The reduced phase space of an open string in the background B-field

Abstract

The problem of an open string in background B-field is discussed. Using the discretized model in details we show that the system is influenced by infinite number of second class constraints. We interpret the allowed Fourier modes as the coordinates of the reduced phase space. This enables us to compute the Dirac brackets more easily. We prove that the coordinates of the string are non-commutative at the boundaries. We argue that in order to find the Dirac bracket or commutator algebra of the physical variables, one should not expand the fields in terms of the solutions of the equations of motion. Instead, one should impose the set of constraints in suitable coordinates.

Symplectic Quantization of Massive Bosonic String in Background B-Field

“…The possibility (22) corresponds to zero mode solutions with sinh k 0 σ and cosh k 0 σ. Traditionally the zero mode solution is denoted as the zero frequency (infinite wave length) limiting term in the Fourier expansions, as shown for example for the massless case of the current problem in [11]. Here, however, we interpret the zero mode solution as a solution which satisfies the boundary conditions not only at the end-points but also throughout all the medium.…”

A Gauged Open 2-Brane String in thep-Brane Background

“…The idea of considering boundary conditions as constraints of a physical system was proposed [27,28,29] by the late 1990s. In fact, boundary conditions in the background magnetic field are first order equations of time variable, whereas the Euler-Lagrange equation is a second order one, which means that the acceleration determines the dynamics of the motion.…”

“…In fact, boundary conditions in the background magnetic field are first order equations of time variable, whereas the Euler-Lagrange equation is a second order one, which means that the acceleration determines the dynamics of the motion. Hence, boundary conditions in Lagrangian equations are called acceleration-free equations [23,24,25,26,29]. So, these objects do not play any role in the dynamics of the system.…”

“…Instead, they set up some identities in the phase space which satisfy Dirac's constrained conditions. Also, the difference between constraints, obtained from boundary conditions equations and other constraints could be evident in Hamiltonian approach [29].…”

Canonical Quantization of Field Theories with Boundaries

“…consistently in the whole space because on the boundaries those BCs are inconsistent with the usual canonical commutation relations [1][2][3][4][5].…”