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arXiv:0709.4312

Universality of quantum symplectic structure

Tulsi Dass

Published Sep 27, 2007• 1 citation
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Abstract

Operating in the framework of `supmech' (a scheme of mechanics which aims at providing a concrete setting for the axiomatization of physics and probability theory as required in Hilbert's sixth problem; integrating noncommutative symplectic geometry and noncommutative probability in an algebraic setting, it associates, with every `experimentally accessible' system, a symplectic algebra and operates essentially as noncommutative Hamiltonian mechanics with some extra sophistication in the treatment of states) it is shown that interaction between systems can be consistently described only if either (i) all system algebras are commutative or (ii) all system algebras are noncommutative and have a quantum symplectic structure characterized by a UNIVERSAL Planck type real-valued constant of the dimension of action.

DOI
10.48550/arxiv.0709.4312

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Operating in the framework of supmech' (a scheme of mechanics which aims at providing a concrete setting for the axiomatization of physics and probability theory as required in Hilbert's sixth problem; integrating noncommutative symplectic geometry and noncommutative probability in an algebraic setting, it associates, with every experimentally accessible' system, a symplectic algebra and operates essentially as noncommutative Hamiltonian mechanics with some extra sophistication in the treatment of states) it is shown that interaction between systems can be consistently described only if either (i) all system algebras are commutative or (ii) all system algebras are noncommutative and have a quantum symplectic structure characterized by a UNIVERSAL Planck type real-valued constant of the dimension of action.
As the first step in an approach to the solution of Hilbert's sixth problem, a general scheme of mechanics, called `supmech', is developed integrating noncommutative symplectic geometry and noncommutative probability theory in an algebraic framework; it has quantum mechanics (QM) and classical mechanics as special subdisciplines and facilitates an autonomous development of QM and satisfactory treatments of quantum-classical correspondence and quantum measurements (including a straightforward \emph{derivation} of the von Neumann reduction rule). The scheme associates, with every `experimentally accessible' system, a symplectic superalgebra and operates essentially as noncommutative Hamiltonian mechanics incorporating the extra condition that the sets of observables and pure states be mutually separating. The latter condition serves to smoothly connect the algebraically defined quantum systems to ilbert space-based ones; the rigged Hilbert space - based Dirac bra-ket formalism naturally appears. The formalism has a natural place for commutative superselection rules. Noncommutative analogues of objects like the momentum map and the Poincareˊ\acute{e}eˊ-Cartan form are introduced and some related symplectic geometry developed.
In this sequence of papers, noncommutative analysis is used to give a consistent axiomatic approach to a unified conceptual foundation of classical and quantum physics. The present Part I defines the concepts of observables, states and ensembles, clarifies the logical relations and operations for them, and shows how they give rise to dynamics and probabilities. States are identified with maximal consistent sets of weak equalities in the algebra of observables (instead of, as usual, with the rays in a Hilbert space). This leads to a concise foundation of quantum mechanics, free of undefined terms, separating in a clear way the deterministic and the stochastic features of quantum physics. The traditional postulates of quantum mechanics are derived from well-motivated axiomatic assumptions. No special quantum logic is needed to handle the peculiarities of quantum mechanics. Foundational problems associated with the measurement process, such as the reduction of the state vector, disappear. The new interpretation of quantum mechanics contains `elements of physical reality' without the need to introduce a classical framework with hidden variables. In particular, one may talk about the state of the universe without the need of an external observer and without the need to assume the existence of multiple universes.
A formalism is presented in which quantum particle dynamics can be developed on its own rather than `quantization' of an underlying classical theory. It is proposed that the unification of probability and dynamics should be considered as the basic feature of quantum theory. Arguments are given to show that when such a unification is attempted at the configuration space level, the wave funtions of Schro¨\ddot{o}o¨dinger theory appear as the natural candidates for the desired unification. A *-algebra AQ\mathcal{A}_{Q}AQ​ of (not necessarily bounded) linear operators acting on an appropriate dense set of these wave functions appears as the arena for quantum kinematics. A simple generalization of an existing formalism in noncommutative geometry is employed to develop the notion of generalized algebraic symplectic structure (GASS) which can accomodate classical and quantum symplectic structures as special cases. Quantum kinematics and dynamics is developed in in the framework of a noncommutative Hamiltonian system employing an appropriate GASS based in $ \mathcal{A}_{Q}$. The Planck constant is introduced at only one place -- in the quantum symplectic form; its appearance at conventional places is then automatic. Unitary Wigner symmetries appear as canonical transformations in the noncommutative Hamiltonian system. A straightforward treatment of quantum - classical correspondence is given in terms of appropriate GASSes.
We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schrodinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.
This paper explores the intersection of quantum algebraic geometry and quantum information theory, focusing on how noncommutative algebraic structures provide a powerful framework for understanding and manipulating quantum systems. We investigate the role of noncommutative rings and schemes in encoding quantum states, quantum operations, and entanglement. Furthermore, we examine the application of tools from algebraic geometry, such as sheaves and cohomology, to characterize the complexity and properties of quantum information processing tasks. Specifically, we delve into the use of noncommutative tori and their classifications, alongside explorations of quantum metrics and preordered semialgebras, to build a bridge between abstract mathematical structures and concrete applications in quantum technologies. Our goal is to highlight the potential of quantum algebraic geometry to offer new insights and techniques for advancing the field of quantum information.