All-order Galilei T-matrix for arbitrary quantum reference-frame states

Establish that the universal T-matrix of the unique quantum (1+1) centrally extended Galilei group with non-trivial central extension provides the algebraic description of Galilei quantum reference frame transformations for arbitrary quantum states of the reference frame, beyond first order in the quantum deformation parameter and without restricting the reference frame to superpositions of semiclassical states.

Background

Within the perspectival approach to quantum reference frames, transformations between inertial quantum reference frames were related to a quantum Galilei group. Specifically, a unique quantum (1+1) centrally extended Galilei group was identified whose universal T-matrix reproduces the desired transformations at first order in the deformation parameter, assuming the reference frame is in a superposition of semiclassical states.

Motivated by this first-order result, the authors explicitly articulate a conjecture extending it to all orders and to arbitrary quantum states of the reference frame, proposing that the full, non-perturbative universal T-matrix encodes the complete algebraic structure of Galilei quantum reference frame transformations.

References

In particular, there exists a unique quantum (1+1) centrally extended Galilei group (with non-trivial central extension) identified in such that its universal $T$-matrix produces the sought Galilei QRF transformations, provided that only the first order in the quantum deformation parameter is regarded and that the QRF is in a quantum superposition of semiclassical states. This result triggered the conjecture that the all-order Galilei $T$-matrix should supply the algebraic description of Galilei QRF transformations involving arbitrary quantum states for the QRF (see for details).

— Universal $T$-matrices for quantum Poincaré groups: contractions and quantum reference frames  (2604.01058 - Ballesteros et al., 1 Apr 2026) in Introduction (Section 1)