Demanding associativity of the momentum translation operator for Schwinger's non-commuting coordinates of massless particles yields the helicity quantization λ=(ℏ/2)n, shown to be dual to Dirac's monopole quantization.
(Non-)commutative closed string on T-dual toroidal backgrounds
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abstract
In this paper we investigate the connection between (non-)geometry and (non-)commutativity of the closed string. To this end, we solve the classical string on three T-dual toroidal backgrounds: a torus with H-flux, a twisted torus and a non-geometric background with Q-flux. In all three situations we work under the assumption of a dilute flux and consider quantities to linear order in the flux density. Furthermore, we perform the first steps of a canonical quantization for the twisted torus, to derive commutators of the string expansion modes. We use them as well as T-duality to determine, in the non-geometric background, a commutator of two string coordinates, which turns out to be non-vanishing. We relate this non-commutativity to the closed string boundary conditions, and the non-geometric Q-flux.
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Schwinger's non-commutative coordinates and duality between helicity and Dirac quantisation conditions
Demanding associativity of the momentum translation operator for Schwinger's non-commuting coordinates of massless particles yields the helicity quantization λ=(ℏ/2)n, shown to be dual to Dirac's monopole quantization.