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Schrodinger Equation for Nonassociative Quantum Mechanics and No Go Theorem

Susumu Ôkubo

Published Jan 1, 1982• 0 citations

Abstract

Within an algebraic framework of non-associative quantum mechanics, a Hilbert space and the corresponding Schroedinger equation have been introduced into the theory as follows. The underlying non-associative operator algebra A is assumed to be flexible and Lie-admissible so that it is compatible with the Heisenberg equation of motion and quantization. A Hilbert space can be introduced into the theory as a faithful representation of the associated Lie algebra A/sup -/ of the Lie-admissible algebra A. However, if A/sup -/ is the standard Heisenberg algebra, and if the representation is irreducible, then it is shown that A must be associative, reproducing the standard associative quantum mechanics. Some discussions to circumvent this No-Go theorem are discussed. Especiallyyu if we are interested in the formulation of infinite component wave equations, then this difficulty can be avoided.

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Within an algebraic framework of non-associative quantum mechanics, a Hilbert space and the corresponding Schroedinger equation have been introduced into the theory as follows. The underlying non-associative operator algebra A is assumed to be flexible and Lie-admissible so that it is compatible with the Heisenberg equation of motion and quantization. A Hilbert space can be introduced into the theory as a faithful representation of the associated Lie algebra A/sup -/ of the Lie-admissible algebra A. However, if A/sup -/ is the standard Heisenberg algebra, and if the representation is irreducible, then it is shown that A must be associative, reproducing the standard associative quantum mechanics. Some discussions to circumvent this No-Go theorem are discussed. Especiallyyu if we are interested in the formulation of infinite component wave equations, then this difficulty can be avoided.

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 Gel’fand–Naimark–Segal constructions. We describe Heisenberg and Schrödinger 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.

A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation

We discuss an alternative version of non- relativistic Newtonian mechanics in terms of a real Hilbert space mathematical framework. It is demonstrated that the physics of this scheme correspondent with the standard formulation. Heisenberg-Schrodinger non-relativistic quantum mechanics is considered adequate and complete. Since the suggested theory is dispersion free, linear superposition principle is not violated but cannot affect results of measurements due to spectral decomposition theorem for self-adjoint operators (the collapse of wave function).

We discuss an alternative version of non-relativistic Newtonian mechanics in terms of real Hilbert space mathematical framework. It is demonstrated that the physics of this scheme is in accordance with the standard formulation. Heisenberg-Schrodinger non-relativistic quantum mechanics is considered adequate and complete. Since the suggested theory is dispersion free, linear superposition principle is not violated but cannot affect results of measurements due to spectral decomposition theorem for self-adjoint operators (the collapse of wave function).

Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.

In this note I discuss some aspects of a formulation of quantum mechanics based entirely on the Jordan algebra of observables. After reviewing some facts of the formulation in the \CS -approach I present a Jordan-algebraic Hilbert space construction (inspired by the usual GNS-construction), thereby obtaining a real Hilbert space and a (Jordan-) representation of the algebra of observables on this space. Taking the usual case as a guideline I subsequently derive a Schrödinger equation on this Hilbert space.