Strings in pp-wave background and background B-field from membrane and its symplectic quantization

The symplectic quantization technique is applied to open free membrane and strings in pp-wave background and background gauge field obtained by compactifying the open membrane in the presence of a background anti-symmetric 3–form field. In both cases, first the Poisson brackets among the Fourier modes are obtained and then the Poisson brackets among the membrane(string) coordinates are computed. The full noncommutative phase-space structure is reproduced in case of strings in pp-wave background and background gauge field. We feel that this method of obtaining the Poisson algebra is more elegant than previous approaches discussed in the literature.