Nonrelativistic spatiotemporal quantum reference frames

Quantum reference frames have attracted renewed interest recently, as their exploration is relevant and instructive in many areas of quantum theory. Among the different types, position and time reference frames have captivated special attention. Here, we introduce and analyze a nonrelativistic framework in which each system contains an internal clock in addition to its external (spatial) degree of freedom and, hence, can be used as a spatiotemporal quantum reference frame. We present expressions for expectation values and variances of relevant observables in different perspectives, as well as relations between these quantities in different perspectives in scenarios with no interactions. In particular, we show that even in these simple scenarios the relative uncertainty between clocks affects the relative spatial spread of the systems.

Figure 1

Illustration of a spatial perspective-dependent description when particle $A$ is chosen to be a reference frame.

Figure 2

Perspective in a spatiotemporal reference frame. In addition to its spatial degree of freedom, each particle is assumed to have an internal degree of freedom identified as a clock. Here, system $A$ is chosen to be a reference frame. In this description, ${\widehat{x}}_B, {\widehat{x}}_C, {\widehat{t}}_B$, and ${\widehat{t}}_C$ are observables, while $t_A$ is a parameter.

Figure 3

Representation of reciprocal temporal descriptions. In general, the clocks are not synchronized but have the same variance in each other's reference frames.

Figure 4

Representation of reciprocal spatial descriptions. Due to the different reference frames for time associated with each system, there is generally a shift that violates the antisymmetric position expectation value and, moreover, the reciprocal spatial variance is not the same.