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QUANTUM ONTOLOGY IN THE LIGHT OF GAUGE THEORIES

Gabriel Catren

Published Mar 5, 2014• 4 citations
Publisher

Abstract

We propose the conjecture according to which the fact that quantum me-chanics does not admit sharp value attributions to both members of a com-plementary pair of observables can be understood in the light of the symplec-tic reduction of phase space in constrained Hamiltonian systems. In order to unpack this claim, we propose a quantum ontology based on two inde-pendent postulates, namely the phase postulate and the quantum postulate. The phase postulate generalizes the gauge correspondence between first-class constraints and gauge transformations to the observables of unconstrained Hamiltonian systems. The quantum postulate specifies the relationship be-tween the numerical values of the observables that permit us to individualize a physical system and the symmetry transformations generated by the oper-ators associated to these observables. We argue that the quantum postulate and the phase postulate are formally implemented by the two independent stages of the geometric quantization of a symplectic manifold, namely the prequantization formalism and the election of a polarization of pre-quantum states respectively.

DOI
10.1142/9789814596299_0002

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We propose the conjecture according to which the fact that quantum me-chanics does not admit sharp value attributions to both members of a com-plementary pair of observables can be understood in the light of the symplec-tic reduction of phase space in constrained Hamiltonian systems. In order to unpack this claim, we propose a quantum ontology based on two inde-pendent postulates, namely the phase postulate and the quantum postulate. The phase postulate generalizes the gauge correspondence between first-class constraints and gauge transformations to the observables of unconstrained Hamiltonian systems. The quantum postulate specifies the relationship be-tween the numerical values of the observables that permit us to individualize a physical system and the symmetry transformations generated by the oper-ators associated to these observables. We argue that the quantum postulate and the phase postulate are formally implemented by the two independent stages of the geometric quantization of a symplectic manifold, namely the prequantization formalism and the election of a polarization of pre-quantum states respectively.
Abstract. We argue that the classical description of a symplectic manifold endowed with a Hamil-tonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry (notably, Souriau’s moment map and the Mardsen-Weinstein symplectic reduction formalism) and group representation theory (notably Kirillov’s orbit method). The proposed argumentative line strongly relies on the conjecture pro-posed by Guillemin and Sternberg according to which “quantization commutes with (symplectic) reduction”. By using the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric un-derpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle. Key words: quantum mechanics; gauge theories; moment map; symplectic reduction. 1.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables: they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role. (orig.)
We propose a new quantum ontology, in which properties are the fundamental building blocks. In this property ontology physical systems are defined as bundles of type-properties (specified by algebras of observables in a Hilbert space). Not all elements of such bundles are associated with definite case-properties, and this accommodates the Kochen and Specker theorem and contextuality. Moreover, we do not attribute an identity to the type-properties, which gives rise to a novel form of the bundle theory. There are no "particles" in the sense of classical individuals in this ontology, although the behavior of such individuals is mimicked in some circumstances. This picture leads in a natural way to the symmetrization postulates for systems of many "identical particles".
As a consequence of Heisenberg indeterminacy principle, quantum states are defined by half the number of variables required in classical mechanics. The main claim of this paper is that this " reduction " in the number of variables required to completely describe a physical system can be understood as a consequence of the same formalism underlying the reduction procedure used in gauge theories, namely the Mardsen-Weinstein symplectic reduction. This fact points towards a gauge-theoretical interpretation of the indeterminacy principle in quantum mechanics.
As a consequence of Heisenberg indeterminacy principle, quantum states are defined by half the number of variables required in classical mechanics.The main claim of this paper is that this "reduction" in the number of variables required to completely describe a physical system can be understood as a consequence of the same formalism underlying the reduction procedure used in gauge theories, namely the Mardsen-Weinstein symplectic reduction.This fact points towards a gauge-theoretical interpretation of the indeterminacy principle in quantum mechanics.