Abstract
This paper aims at investigating the influence of space-time curvature on the uncertainty relation. In particular, relying on previous findings, we assume the quantum wave function to be confined to a geodesic ball on a given spacelike hypersurface whose radius is a measure of the position uncertainty. On the other hand, we concurrently work out a viable physical definition of the momentum operator and its standard deviation in the nonrelativistic limit of the formalism. Finally, we evaluate the uncertainty relation, which to second order depends on the Ricci scalar of the effective 3-metric and the corresponding covariant derivative of the shift vector. For the sake of illustration, we apply our general result to a number of examples arising in the context of both general relativity and extended theories of gravity.
- Received 18 January 2021
- Accepted 23 April 2021
DOI:https://doi.org/10.1103/PhysRevD.103.104061
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