Quantum reference frames for general symmetry groups

Abstract

A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry group G, we define a general operator for reversibly changing between quantum reference frames associated to a group G. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are `in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations of G. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry group G is a semi-direct product G=N⋊P or a direct product G=N×P, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

“…This method is then used to explore an indirect self-reference phenomenon that arises when transforming between clock perspectives and to reveal the temporal frame and state dependence of comparing or even synchronizing the readings of different quantum clocks. This result adds to the growing list of quantum reference frame dependent physical properties, such as entanglement [70,72,74], spin [73], classicality [72] or objectivity [79,80] of a subsystem, superpositions [70,72], certain quantum resources [78], measurements [70,76], causal relations [47,83], temporal locality [7,47], and even spacetime singularity resolution [77]. The temporal frame changes may also be employed to extend recent proposals for studying time dilation effects of quantum clocks [45,136,137] (see also [137][138][139][140]).…”

“…Our proposal is thus to turn the multiple choice problem into a feature by having a multitude of quantum time choices at our disposal, which we are able to connect through quantum temporal frame transformations. This is in line with developing a genuine quantum implementation of general covariance [7,30,31,38,[70][71][72][73][74]. This proposal is part of current efforts to develop a general framework of quantum reference frame transformations (and study their physical consequences [75][76][77][78][79][80][81][82][83][84][85]), and should be contrasted with other attempts at resolving the multiple choice problem by identifying a preferred choice of clock [53] (see [7] for further discussion of this proposal).…”

“…In Castro-Ruiz et al [47] the clock change method was exhibited for state transformations in the relational Schrödinger picture for ideal clocks whose Hamiltonian coincides with the clock momentum itself. Our discussion can also be viewed as a full quantum extension of the semiclassical method in Bojowald et al [32,33] and Höhn et al [34] which is equivalent at semiclassical order, however, also applies to clock functions which are nonmonotonic, i.e., have turning points, in contrast to the other works mentioned (See [70][71][72]74] for related spatial quantum frame changes). 24 Indeed, the Hamiltonian constraint of the vacuum Bianchi I and II models can be written in the form [116]…”

“…This is advantageous for quantum gravity, where classical spacetime coordinates are a priori absent. A quantum version of general covariance should be formulated in terms of dynamical reference degrees of freedom [7,30,31,38,[70][71][72][73][74].…”

The group structure of dynamical transformations between quantum reference frames

“…From a foundational perspective, quantum mechanics can be formulated in relational terms [23] from the perspective of QRFs [24][25][26][27]. Recently, a relational formulation of QRFs has been used to perform QRFs transformations in different contexts, such as Galilean [28][29][30][31] and special-relativistic [32,33] quantum physics, quantum clocks models [34][35][36][37], cosmology [38], finite-dimensional quantum systems [39,40], and superpositions of curved spacetimes [22].…”