Einstein's Equivalence principle for superpositions of gravitational fields and quantum reference frames

Abstract

The Einstein Equivalence Principle (EEP), stating that all laws of physics take their special-relativistic form in any local inertial (classical) reference frame, lies at the core of general relativity. Because of its fundamental status, this principle could be a very powerful guide in formulating physical laws at regimes where both gravitational and quantum effects are relevant. The formulation of the EEP only holds when both matter systems and gravity are classical, and we do not know whether we should abandon or modify it when we consider quantum systems in a-possibly nonclassical-gravitational field. Here, we propose that the EEP is valid for a broader class of reference frames, namely Quantum Reference Frames (QRFs) associated to quantum systems. By imposing certain restrictions on the type of nonclassicality of the gravitational field, we develop a framework that enables us to formulate an extension of the EEP for such gravitational fields. This means that the EEP is valid in a much wider set of physical situations than what it is currently applied to, including those in which the gravitational field is in a quantum superposition state.

Figures (3)

• Figure 1: Illustration of the generalisation of the Einstein Equivalence Principle (EEP) to quantum reference frames (QRFs). a) A mass in a spatial quantum superposition sources a superposition of classical gravitational fields. A clock in such superposition of gravitational fields, as seen from a general reference frame, is thus entangled with the gravitational field, and displays a different time according to the position of the mass. b) With a QRF transformation to the Quantum Locally Inertial Frame (QLIF) of the clock, the metric which was originally in a superposition can be cast as a locally minkowskian metric at the location of the clock. The clock then ticks according to its proper time, which is well-defined, and decouples from the gravitational field, which can still be in a quantum superposition in the neighbourhood of the clock.
• Figure 2: We formally describe the nonclassical structure of spacetime as a superposition of classical backgrounds enumerated as a set of manifolds $\mathcal{M} = \lbrace \mathcal{M}_i\rbrace_{i=1,\cdots, N}$ (with $N=3$ in the figure). On each classical manifold $\mathcal{M}_i$, we choose a coordinate system. Different points $X^{(i)}, \forall i=1, \cdots, N$ belonging to different manifolds are operationally identified with a "physical point" in the superposition of manifolds if a quantum system living in a superposition of those manifolds can be localised at those points $X^{(i)}$. In the figure, we represent with the shaded area the subregion of each classical manifold where the wavefunction of such quantum system has support. In the idealised case, the support of the quantum system reduces to a single point (red point in the figure) in every classical manifold. In the case considered, to each classical manifold $\mathcal{M}_i$ there corresponds a classical state of the gravitational field. In general, we do not know how to assign values to such gravitational field on a superposition of manifolds. We describe the gravitational field by correlating the spacetime point at which the field is defined with a quantum system, and we write this operation as $\sum_i \ket{g_i\triangleright \phi_i}$. This identification is natural, because tests of the gravitational field can be performed either directly, by measuring the gravitational radiation, or indirectly, as we choose here, via a probe particle. In the latter case, we can perform a measurement on the physical system to learn information about the gravitational field in the superposition of manifolds.
• Figure 3: Two different ways of identifying points across a superposition of classical manifolds. Using projector $\hat{\Pi}_1 = \ket{x_0^{(1)}}\bra{x_0^{(1)}} + \ket{y_1^{(2)}}\bra{y_1^{(2)}} + \ket{z_1^{(3)}}\bra{z_1^{(3)}}$, point $x_0$ in the first manifold $\mathcal{M}_1$ is identified with point $y_1$ in the manifold $\mathcal{M}_2$ and with point $z_1$ in the manifold $\mathcal{M}_3$. With the second projector $\hat{\Pi}_2 = \ket{x_0^{(1)}}\bra{x_0^{(1)}} + \ket{y_2^{(2)}}\bra{y_2^{(2)}} + \ket{z_2^{(3)}}\bra{z_2^{(3)}}$, point $x_0$ in the first manifold $\mathcal{M}_1$ is identified with point $y_2$ in the manifold $\mathcal{M}_2$ and with point $z_2$ in the manifold $\mathcal{M}_3$. The different physical identification of points corresponds to a different choice of measurement apparatus, where the projector is a mathematical abstraction representing the position of a measurement apparatus.