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arXiv:1511.02777

Geometric phase in the G3+ quantum state evolution

Alexander Soiguine

Published Oct 23, 2015• 2 citations
arXivHTMLPDF

Abstract

When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.

DOI
10.48550/arxiv.1511.02777

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When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.
Quantum mechanical qubit states as elements of two dimensional complex Hilbert space can be generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to special maps of the geometric algebra elements to the unit sphere in 3D generated by arbitrary unit value bivectors. Analysis of the structure of the map of the even subalgebra to the Hilbert space demonstrates that quantum state evolution in the latter gives only restricted information compared to that in geometric algebra.
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that case. Generalization of formal complex plane to an an arbitrary plane in three dimensions and of usual Hopf fibration to the map generated by an arbitrary unit value element of even sub-algebra of the three-dimensional Geometric Algebra are resulting in more profound description of qubits compared to quantum mechanical Hilbert space formalism.
Recently suggested scheme [1] of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.
The Geometric Algebra formalism allows for the development of a theory to replace conventional quantum mechanics. Generalizations resulting from the replacement of complex numbers with geometrically feasible three-dimensional objects, followed by unambiguous definitions of states, observables, and measurements, bring into reality clear explanations of strange quantum mechanical features, such as primitively considering atoms as a type of solar system. The three-sphere S\mathbb{S}S3 is transformed into a playground for torsion kind states that do not require abstract Hilbert space vectors. The points S\mathbb{S}S3 evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.