Quantization of the open string on plane-wave limits of and non-commutativity outside branes

Abstract

The open string on the plane-wave limit of dS n × S n with constant B 2 and dilaton background fields is canonically quantized. This entails solving the classical equations of motion for the string, computing the symplectic form, and defining from its inverse the canonical commutation relations. Canonical quantization is proved to be perfectly suited for this task, since the symplectic form is unambiguously defined and non-singular. The string position and the string momentum operators are shown to satisfy equal-time canonical commutation relations. Noticeably the string position operators define non-commutative spaces for all values of the string worldsheet parameter σ, thus extending non-commutativity outside the branes on which the string endpoints may be assumed to move. The Minkowski spacetime limit is smooth and reproduces the results in the literature, in particular non-commutativity gets confined to the endpoints.

Symplectic Quantization of Massive Bosonic String in Background B-Field

“…A modified version of Faddeev-Jackiw formulation recognized as "symplectic quantization" is more or less used by some authors, when they want to build a quantum theory out of a given classical action [7,8,9,10]. As we will see, the most essential point in this process is finding the complete set of physical modes and the correct expansions of the fields in terms of them.…”

D-branes with Lorentzian signature in the Nappi-Witten model

“…Two important examples of the latter are provided by (i) the family of pp-wave geometries [5,6] with also a globally defined constant B that describe the Penrose limits of AdS n × S m and dS n × S m , and (ii) an S 3 background with nonzero H [7,8]. In the first case, the three-form H vanishes and noncommutativity at the string endpoints can be established through canonical quantization.…”

Geometric construction of D-branes in WZW models

“…There are many reasons for this. Among them, the evidence that D-branes provide soliton and bound states in string backgrounds [1] and the realization that they become upon quantization noncommutative spacetimes [2][3][4][5][6][7].…”