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arXiv:funct-an/9701002

Derivation of Local Gauge Freedom from a Measurement Principle

Felix Finster

Published Jan 6, 1997• 4 citations
arXivHTMLPDF

Abstract

We define operator manifolds as manifolds on which a spectral measure on a Hilbert space is given as additional structure. The spectral measure mathematically describes space as a quantum mechanical observable. We show that the vectors of the Hilbert space can be represented as functions on the manifold. The arbitrariness of this representation is interpreted as local gauge freedom. In this way, the physical gauge principle is linked with quantum mechanical measurements of the position variable. We derive the restriction for the local gauge group to be U(m), where m is the number of components of the wave functions.

DOI
10.48550/arxiv.funct-an/9701002

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We define operator manifolds as manifolds on which a spectral measure on a Hilbert space is given as additional structure. The spectral measure mathematically describes space as a quantum mechanical observable. We show that the vectors of the Hilbert space can be represented as functions on the manifold. The arbitrariness of this representation is interpreted as local gauge freedom. In this way, the physical gauge principle is linked with quantum mechanical measurements of the position variable. We derive the restriction for the local gauge group to be U(m), where m is the number of components of the wave functions.
The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with positions and displacements on the manifold. The guiding, physically motivated, principles are (i) locality, i.e., the generating role of the algebras localized in small, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, “topological,” subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering M̃ of M and on the decomposition of L2(M̃) according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the permutation group in particle statistics are discussed.
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of quantum theory are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then explain the origin of physical dimensions, provide a geometric interpretation of Planck’s constant, and find a simple interpretation of the two-slit experiment and the process of measurement.
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.)
In this thesis we shall demonstrate that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the formalism used entails a description that is non-local in that it requires all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary to specify quantum states completely. The remainder of the thesis, will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic field found by Susskind. It will be argued that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, with an interpretation in terms of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions make explicit the additional structure which is encoded more subtly in the non-local description. Additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory.
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.