## Introduction

The general rules according to which gravity couples to matter (quantum or classical) are given in Einstein’s Equivalence Principle. These rules presuppose that the dynamical laws for the matter without gravity are given in a form that is compatible with Special Relativity, that is, that they are Poincaré invariant. However, that is not the case with ordinary Quantum Mechanics. Hence it is necessary to translate the general (relativistic) coupling rules to the non-relativistic setting.

Einstein’s Equivalence Principle does not apply to genuine quantum systems. We aim to formulate a Quantum-Equivalence-Principle that entails a general scheme according to which the coupling of quantum-systems to all components the Einsteinian gravitational field can be deduced unambiguously. This is important to properly set up and interpret current and future tests of the Equivalence principle involving quantum systems.

## Results

In a recent paper (Schwartz, Giulini: “Post-Newtonian corrections to Schrödinger equations in gravitational fields”, Class. Quantum Grav. 36 095016 (2019)) we extend the WKB-like non-relativistic expansion of the minimally coupled Klein-Gordon equation to arbitrary order in c, leading to Schrödinger equations describing a quantum particle in a general gravitational field, and compare the results with canonical quantisation of a free particle in curved spacetime. Furthermore, using a more operator-algebraic approach, the Klein-Gordon equation and the canonical quantisation method are shown to lead to essentially the same results in the Hamiltonian describing a single particle in a general stationary spacetime, without any non-relativistic expansion.

## Future

Currently we (Giulini-Schwartz) investigate possible extension of previous results mentioned above to many-particle systems. In addition, we implement recent proposals concerning tests of the Universality of Free Fall (known as “UFF”, which is part of the Equivalence Principle) into our framework. We are in the process of writing up results concerning recent discussions in the literature on the relation between non-violation of UFF and the symmetries of Einstein’s Equations (diffeomorphism invariance), in particular, the relativistic virial theorem and Laue’s theorem. (For the latter, see: Domenico Giulini, Int. Jour. Geom. Meth. Phys. 15, 1850182 (2018))

Our results are relevant to understanding the coupling between gravity and matter, and in particular concerning experimental tests of the principles underlying that coupling, like in B07. Eventually this understanding should be formalised in a formulation of a Quantum-Equivalence-Principle (QEP). Without such a formalisation we will have no hard criteria by means which we can unambiguously recognise possible violations of the equivalence principle in the quantum domain. Conversely, such a principle will lead us to new proposals of tests of the equivalence principle, which lies at the heart of our current understanding of gravity and the structure of space and time.

Our work will actively contribute to answer the following questions posed in B07:

- Detectability and interpretation of gravitational redshift in atom-interferometers;
- the contribution of gravitation to dephaksing in atom-interferometry;
- gravity-induced limitations to atomic clocks,
- gravitational-wave detection with atom-interferometry and Einstein-Bose condensates.

## Publications

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