Poisson Brackets, Strings and Membranes

TL;DR

This paper develops a method to construct compatible Poisson brackets for open strings and membranes with background fields, simplifying the constraint analysis needed to determine their boundary dynamics.

Contribution

It introduces a systematic approach to derive Poisson brackets at boundaries of strings and membranes, using primary and secondary constraints, and applies canonical transformations for membranes.

Findings

Poisson brackets at string boundaries are explicitly constructed.

Only two constraints are needed to determine the brackets.

The method is extended to membranes via canonical transformations.

Abstract

We construct Poisson brackets at boundaries of open strings and membranes with constant background fields which are compatible with their boundary conditions. The boundary conditions are treated as primary constraints which give infinitely many secondary constraints. We show explicitly that we need only two (the primary one and one of the secondary ones) constraints to determine Poisson brackets of strings. We apply this to membranes by using canonical transformations.