foreXiv

Now I would like to apply the reasoning of the last post to the case of verifying macroscopic superpositions of the metric.  It’s been 4 years since I’ve touched GR, so I’m going to rely heavily on E&M concepts and pray I don’t miss any key changes in the translation to gravity.

In the two-slit experiment with light, we don’t take the visibility of interference fringes as evidence of quantum mechanics when there are many photons.  This is because the observations are compatible with a classical field description. We could interfere gravitational waves in a two-slit set up, and this would also have a purely classical explanation.

But in this post I’m not concentrating on evidence for pure quantum mechanics (i.e. a Bell-like argument grounded in locality), or evidence of the discrete nature of gravitons. Rather, I am interested in superpositions of two macroscopically distinct states of the metric as might be produced by a superposition of a large mass in two widely-separated positions.  Now, we can only call a quantum state a (proper) superposition by first identifying a preferred basis that it can be a superposition with respect to.  For now, I will wave my hands and say that the preferred states of the metric are just those metric states produced by the preferred states of matter, where the preferred states of matter are wavepackets of macroscopic amounts of mass localized in phase space (e.g. L/R).  Likewise, the conjugate basis states (e.g. L+R/L-R) are proper superpositions in the preferred basis, and these two bases do not commute.

There are two very distinct ways to produce a superposition with different states of the metric: (1) a coherent superposition of just gravitational radiation Note that we expect to produce this superposition by moving a macroscopic amount of matter into a superposition of two distinct position or momentum states.  … [continue reading]

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]