$\mathfrak{D}$ -Differentiation in Hilbert Space and the Structure of Quantum Mechanics

Foundations of Physics 39 (5):433-473 (2009)
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Abstract

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of $\mathcal{C}$ - and $\mathfrak{D}$ -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space is flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect

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David Hurley
Portland State University

References found in this work

Quantum Mechanics: Myths and Facts.Nikolic Hrvoje - 2007 - Foundations of Physics 37 (11):1563-1611.
A geometric approach to quantum mechanics.J. Anandan - 1991 - Foundations of Physics 21 (11):1265-1284.

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