Abstract We ask the question of how the (weak) equivalence principle established in classical gravitational physics should be reformulated and interpreted for massive quantum objects that may also have internal degrees of freedom (dof). This inquiry is necessary because even elementary concepts like a classical trajectory are not well defined in quantum physics—trajectories originating from quantum histories become viable entities only under stringent decoherence conditions. From this investigation we posit two logically and operationally distinct statements of the equivalence principle for quantum systems. Version A: the probability distribution of position for a free-falling particle is the same as the probability distribution of a free particle, modulo a mass-independent shift of its mean. Version B: any two particles with the same velocity wave-function behave identically in free fall, irrespective of their masses. Both statements apply to all quantum states, including those without a classical correspondence, and also for composite particles with quantum internal dof. We also investigate the consequences of the interaction between internal and external dof induced by free fall. For a class of initial states, we find dephasing occurs for the translational dof, namely, the suppression of the off-diagonal terms of the density matrix, in the position basis. We also find a gravitational phase shift in the reduced density matrix of the internal dof that does not depend on the particle’s mass. For classical states, the phase shift has a natural classical interpretation in terms of gravitational red-shift and special relativistic time-dilation.
Abstract We ask the question of how the (weak) equivalence principle established in classical gravitational physics should be reformulated and interpreted for massive quantum objects that may also have internal degrees of freedom (dof). This inquiry is necessary because even elementary concepts like a classical trajectory are not well defined in quantum physics—trajectories originating from quantum histories become viable entities only under stringent decoherence conditions. From this investigation we posit two logically and operationally distinct statements of the equivalence principle for quantum systems. Version A: the probability distribution of position for a free-falling particle is the same as the probability distribution of a free particle, modulo a mass-independent shift of its mean. Version B: any two particles with the same velocity wave-function behave identically in free fall, irrespective of their masses. Both statements apply to all quantum states, including those without a classical correspondence, and also for composite particles with quantum internal dof. We also investigate the consequences of the interaction between internal and external dof induced by free fall. For a class of initial states, we find dephasing occurs for the translational dof, namely, the suppression of the off-diagonal terms of the density matrix, in the position basis. We also find a gravitational phase shift in the reduced density matrix of the internal dof that does not depend on the particle’s mass. For classical states, the phase shift has a natural classical interpretation in terms of gravitational red-shift and special relativistic time-dilation.
The weak equivalence principle of gravity is examined at the quantum level in two ways.First, the position detection probabilities of particles described by a non-Gaussian wave packet projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated.These probabilities exhibit mass dependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle.Second, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be mass dependent.Such a mass dependence is shown to be enhanced by increasing the non-Gaussianity parameter of the wave packet, thus signifying a stronger violation of the weak equivalence principle through a greater departure from Gaussianity of the initial wave packet.The mass dependence of both the position detection probabilities and the mean arrival time vanishes in the limit of large mass.Thus, compatibility between the weak equivalence principle and quantum mechanics is recovered in the macroscopic limit of the latter.A selection of Bohm trajectories is exhibited to illustrate these features in the free fall case.
Pikovski et al. [Nat. Phys. 11, 668 (2015)] show that a composite particle prepared in a pure initial quantum state and propagated in a uniform gravitational field undergoes a decoherence process at a rate determined by the gravitational acceleration. By assuming Einstein's equivalence principle to be valid, we analyze a physical realization of the (1+1)D thought experiment of Pikovski et al. to demonstrate that the dephasing between the different internal states arises not from gravity but rather from differences in their rest mass, and the mass dependence of the de Broglie wave's dispersion relation. We provide an alternative view to the situation considered by Pikovski et al., where we propose that gravity plays a kinematic role by providing a relative velocity to the detector frame with respect to the particle; visibility can be easily recovered by giving the screen an appropriate uniform velocity. We then apply this insight to their thought experiment in (1+1)D to draw a direct correspondence, and obtain the same mathematical result for dephasing. We finally propose that dephasing due to gravity may in fact take place for certain modifications to the gravitational potential where the equivalence principle is violated.
