Suppose we are given an ensemble of systems which are believed to contain coherent superposition of the metric. How would we confirm this?

Well, in order to verify that an arbitrary system is in a coherent superposition, which is always relative to a preferred basis, it’s well known that we need to make measurements with respect to (at least?) two non-commuting bases. If we can make measurement *M* we expect it to be possible to make measurement *M*` = *RM* for some symmetry *R*.

I consider essentially two types of Hilbert spaces: the infinite-dimensional space associated with position, and the finite-dimensional space associated with spin. They have a very different relationship with the fundamental symmetries of spacetime.

For spin, an arbitrary rotation in space is represented by a unitary which can produce proper superpositions. Rotating 90 degrees about the y axis takes a z-up eigenstate to an equal superposition of z-up and z-down. The rotation takes one basis to another with which it does not commute.

In contrast, for position, the unitary representing spatial translation is essentially just a permutation on the space of position eigenstates. It does not produce superpositions from non-superpositions with respect to this basis.

You might think things are different when you consider more realistic measurements with respect to the over-complete basis of wavepackets. (Not surprisingly, the issue is one of preferred basis!) If you imagine the wavepackets as discretely tiling space, it’s tempting to think that translating a single wavepacket a half-integer number of tile spacing will yield an approximate superposition of two wavepackets. But the wavepackets are of course not discrete, and a POVM measurement of “fuzzy” position (for any degree of fuzziness σ) is invariant under spatial translations.… [continue reading]