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The open string on the plane-wave limit of $dS_n\times S^n $ with constant  $B_2$ and dilaton background fields is canonically quantized. This entailssolving the classical equations of motion for the string, computing thesymplectic form, and defining from its inverse the canonical commutationrelations. Canonical quantization is proved to be perfectly suited for thistask, since the symplectic form is unambiguously defined and non-singular. Thestring position and the string momentum operators are shown to satisfyequal-time canonical commutation relations. Noticeably the string positionoperators define non-commutative spaces for all values of the stringworld-sheet parameter $\sig$, thus extending non-commutativity outside thebranes on which the string endpoints may be assumed to move. The Minkowskispacetime limit is smooth and reproduces the results in the literature, inparticular non-commutativity gets confined to the endpoints.

Quantization of the open string on plane-wave limits of dS_n x S^n and non-commutativity outside branes