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arXiv:0909.1797 · International Journal of Geometric Methods in Modern Physics

HERMITIAN VECTOR FIELDS AND COVARIANT QUANTUM MECHANICS OF A SPIN PARTICLE

Daniel Canarutto

Published Jun 1, 2010• 30 citations
arXivHTMLPDF

Abstract

In the context of Covariant Quantum Mechanics for a spin particle, we classify the "quantum vector fields", i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called "special phase functions". This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.

DOI
10.1142/s0219887810004464

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In the context of Covariant Quantum Mechanics for a spin particle, we classify the "quantum vector fields", i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called "special phase functions". This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.
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We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces [Formula: see text], by explicitly constructing Hilbert-space vector bundles over [Formula: see text]. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle [Formula: see text], plus a complex line bundle [Formula: see text]. Quantum states (except the vacuum) appear as tangent vectors to [Formula: see text]. The vacuum state appears as the fibrewise generator of [Formula: see text]. Holomorphic line bundles [Formula: see text] are classified by the elements of [Formula: see text], the Picard group of [Formula: see text]. In this way [Formula: see text] appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when [Formula: see text] is complex projective space CPn.
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of operators belonging to the enveloping Lie algebra that satisfy an approximate symmetry and have the properties required by physics.
We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite–dimensional, compact, classical phase spacesC, by explicitly constructing Hilbert–space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum states (except the vacuum) appear as tangent vectors toC. The vacuum state appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are classified by the elements of Pic (C), the Picard group of C. In this way Pic (C) appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case whenC is complex projective space CP n.
According to some generalized correspondence principle the classical limit of a non-Hermitian Quantum theory describing quantum degrees of freedom is expected to be well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity has been developed by Einstein merely for real-valued space-time and four-momentum we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, yet also to the non-Hermitian Klein-Gordon and Dirac equations which are to lay the foundations of a non-Hermitian quantum theory.
According to some generalized correspondence principle the classical limit of a non-Hermitian quantum theory describing quantum degrees of freedom is expected to be the well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity was developed by Einstein merely for real-valued space-time and four-momentum, we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, but also to the non-Hermitian Klein-Gordon and Dirac equations, which are to lay the foundations of a non-Hermitian quantum theory.