Unitary cocycle representations of the Galilean line group: Quantum mechanical principle of equivalence

Abstract

We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these representations determine the structure of quantum mechanics in non-inertial reference frames. Our representations of the Galilean line group contain the usual unitary projective representations of the Galilei group, but have a more intricate cocycle structure. The transformation formula for the Hamiltonian under the Galilean line group shows that in a non-inertial reference frame it acquires a fictitious potential energy term that is proportional to the inertial mass, suggesting the equivalence of inertial mass and gravitational mass in quantum mechanics.

Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations

“…As such, loops are sometimes referred to as non-associative groups. While there exist no two-cocycles of the full Galilean line group (including time dependent rotations) taking values on the abelian group of real analytic functions and reducing to a two-cocycle of the Galilei group [2,3], we show here that there do exist three-cocycles fulfilling the reduction criterion and that these three cocycles lead to loop prolongations of the Galilean line group which contain central extensions of the Galilei group. The main technical result we report in this paper is the construction of a certain class of representations of loop prolongations of the Galilean line group.…”

“…Alternatively, omitting ϕ, we may view the representations (1.1) as representations of the Galilean line group with an intricate cocycle structure so that they do not correspond to vector representations of a group extension of the Galilean line group. From the results reported in [1][2][3] and what is developed below in this paper, we note these representations completely describe the effects of fictitious forces due to both linear and rotational accelerations of reference frames. Consequently, our main proposition is that Galilean quantum mechanics in accelerating reference frames, both linearly and rotationally, is grounded in the representations (1.1).…”

“…Since projective unitary representations of a group are equivalent to true (vector) unitary representations of the central extensions of the group, we may recast this restriction on the admissible representations as an embedding criterion: construct a suitable extension of the Galilean line group that embeds a given central extension of the Galilei group. However, as shown in [2], while the Galilean line group naturally contains the Galilei group as a subgroup, it has no central extensions that contains a central extension of the Galilei group. This motivates us to consider more general, non-central extensions of the line group.…”

“…violations always disappear [3]. There also exist two-cocycles of the linear acceleration group which uphold the equivalence principle [1,2].…”

Realization of uniform synthetic magnetic fields by periodically shaking an optical square lattice

“…Our first task is to define a unitary transformation between a Hamiltonian given in the laboratory frame, and a Hamiltonian where the (accelerated) lattice appears to be at rest. The authors of [65,66] constructed a unitary representation of the Galilean line group, which is the source for the unitary transformations we will use. A concise summary (in first quantization) of the rules of transformation between accelerated frames can be found in [67].…”