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Holomorphic quantization on the torus and finite quantum mechanics

Gregory G. Athanasiu, Emmanuel G. Floratos, Stam Nicolis

Published Sep 18, 1995• 27 citations
OpenAlex

Abstract

We construct explicitly the quantization of classical linear maps of SL(2,{\\Bbb R}) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2,{\\Bbb Z}) to the subgroup SL(2,{\\Bbb Z})/\\Gamma_l, \\Gamma_l being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, O_{l}(2)\\subset SL(2,l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.

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Quantum Mechanics as a Gauge Theory of Metaplectic Spinor Fields

Reuter, M., Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)

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.)

•Jan 1, 1997•11 citations
We construct explicitly the quantization of classical linear maps of SL(2,{\Bbb R}) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2,{\Bbb Z}) to the subgroup SL(2,{\Bbb Z})/\Gamma_l, \Gamma_l being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, O_{l}(2)\subset SL(2,l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.
We explicitly construct the quantization of classical linear maps of on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that finite quantum mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of to the subgroup , being the principal congruent subgroup , on a finite dimensional Hilbert space. The generators of the `rotation group' , , for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.
A finite-dimensional system with a quadratic Hamiltonian constraint is Dirac quantized in holomorphic, antiholomorphic, and mixed representations. A unique inner product is found by imposing Hermitian conjugacy relations on an operator algebra. The different representations yield drastically different Hilbert spaces. In particular, all the spaces obtained in the antiholomorphic representation violate classical expectations for the spectra of certain operators, whereas no such violation occurs in the holomorphic representation. A subset of these Hilbert spaces is also recovered in a configuration space representation. A propagation amplitude obtained from an (anti)holomorphic path integral is shown to give the matrix elements of the identity operators in the relevant Hilbert spaces with respect to an overcomplete basis of representation-dependent generalized coherent states. The relation to quantization of spatially homogeneous cosmologies is discussed in view of the no-boundary proposal of Hartle and Hawking and the new variables of Ashtekar.
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.)