Fisher information and the weak equivalence principle of a quantum particle in a gravitational wave
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Regular Article - Theoretical Physics
Fisher information and the weak equivalence principle of a quantum particle in a gravitational wave James Q. Quacha Institute for Photonics and Advanced Sensing and School of Chemistry and Physics, The University of Adelaide, Adelaide, SA 5005, Australia
Received: 31 July 2020 / Accepted: 6 October 2020 © The Author(s) 2020
Abstract We show that the weak equivalence principle (WEP) is violated for a quantum particle in a gravitational wave (GW) background, in the sense that extra mass information can be extracted in the presence of the GW. We quantify the degree of violation with the Fisher information of mass. This provides a precise characterisation of WEP violation by quantum systems in a GW, that should be useful in formalising other works that have argued for such violations heuristically.
1 Introduction The WEP states that point particles in free-fall will follow trajectories that are independent of their mass. This principle underpins classical gravitational theory. In the context of classical theory the WEP is well defined; in quantum theory however, the WEP is ill-defined. This is because under the Heisenberg’s uncertainty principle, point particles and trajectories are ambiguous concepts. The problem is further highlighted when one compares the classical action of a particle with mass m in a gravitational field with the quantum action of a wavefunction ψ of a massive spinless particle. The classical action is SC = −mc
ds ,
(1)
where ds 2 = gμν d x μ d x ν . As m appears simply as a multiplicative factor, it does not feature in the equations of motion. This is consistent with the WEP. In comparison, the quantum action is SQ = −
a e-mail:
h¯ 2 2m
m 2 c2 √ μν g g Dμ Φ † Dν Φ + 2 Φ † Φ d4 x , (2) h¯
[email protected] (corresponding author)
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where Dμ is the covariant derivative in curved space-time. In this case m simply is not a multiplicative factor, and features in the Klein–Gordon equation. In this background, there have been suggestions that the properties of quantum fluids (superconductors, superfluids, quantum Hall fluids, Bose–Einstein condensates) may enhance the interaction with GW, leading to superfluids as a medium for gravitational antennae [1–7], superconducting circuits as GW detectors [8], transducers [9,10] and mirrors [11–13]. These idea have not been met without controversy [14–16]. The reason for this is that many of these ideas heuristically apply the notion that quantum particles violate the WEP. This motivates us to provide a more rigorous characterisation of the WEP for quantum particles in GWs. The WEP’s notion that free-falling trajectories should be independent of mass, can be reformulated as the statement that the Fisher information of a free-falling object is invariant with mass [17]. In this information-theoretic framework, violation of the WEP means that one may extract information about an object’s mass in free-fall. This informationtheoretic formulation of the WEP has the advantage that it is e
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