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 non-commutative 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.