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Hadronic J.; (United States)

Non-associative algebras of quantum-mechanical observables

E. Kapuścik

Published Apr 1, 1981• 0 citations
OpenAlex

Abstract

A critical examination of the original Heisenberg approach to quantum mechanics is presented. It is shown that, using the same kind of arguments as Heisenberg, it is possible to construct new non-associative algebras of quantum-mechanical observables. Two examples of such algebras are explicitly constructed.

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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 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.
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.
In 1925, Werner Heisenberg formulated quantum mechanics through non-commuting arrays of numbers, which were soon systematized into matrix mechanics and eventually superseded in popularity by Schrödinger's wave-function formalism. This paper explores an alternative foundational branch in the history of physics: What if Heisenberg had bypassed concrete matrix representations and directly proposed that the physical reality of quantum mechanics lies in the algebraic relations between observables rather than their numerical values? We demonstrate that this shift from a state-first paradigm to a relation-first paradigm would have restructured the development of modern theoretical physics. Under this "Algebra of Observables" framework, concepts such as C*-algebras, the GNS construction, operator algebras, and noncommutative geometry would have emerged as the foundational language of quantum theory from its inception, rather than as later mathematical abstractions. Furthermore, this approach eliminates classical measurement paradoxes associated with wave-function collapse by treating physical states merely as positive linear functionals over the observable algebra. Finally, we analyze how a background-independent algebraic foundation naturally resolves the problem of time through modular automorphism groups, offering a direct path toward quantum gravity and emergent spacetime without relying on fixed spatial backgrounds.
A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum mechanics, and to study the limits of its applicability. A short outline is given of the necessary material from the theory of algebras and probability theory. It is described how the mathematical scheme of the paper agrees with the theory of quantum measurements, and avoids quantum paradoxes.
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