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Probability + Geometry > Quantum Probability
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Alexandre Harvey-Tremblay

2021
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figshare.com
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I derive the 'general linear probability amplitude' and its associated probability rule. Said probability amplitude, of which the familiar complex probability amplitude is a special case, constitutes the largest possible group of LINEAR probability amplitudes, and as such admits remarkable applications --- including that of retaining its linearity in the general relativity (GR) scenario. To properly motivate, I investigate a two-state (thought) experiment that produces a \textit{general linear
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... nterference pattern} exceeding what is possible with mere complex interference, thus suggesting, via a simple system, that geometric probabilities (GP) are realized/realizable in nature. The derivation begins with the wave-function of GP, bluntly obtained as the probability measure that maximizes the entropy constrained by an ensemble of linear transformations, thereby superseding the Gibbs ensemble, and inheriting from statistical physics the statistical ensemble interpretation as its only tenable (quantum mechanical) interpretation. Properly parametrized, the wave-function of GP in GL(4,ℝ) is interpreted as the probability density of measuring an event (in space-time). As a geometric sampling space, GP provides a simple origin for the notoriously elusive 'wave-function collapse' as an automatic consequence of performing any measurement on the statistical system --- something that quantum probabilities (QP), a subset of GP that excludes the relevant general linear amplitudes, cannot provide. Taking the gauge covariant derivative of the GP space-time wave-function, as it transforms with respect to the GL(4,ℝ) group, is able to support in the general case a gauge theory of GR on the T(4) x GL(4,ℝ) affine group (or its structure reduction to the T(4) x O(4) Poincaré group), where the Christoffel symbols Γ^μ are viewed as a GL(4,ℝ)-valued gauge field, and where the quantum behaviour is provided in the form of an affine probability amplitude. Finally, we extend the last result by adding the gauge group of the Standard model U( [...]

doi:10.6084/m9.figshare.13012754.v17
fatcat:5ctcepvmyza3hbsl7dsvdkew2e