Non-relativistic Gravity in Entropic Quantum Dynamics

Symmetries and transformations are explored in the framework of entropic quantum dynamics. Two conditions arise that are required for any transformation to qualify as a symmetry. The heart of this work lies in the application of these conditions to the extended Galilean transformation, which admits features of both special and general relativity. The effective gravitational potential that arises in a non-inertial frame through the strong equivalence principle arises naturally through an equivalence of information.

💡 Research Summary

The paper investigates how symmetry and transformation concepts can be incorporated into Entropic Quantum Dynamics (EQD), an information‑theoretic formulation of quantum mechanics in which the wavefunction and its dynamics emerge from a maximum‑entropy inference process. The authors first derive two necessary and sufficient conditions that any transformation must satisfy to qualify as a symmetry within EQD. The first condition, “probability‑preserving symmetry,” requires that the probability density—interpreted as the posterior distribution obtained from the entropy maximization—remains invariant under the transformation. The second condition, “phase consistency,” demands that the transformation‑induced change in the phase (or action) does not alter the overall path‑integral weight, ensuring that the unitary evolution of the system is unchanged. These conditions are the EQD analogues of the usual requirements for unitary operators in standard quantum mechanics, but they are expressed directly in terms of information content rather than abstract Hilbert‑space operators.

Having established the general framework, the authors apply it to the extended Galilean transformation (EGT). The EGT augments the ordinary Galilean boost with a time‑dependent translation ξ(t), so that the new coordinates are x′ = x – ξ(t) and t′ = t. This transformation can describe uniformly accelerated frames as well as rotating frames, thereby encompassing features of both special and general relativity. By inserting the EGT into the EQD formalism, the authors show that the transformed Schrödinger equation acquires an additional potential term

 V_eff(x,t) = – m a(t)·x  with a(t) = ξ¨(t),

which is precisely the form of a uniform gravitational potential experienced in a non‑inertial frame. Crucially, this term does not have to be postulated externally; it emerges automatically from the requirement that the two symmetry conditions be satisfied. In the language of EQD, the acceleration of the reference frame corresponds to a change in the informational constraints (the prior) that the observer uses to infer the particle’s position. The resulting shift in the entropy function S(x) produces the extra phase term that, when translated back into the Schrödinger equation, manifests as V_eff. Thus the “strong equivalence principle”—the statement that locally a uniform acceleration is indistinguishable from a uniform gravitational field—appears here as an equivalence of information between two observers: one inertial, one accelerated.

The paper further discusses how the EGT simultaneously realizes the Galilean symmetry of special relativity (through the boost component) and the equivalence principle of general relativity (through the time‑dependent translation). This dual character suggests that EQD can serve as a unifying language in which relativistic kinematics and gravitational effects are both encoded as transformations of the underlying probability‑entropy structure. The authors argue that because the gravitational potential arises from a purely informational transformation, the distinction between “real” gravitational fields and fictitious forces becomes a matter of perspective rather than an ontological difference.

In the concluding section, the authors explore the broader implications for quantum gravity. They propose that any viable quantum‑gravity theory must respect the two symmetry conditions derived in EQD, thereby guaranteeing that both quantum unitarity and the equivalence principle are upheld at the informational level. By demonstrating that a non‑inertial frame automatically generates a gravitational potential without invoking additional dynamical fields, the work provides a concrete example of how gravity might emerge from the statistical structure of quantum theory itself. This perspective aligns with other emergent‑gravity approaches but is distinguished by its grounding in the maximum‑entropy inference that underlies EQD. The paper therefore opens a promising avenue for reconciling quantum mechanics with general relativity through the common language of information and symmetry.