Boundary conditions as Dirac constraints

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 infinite chain of second class constraints which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are noncommutative.

Dynamical rearrangement of gauge symmetry on the orbifold S1/Z2

“…We show that this conjecture is valid in all examples that we know and inclusive we check that in the example of PP-waves [9] our result is fully consistent whereas the previous result it is not. In the literature exist several proposals [10,11,12,13,14,15,16,17,18,19] to solve the problems associated to the procedure of consider the boundary conditions as Dirac's constraints. However, all the previous proposals are not useful in at least in one of the examples that we present, whereas our procedure is fully consistent and systematic in all these examples.…”

Open membranes in a constant C-field background and noncommutative boundary strings

“…Boundary condition as constraint According to [9], we can treat the boundary conditions of an open string as constraints. The consistency conditions of these constraints lead to an infinite chain of secondary constraints, which are all second class.…”

“…where mixed directions are named for their mixtures of some directions [9] and we shall only consider below the directions obeying these mixed boundary conditions. We now go on to the canonical formalism.…”

Noncommutativity in open string: new results in a gauge-independent analysis

“…Also, an alternative Hamiltonian (Dirac [5]) approach based on regarding the Boundary Conditions (BC) as constraints was given in [6]; the corresponding Lagrangian (symplectic) version being done in [7]. The interpretation of the BC as primary constraints usually led to an infinite tower of second class constraints [8], in contrast to the usual Dirac formulation of constrained systems [5,9]. Some other approaches to this problem have been discussed in [10,11].…”