Abstract
We show that the precision of an angular measurement or rotation (e.g., on the orientation of a qubit or spin state) is limited by fundamental constraints arising from quantum mechanics and general relativity (gravitational collapse). The limiting precision is r−1 in Planck units, where r is the physical extent of the (possibly macroscopic) device used to manipulate the spin state. This fundamental limitation means that spin states S1 and S2 cannot be experimentally distinguished from each other if they differ by a sufficiently small rotation. Experiments cannot exclude the possibility that the space of quantum state vectors (i.e., Hilbert space) is fundamentally discrete, rather than continuous. We discuss the implications for finitism: does physics require infinity or a continuum?
Highlights
FUNDAMENTAL LIMITS ON MEASUREMENTGedanken experiments can reveal fundamental limitations on measurements or other experimental procedures that arise from the laws of physics, see e.g. [1,2,3,4,5,6,7]
It has been shown that discreteness of space-time on length scales smaller than the Planck length cannot be detected due to limitations on measuring devices which arise from quantum mechanics and general relativity [8,9,10]
This suggests, but does not prove, that models of quantum gravity that are consistent with what is currently known about low energy physics will incorporate minimal length in some fundamental way
Summary
Gedanken experiments can reveal fundamental limitations on measurements or other experimental procedures that arise from the laws of physics, see e.g. [1,2,3,4,5,6,7]. It has been shown that discreteness of space-time on length scales smaller than the Planck length cannot be detected due to limitations on measuring devices which arise from quantum mechanics and general relativity [8,9,10] This suggests, but does not prove, that models of quantum gravity that are consistent with what is currently known about low energy (long distance) physics will incorporate minimal length in some fundamental way. We give a more complete derivation of the result: we consider the angular displacement operator φ(t) − φ(0) and examine limits on related experimental procedures This is analogous to the approach used in [8] to deduce minimal length
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