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arXiv:1905.04189 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

Gerd Niestegge

Published Jan 1, 2020• 7 citations
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Abstract

Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.

DOI
10.1098/rspa.2019.0604

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Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.
I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.
The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system A can be paired with an isomorphic conjugate system, A¯ , by means of a non-signaling bipartite state ηA perfectly and uniformly correlating each basic measurement on A with its counterpart on A¯ . In the case of a quantum-mechanical system associated with a complex Hilbert space H , the conjugate system is that associated with the conjugate Hilbert space H , and ηA corresponds to the standard maximally entangled EPR state on H⊗H¯ . A second ingredient is the notion of a reversible filter , that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the ``no restriction" hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. (An appendix shows how a version of Hardy's ``subpace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.)
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.
The well-known limitations on the field of the quantum mechanical vector space, which, within the framework of the propositional logic of Birkhoff and von Neumann, can be obtained only for systems of dimension greater than or equal to four, are obtained here for all dimensions. The propositional logic fails to provide any information about the quantum systems of dimension less than four and, moreover, one does not know the empirical meaning of one of its basic assumptions. The Jordan algebra approach, on the other hand, which provides the same limitations on the field, also suffers from anomalies in low dimensions and is altogether formal rather than physical. In the present work, which is based on a recent study of the topological properties of quantum states, there are no low-dimensional anomalies and the assumptions have clear empirical meaning.