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Abstract
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum $j$ and introduce the transformations between them by upgrading the Euler angles occurring in classical $\text{SO}(3)$ spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large $j$, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.
Highlights
The description of physical systems relies heavily on the choice of the reference frame used
As a starting point we imagine ourselves “sitting” in the frame C and assume that from this perspective we describe the physical properties of both A and B
The key questions we address in this paper are how spin degrees of freedom are encoded in the physical structure of the quantum reference frames (QRFs) A and C, and how such a change of perspective between them induces a transformation of the description of B and the remaining QRF
Summary
The description of physical systems relies heavily on the choice of the reference frame used. A recent work by Giacomini, Castro-Ruiz, and Brukner [22] outlines a formalism to treat reference systems with quantum degrees of freedom and provides an approach to the transformation of spatial and momentum variables in QRFs that is genuinely relational by construction. We “quantize” rotational reference frames by treating the Euler angles entering the classical SO(3) spin transformations as quantum mechanical operators This extends the group of transformations between QRFs from rotations to “superpositions of rotations.”. For an active QRF transformation, one would need to prepare a macroscopic (reference frame) system in a quantum state with respect to the laboratory and consider measurements on other systems relative to that system. The same method was recently used to define the spin operator of a massive particle that moves in a superposition of relativistic velocities in a laboratory reference frame. IV we conclude with a physical discussion of the results and some possible generalizations
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