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.
An algebraic formulation of nonassociative quantum mechanics
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abstract
We develop a version of quantum mechanics that can handle nonassociative algebras of observables and which reduces to standard quantum theory in the traditional associative setting. Our algebraic approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra to introduce a generalized notion of associative composition product. We formulate properties of states together with notions of trace, and use them to develop GNS constructions. We describe Heisenberg and Schr\"odinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra.
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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.