Vanzella, Daniel A. Turolla and Butterfield, Jeremy (2024) A frame-bundle formulation of quantum reference frames: from superposition of perspectives to superposition of geometries. [Preprint]
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Abstract
Recent experimental advances suggest we may soon be able to probe the gravitational field of a mass in a coherent superposition of position states---a system which is widely believed to lie outside the scope of classical and semiclassical gravity. The recent theoretical literature has applied the idea of quantum reference frames (QRFs), originally introduced for non-gravitational contexts, to such a scenario.
Here, we provide a possible fully geometric formulation of the core idea of QRFs as it has been applied in the context of gravity, freeing its definition from unnecessary (though convenient) ingredients, such as coordinate systems. Our formulation is based on two main ideas. First, a QRF encodes uncertainty about what is the observer’s (and, hence, the measuring apparatus's) perception of time and space at each spacetime point (i.e., event). For this, an observer at an event $p$ is modeled, as usual, as a tetrad in the tangent space $T_p$. So a QRF at an event $p$ is a complex function on the tetrads at $p$. Second, we use the result that one can specify a metric on a given manifold by stipulating that a basis one assigns at each tangent space is to be a tetrad in the metric one wants to specify. Hence a spacetime, i.e. manifold plus metric, together with a choice of ``point of view'' on it, is represented by a section of the bundle of bases, understood as taking the basis assigned to each point to be a tetrad. Thus a superposition of spacetimes gets represented as, roughly speaking, an assignment of complex amplitudes to sections of this bundle. A QRF, defined here as the collection of complex amplitudes assigned to bases at events---i.e., a complex function defined on the bundle of bases of the manifold---can describe, in a local way (i.e., attributing the amplitudes to bases at events instead of to whole sections), these superpositions.
We believe that this formulation sheds some light on some conceptual aspects and possible extensions of current ideas about QRFs. For instance, thinking in geometric terms makes it clear that the idea of QRFs applied to the gravitational scenarios treated in the literature (beyond linear approximation) lacks predictive power due to arbitrariness which, we argue, can only be resolved by some further input from physics.
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