Modified uncertainty relations from classical and quantum gravity

TL;DR

This paper explores the connections between modified uncertainty principles, curved spaces, and quantum gravity, deriving new relations and demonstrating the consistency of quantum mechanics on curved phase spaces.

Contribution

It establishes a novel link between GUPs, EUPs, and curved spaces, including a new EUP relating geodesic radius to momentum uncertainty and a formulation of quantum mechanics on curved cotangent bundles.

Findings

Derived a new extended uncertainty principle relating geodesic radius to momentum standard deviation.

Established a correspondence between GUP theories and quantum dynamics on non-Euclidean momentum space.

Showed quantum mechanics can be consistent on arbitrarily curved cotangent bundles, illustrating Born reciprocity.

Abstract

A good hundred years after the necessity for a quantum theory of gravity was acknowledged by Albert Einstein, the search for it continues to be an ongoing endeavour. Nevertheless, the field still evolves rapidly as manifested by the recent rise of quantum gravity phenomenology supported by an enormous surge in experimental precision. In particular, the minimum length paradigm ingrained in the program of generalized uncertainty principles (GUPs) is steadily growing in importance. The present thesis is aimed at establishing a link between modified uncertainty relations, derived from deformed canonical commutators, and curved spaces - specifically, GUPs and nontrivial momentum space as well as the related extended uncertainty principles (EUPs) and curved position space. In that vein, we derive a new kind of EUP relating the radius of geodesic balls, assumed to constrain the wave functions in the underlying Hilbert space, with the standard deviation of the momentum operator. This result is gradually generalized to relativistic particles in curved spacetime in accordance with the 3+1 decomposition. In a sense pursuing the inverse route, we find an explicit correspondence between theories yielding a GUP and quantum dynamics set on non-Euclidean momentum space. Quantitatively, the coordinate noncommutativity translates to momentum space curvature in the dual description, allowing for an analogous transfer of constraints from the literature. Finally, we find a formulation of quantum mechanics which proves consistent on the arbitrarily curved cotangent bundle. Along these lines, we show that the harmonic oscillator can not be used as a means to distinguish between curvature in position and momentum space, thereby providing an explicit instantiation of Born reciprocity in the context of curved spaces.