O ct 1 99 8 The equivalence principle in classical mechanics and quantum mechanics ∗

We discuss our understanding of the equivalence principle in both classical mechanics and quantum mechanics. We show that not only does the equivalence principle hold for the trajectories of quantum particles in a background gravitational field, but also that it is only because of this that the equivalence principle is even to be expected to hold for classical particles at all. While the equivalence principle stands at the very heart of general relativity, there appear to be some aspects of it that are not quite as secure as they might be. In particular, while there is no disputing the fact that classical geodesics in a background gravitational field exhibit the equivalence principle, the question of whether the motions of real physical systems can explicitly be associated with such geodesics is actually a logically independent issue. Moreover, so also is the further question of what is supposed to happen when the physical systems are to be described by quantum mechanics, a situation which has actually been explored experimentally in the landmark Colella-Overhauser-Werner (COW) study [1, 2, 3, 4] of a quantum-mechanical beam of neutrons traversing an interferometer located in an external gravitational 1 field. In this paper we shall examine both of these issues to show that not only is the equivalence principle actually found to hold for quantum-mechanical particles, but that classical-mechanical particles actually inherit the classical-mechanical equivalence principle from them. While the standard road to the classical-mechanical equivalence principle is of course completely familiar, it is nonetheless pedagogically instructive to quickly recall the steps. Suppose we begin with a standard, free, spin-less, classical-mechanical Newtonian particle of non-zero kinematic mass m moving in flat spacetime according to the special relativistic generalization of Newton's second law of motion m d 2 ξ α dτ 2 = 0 , R µνστ = 0 (1) where dτ = (−η αβ dξ α dξ β) 1/2 is the proper time and η αβ is the flat spacetime metric, and where we have indicated explicitly that the Riemann tensor is (for the moment) zero. Now let us transform to an arbitrary coordinate system x µ. Using the definitions