We adapt to the context of quantum theory the treatment, familiar in relativity theory, of a reference frame as a tetrad. Since a tetrad is a set of four orthonormal vectors at a spacetime point, it encodes a timelike direction, three orthogonal spatial directions, and units; and so it gives a configuration in spacetime (a spatial orientation and state of motion) of an idealized infinitesimal apparatus (``rulers and a clock"). So a natural ``first guess" is that the state of a quantum reference frame should be an assignment of complex amplitudes to a set of tetrads. We develop this idea, in terms of bundles over spacetime of bases of the tangent spaces.
This paper is a companion to two other papers. It emphasises the main conceptual aspects of our proposal, while the other two papers give physical details. It also explains the main contrast between those two papers. Namely: (i) the first considers a single spacetime geometry, so that a quantum reference frame, on our proposal, gives a superposition of perspectives on that geometry; while (ii) the second considers the spacetime manifold with no geometry given a priori, which will imply that a quantum reference frame, on our proposal, can describe a superposition of perspectives on a superposition of geometries---not ``just" a superposition of perspectives on a single geometry. The main mathematical idea for representing this contrast will be choosing between (i) the orthonormal frame bundle, for a given spacetime geometry; (ii) the frame bundle in the usual sense of containing all bases of all spacetime points' tangent spaces.
We adapt to the context of quantum theory the treatment, familiar in relativity theory, of a reference frame as a tetrad. Since a tetrad is a set of four orthonormal vectors at a spacetime point, it encodes a timelike direction, three orthogonal spatial directions, and units; and so it gives a configuration in spacetime (a spatial orientation and state of motion) of an idealized infinitesimal apparatus (rulers and a clock"). So a natural first guess" is that the state of a quantum reference frame should be an assignment of complex amplitudes to a set of tetrads. We develop this idea, in terms of bundles over spacetime of bases of the tangent spaces.
This paper is a companion to two other papers. It emphasises the main conceptual aspects of our proposal, while the other two papers give physical details. It also explains the main contrast between those two papers. Namely: (i) the first considers a single spacetime geometry, so that a quantum reference frame, on our proposal, gives a superposition of perspectives on that geometry; while (ii) the second considers the spacetime manifold with no geometry given a priori, which will imply that a quantum reference frame, on our proposal, can describe a superposition of perspectives on a superposition of geometries---not ``just" a superposition of perspectives on a single geometry. The main mathematical idea for representing this contrast will be choosing between (i) the orthonormal frame bundle, for a given spacetime geometry; (ii) the frame bundle in the usual sense of containing all bases of all spacetime points' tangent spaces.
We provide a possible fully geometric formulation of the core idea of quantum reference frames (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 Tp. 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.
Quantum permutations, or magic unitaries, have in recent years been explored in the context of identifying `genuinely quantum' isometries of graphs. Here, we import this tool in physics, showing that quantum permutations yield a generalisation of quantum reference frames in a discrete setting that is reminiscent of the passage from special to general relativity. We show that the typical quantum reference frames framework corresponds to quantum permutations classified as `classical' in the mathematical literature, and which we demonstrate are quantum controlled transformations (superpositions of classical coordinate maps). Genuinely quantum permutations (i) allow to construct \emph{non--commuting} quantum reference frames (ii) correspond to \emph{local}, as opposed to global, superpositions of transformations. Strikingly, we find that the non-commutativity of quantum fields, when used as reference systems, is exactly what implies that the change of frame is achieved through a genuine quantum permutation. We illustrate the above with several examples in both first and second quantization formalism, which demonstrate (a) simultaneous control on non--commuting variables, (b) the existence of bipartite states that can be localized with a genuine quantum permutation and cannot be localized with the usual quantum reference frame transformations (without introducing additional degrees of freedom), (c) extension of the Ising model symmetries to genuinely quantum permutations, and (d) extension of the symmetries of a scalar field action on curved spacetime to genuinely quantum permutations. While we have in mind applications in quantum gravity, we expect our formalism to be of interest in a wide range of topics in quantum information.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
Abstract Quantum reference frame as in the example of observing a quantum particle from another has been a topic of much recent interest. Quantum spatial translations, quantum rotations, and quantum Lorentz boosts in the sense have been studied to some extent. The article aims at using a consistent formulation of all that to give a full picture of what would be the symmetry of Special Quantum Relativity as special relativity as seen from quantum particles. We first look at a direct formulation of the mathematical quantum analog of any classical one-parameter Lie group of such transformations suggested by our Heisenberg picture thinking of quantum mechanics. A quantum rotation and a quantum Lorentz boost are presented. Analysis of the compositions of two such transformations shows unnatural results with various problems, especially with their commutators. From a physical point of view, we argue for a modified formulation. A quantum reference frame transformation, as the description of firstly the spacetime position of an object, idealized as a particle, as seen from another has to involve a full set of four-coordinate quantum quantities instead of one in a single coordinate only. Unlike the classical case, we illustrate that the analog of a change by the zero classical coordinate values has nontrivial effects in the transformation. A consistent full picture of the symmetry is then obtained. Results that could be interpreted essentially as the quantum analogs of the translation of a single coordinate observable, including angular ones for a rotation or a boost, can be retrieved from that but only as approximations. Under the traditional perspective of relativity symmetry being about spacetime reference frames, our task is then completed. We further look at the quantum version of Heisenberg–Weyl symmetry and perspectives on its relation with the subject matter and argue for the necessity of our background noncommutative geometric perspective for a conceptual consistent story.
Recent efforts to formulate quantum mechanics in a way that treats space and time on a more equal footing have led to a large variety of spacetime-oriented approaches. In this work we present a detailed study of spacetime states, the objects that play the role of quantum states in the recently introduced framework of spacetime quantum mechanics, and show that the main proposals in the literature are different manifestations of the same underlying object. Path integrals, quantum states over time, pseudo-density matrices, the Page and Wootters mechanism, superdensity operators, and timelike-entanglement proposals all arise from spacetime states through particular evaluations, reduced information, linear maps, or quantum channels. This unification provides explicit mathematical representations of these formalisms, reveals relations among them, and clarifies the spacetime information each one captures. We also study the broader relevance of the spacetime-state point of view for Leggett-Garg inequalities, OTOCs, temporal tensor networks, fermionic systems, relativistic QFTs, quantum reference frames, and classical physics, together with additional insights and perspectives revealed by the common unifying framework.
The topic of quantum reference frames (QRFs) has attracted a great deal of attention in the recent literature. Potentially, the correct description of such frames is important for both the technological applications of quantum mechanics and for its foundations, including the search for a future theory of quantum gravity. In this letter, we point out potential inconsistencies in the mainstream approach to this subject and propose an alternative definition that avoids these problems. Crucially, we reject the notion that transformations between QRFs can be represented by unitary operators and explain the clear physical reasons for this. An experimental protocol, capable of empirically distinguishing between competing definitions of the term, is also proposed. The implications of the new model, for uncertainty relations, spacetime symmetries, gauge symmetries, the quantisation of gravity, and other foundational issues are discussed, and possible directions for future work in this field are considered.