories in the presence of boundaries: Maxwell+Pontryagin

We study the canonical Hamiltonian analysis of gauge theories in the presence of boundaries. While the implementation of Dirac's program in the presence of boundaries, as put forward by Regge and Teitelboim, is not new, there are some instances in which this formalism is incomplete. Here we propose an extension to the Dirac formalism-together with the Regge-Teitelboim strategy,-that includes generic cases of field theories. We see that there are two possible scenarios, one where there is no contribution from the boundary to the symplectic structure and the other case in which there is one, depending on the dynamical details of the starting action principle. As a concrete system that exemplifies both cases, we consider a theory that can be seen both as defined on a four dimensional spacetime region with boundaries-the bulk theory-, or as a theory defined both on the bulk and the boundary of the region-the mixed theory-. The bulk theory is given by the 4-dimensional Maxwell + U (1) Pontryagin action while the mixed one is defined by the 4-dimensional Maxwell + 3-dimensional U (1) Chern-Simons action on the boundary. Finally, we show how these two descriptions of the same system are connected through a canonical transformation that provides a third description. The focus here is in defining a consistent formulation of all three descriptions, for which we rely on the geometric formulation of constrained systems, together with the extension of the Dirac-Regge-Teitelboim (DRT) formalism put forward in the manuscript.

Canonical structure and boundary conditions in Yang-Mills theory †)

arXiv: High Energy Physics - Theory, 1999

The canonical structure of pure Yang-Mills theory is analysed in the case when Gauss’ law is satisfied identically by construction. It is shown that the theory has a canonical structure in this case, provided one uses a special gauge condition, which is a natural generalisation of the Coulomb gauge condition of electrodynamics. The emergence of a canonical structure depends critically also on the boundary conditions used for the relevant field variables. Possible boundary conditions are analysed in detail. A comparison of the present formulation in the generalised Coulomb gauge with the well known Weyl gauge (A0 = 0) formulation is made. It appears that the Hamiltonians in these two formulations differ from one another in a non-trivial way. It is still an open question whether these differences give rise to truly different structures upon quantisation. An extension of the formalism to include coupling to fermionic fields is briefly discussed.

HIP-1999-33/TH Canonical structure and boundary conditions in Yang-Mills theory †)

1999

The canonical structure of pure Yang-Mills theory is analysed in the case when Gauss ’ law is satisfied identically by construction. It is shown that the theory has a canonical structure in this case, provided one uses a special gauge condition, which is a natural generalisation of the Coulomb gauge condition of electrodynamics. The emergence of a canonical structure depends critically also on the boundary conditions used for the relevant field variables. Possible boundary conditions are analysed in detail. A comparison of the present formulation in the generalised Coulomb gauge with the well known Weyl gauge (A0 = 0) formulation is made. It appears that the Hamiltonians in these two formulations differ from one another in a non-trivial way. It is still an open question whether these differences give rise to truly different structures upon quantisation. An extension of the formalism to include coupling to fermionic fields is briefly discussed.

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

Classical and Quantum Gravity, 2014

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 boundaries. We carefully discuss the implementation of the geometric constraint algorithm of Gotay, Nester and Hinds with special emphasis on the relevant functional analytic aspects of the problem. This is an important step towards the rigorous understanding of general field theories in the presence of boundaries, very especially when these fail to be regular. The geometric approach developed in the paper is also useful with regard to the interpretation of the physical degrees of freedom and the nature of the constraints when both gauge symmetries and boundaries are present.

On the Hamiltonian formulation of Yang--Mills gauge theories

2005

The Hamiltonian formulation of the theory of J-bundles is given both in the Hamilton--De Donder and in the Multimomentum Hamiltonian geometrical approaches. (3+3) Yang-Mills gauge theories are dealt with explicitly in order to restate them in terms of Einstein-Cartan like field theories.

Boundary structure of gauge and matter fields coupled to gravity

arXiv (Cornell University), 2022

The boundary structure of 3 + 1-dimensional gravity (in the Palatini-Cartan formalism) coupled to to gauge (Yang-Mills) and matter (scalar and spinorial) fields is described through the use of the Kijowski-Tulczijew construction. In particular, the reduced phase space is obtained as the reduction of a symplectic space by some first class constraints and a cohomological description (BFV) of it is presented. 2 Preliminaries In this section we describe some of the mathematical background required in the rest of the paper. In particular, Section 2.1 is devoted to the Kijowski-Tulczijew (KT) construction, Section 2.2 to the BFV formalism and Section 2.3 to the Palatini-Cartan gravity theory. 2.1 The KT construction and the reduced phase space We describe here the Kijowski-Tulczijew [KT79] construction that we will use in the main part of the paper to describe the reduced phase space of the field theories considered. Remark 2. In order to keep the description simple, we describe the construction without details which are collected in the footonotes. Let M be an an N-dimensional manifold with boundary ∂M =: Σ and let F be a vector bundle on M. For a large variety of theories-and in particular the ones at hand-the space of fields F M is in general defined as the space of smooth local sections φ on F , i.e. F M := Γ(M, F), which is in general an infinite-dimensional manifold (inheriting the structure of a Fréchet space) on which we assume that Cartan calculus is defined. A field theory on M is then specified by an action functional S M , obtained by integrating a Lagrangian density L(φ). 2

Quantization of field theories in the presence of boundaries

1996

This paper reviews the progress made over the last five years in studying boundary conditions and semiclassical properties of quantum fields about 4-realdimensional Riemannian backgrounds. For massless spin-1 2 fields one has a choice of spectral or supersymmetric boundary conditions, and the corresponding conformal anomalies have been evaluated by using zeta-function regularization. For Euclidean Maxwell theory in vacuum, the mode-by-mode analysis of BRST-covariant Faddeev-Popov amplitudes has been performed for relativistic and non-relativistic gauge conditions. For massless spin-3 2 fields, the contribution of physical degrees of freedom to one-loop amplitudes, and the 2-spinor analysis of Dirac and Rarita-Schwinger potentials, have been obtained. In linearized gravity, gauge modes and ghost modes in the de Donder gauge have been studied in detail. This program may lead to a deeper understanding of different quantization techniques for gauge fields and gravitation, to a new vision of gauge invariance, and to new points of view in twistor theory.

On covariant and canonical Hamiltonian formalisms for gauge theories

2023

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, specially when the gauge theories are defined over regions with boundaries. Here we consider this issue, by first making precise what we mean by equivalence between the two formalisms. Then we explore several first order gauge theories, and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature for gauge field theories. These results call for a deeper understanding of the subject.

A new geometric setting for classical field theories

Classical and Quantum Integrability, 2003

A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists. The main advantage of this algorithm is that the second order condition is automatically included.