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 groupG, we define a general operator for reversibly changing between quantum reference frames associated to a groupG. 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 ofG. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry groupGis a semi-direct productG=N⋊Por a direct productG=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.
Highlights
In quantum mechanics, physical systems are implicitly described relative to some set of measurement devices
Once we introduce the notion of symmetry groups and how they enter into the formalism, we will see that this default zero-state corresponds to the identity element of the group that describes the transformations of the system
In the case where all the reference systems have configuration space R3, it makes sense to assign to each point x ∈ R3 a unique coordinate system centred on that point
Summary
Physical systems are implicitly described relative to some set of measurement devices. Most physically meaningful quantities are relational, i.e. they only take on well defined values once we agree on the reference system (or the observer) relative to which they are described In his papers [1, 2], Rovelli suggested that quantum mechanics is a complete theory about the description of physical systems relative to other physical systems. Combining these we define changes of reference frame under a group G for classical systems with configuration space X ∼= G.
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