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arXiv:1710.03751

Bosonic Pairings

P. L. Robinson

Published Oct 10, 2017• 0 citations
arXivHTMLPDF

Abstract

We extend the inner product from bosonic Fock space to a pairing between suitable antifunctionals on the symmetric algebra. Our account is illustrated by the (Gaussian) half-form pairing between positive polarizations in the form needed for geometric quantization.
DOI
10.48550/arxiv.1710.03751

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Abstract The half-form bundles are also defined for complex polarizations. By including them in the quantization construction, we arrive at a more useful pairing, which can be applied to a much wider class of nonnegative polarizations. It seems to give the correct physical relationship between the wave functions in the Hilbert spaces of different polarizations. In the case of two Fock spaces, the new pairing is unitary. When P is complex, δp may be nontrivial, and its inclusion in the construction can modify quantization conditions and alter the form of the quantum observables. In the case of the harmonic oscillator, the modification is precisely that needed to get the correct zero-point energy (see §9.2).
In the previous parts of this work, we established the Prequantum Groupoid Tω\mathbf{T}_ωTω​ as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator, E0=nℏ/2E_0 = n\hbar/2E0​=nℏ/2. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.
We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)