Abstract
AbstractNonassociative structures have appeared in the study of D‐branes in curved backgrounds. In recent work, string theory backgrounds involving three‐form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non‐vanishing three‐cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson‐Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non‐linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string‐field theoretic generalization of the AdS/CFT‐like (holographic) duality.
Highlights
Noncommutative and associative algebraic structures are hallmarks of quantum physics
In this paper we first reviewed the work of Stuckelberg on the generalization of Poisson brackets in classical mechanics while preserving Boltzmann’s Htheorem that, in general, violate the Jacobi identity
We proposed that the nonlinear extension of the corresponding quantum mechanics envisaged by Stueckelberg, which involves the fundamental length, should be formulated as a nonassociative extension of quantum mechanics
Summary
Noncommutative and associative algebraic structures are hallmarks of quantum physics. We generalize nonassociative Malcev algebras that naturally appear in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge, to include electric as well as magnetic charges. These results find their classical counterpart in the theory of Poisson-Malcev algebras [33], which we identify as the underlying nonassociative structures behind Stueckelberg’s pioneering work on generalized Poisson brackets that do not obey the Jacobi identity. In view of these facts, we close our paper with a brief discussion of the general role of nonassociativity in string theory (see the discussion in [25]), and in particular, we discuss the role of nonassociativity in closed string field theory [14, 15], which leads us to conjecture a string-field theoretic generalization of the AdS/CFT-like (holographic) duality
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