Investigation into the applicability of the equivalence principle in quantum mechanics has taken many forms, with varying conclusions. Here, a dynamical semiclassical description of a wave packet in terms of its center of mass and higher quantum fluctuations is applied to the case of a quantum particle in gravitational free fall. The analysis provides an intuitive account of the origin of mass dependence in quantum-gravitational dynamics through an effective potential that enforces the uncertainty principle. This potential has two implications: (i) The lowest order quantum fluctuations encoding the width and spreading of the wave packet obey an uncertainty relation whose observance is mass dependent. (ii) In an inhomogeneous gravitational field tidal effects couple the center of mass motion to the quantum fluctuations. The combined effect results in a clear demonstration of how some conceptions of the weak equivalence principle, based on mass dependence, are violated. The size of this violation is within sensitivities of current E\"otv\"os and clock-based return time experiments.
The Einstein equivalence principle (EEP) links the universality of free fall, local Lorentz invariance, and local position invariance to the geometric description of gravity. In quantum systems, however, internal energy is represented by an operator and may occupy coherent superpositions; consequently, equality of classical mass parameters is not sufficient to establish equivalence at the operator level. This work develops a compact operator-level Quantum Equivalence Principle (QEP) residual and performs a literature-based experimental cross-validation using independent high-precision experiments. The theoretical basis follows the operator formulation of Zych and Brukner and its relativistic extension by Das, Fridman, and Lambiase. The quantum weak equivalence sector is tested using the 87Rb Bragg atom-interferometry experiment of Rosi et al., which compared two hyperfine internal-energy eigenstates and a coherent superposition, finding Eotvos ratios consistent with zero and constraining the off-diagonal quantum-WEP parameter to 5 x 10^-8. A classical diagonal WEP benchmark is supplied by the final MICROSCOPE result, eta(Ti,Pt) = [-1.5 +/- 2.3(stat) +/- 1.5(syst)] x 10^-15. The local Lorentz-invariance sector is cross-checked using fast 7Li+ optical clocks, which verified relativistic time dilation at 2 x 10^-10 accuracy, while the local-position-invariance sector is cross-checked using millimetre-scale 87Sr optical-clock gravitational-redshift measurements with fractional-frequency uncertainty 7.6 x 10^-21. Finally, published 2025 gravitational-wave QEP bounds are evaluated, with GW190814 providing the intervals [-0.59,0.54], [-0.40,0.37], and [-0.19,0.17] for WEP-, LLI-, and LPI-associated mass-energy differences, respectively. No statistically significant violation is identified in any analyzed sector. The combined evidence supports operator-level QEP consistency over laboratory quantum superpositions, relativistic clock kinematics, gravitational redshift, precision free fall, and strong-field gravitational-wave phenomenology, while also isolating the remaining experimental gap: direct off-diagonal tests of quantum LLI and quantum LPI.
We use the Feynman path integral approach to nonrelativistic quantum mechanics twofold. First, we derive the lagrangian for a spinless particle moving in a uniformly but not necessarily constantly accelerated reference frame; then, applying the strong equivalence principle (SEP) we obtain the Schroedinger equation for a particle in an inertial frame and in the presence of a uniform and constant gravity field. Second, using the associated Feynman propagator, we propagate an initial gaussian wave packet, with the final wave function and probability density depending on the ratio m/hbar, where m is the inertial mass of the particle, thus exhibiting the fact that the weak equivalence principle (WEP) is violated by quantum mechanics. Although due to rapid oscillations the wave function does not exist in the classical limit, the probability density is well defined and mass independent when hbar goes to 0, showing the recovery of the WEP. Finally, at the quantum level, a heavier particle does not necessarily falls faster than a lighter one; this depends on the relations between the initial and final common positions and times of the particles.
The computation of the phase shift in a symmetric atom interferometer in the presence of a gravitational field is reviewed. The difference of action-phase integrals between the two paths of the interferometer is zero for any Lagrangian which is at most quadratic in position and velocity. We emphasize that in a large class of theories of gravity the atom interferometer permits a test of the weak version of the equivalence principle (or universality of free fall) by comparing the acceleration of atoms with that of ordinary bodies, but is insensitive to that aspect of the equivalence principle known as the gravitational redshift or universality of clock rates.