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arXiv:1406.3751

What quantum state really is?

Alexander Soiguine

Published Jun 14, 2014• 6 citations
arXivHTMLPDF

Abstract

To get out of logical deadlock in interpreting gedanken experiments like Schrodinger cat, actual meaning of a wave function, or a state, in the case of complex two-dimensional Hilbert space, is shown to represent transformations executing by elements of even subalgebra of geometric algebra over three-dimensional Euclidian space.
DOI
10.48550/arxiv.1406.3751

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The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.
The wave function gives probabilities, but is it a real physical quantity, as in the ontic interpretation of quantum mechanics, or just a measure of human belief about a physical system, as in an epistemic interpretation? These questions are introduced by treating Schrödinger’s cat and Wigner’s friend. An important question is whether quantum mechanics applies to macroscopic objects, for example, cats. SQUID experiments have been performed that see quantum interference in very large objects, whereas with most macroscopic objects quantum interference disappears because of decoherence. Various views of the meaning of quantum mechanics are discussed: the Copenhagen interpretation, continuous spontaneous localization, Everett’s many-worlds interpretation, Bohm’s hidden-variable model, and quantum Bayesianism. The latter holds that the wave function represents one’s best belief about the state of a quantum system, rather than an actual physical property. This leads to discussions of recent theories and experiments explicitly designed to test wave function reality.
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.
O NE OF the most basic ideas of quantum mechanics is that the condition of a system is represented mathematically by a wavefunction . This is a complex valued mathematical function of that tells us everything we can know about the system. In order to make predictions we apply mathematical operations to the wavefunction that tell us the probability of various outcomes that we can compare to experimental results. The wavefunction is a complex valued function. In other words when you evaluate it for a particular set of inputs (e.g., position, time, momentum, energy, etc.) the output is a complex number. The magnitude of the complex number gives information about the relative likelihood of a particular outcome. That seems pretty odd I'm sure. The best way to get comfortable with the notion is to play around with it. Let's consider an example.
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.