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As I understand, we do not have yet a unified theory covering at once both general relativity and quantum mechanics. However, do we have a theoretical framework completely covering both classical Newtonian gravitation (i.e. without space-time curvature or for not so massive gravitational sources) and quantum mechanics? Is there any reference on this question?

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Since both Newtonian gravitation and electrostatic interaction (Coulomb's law) follow an inverse square law, Newtonian gravitation is as compatible with QM as electrostatics is. We do not need anything new to account for this interaction. An article in Physics Today describes an experiment where the gravitationally bound states of neutrons in a box were measured. Here is a more recent review.

Since an expansion in $v/c$ or the curvature is possible, one should be able to incorporate even weakly relativistic effects, e.g., the $1/r^3$ correction to the potential that accounts for the anomalous precession of the perihelion of Mercury. This is analogous to the spin-orbit, Thomas precession, and "mass shift" correction terms -- all relativistic of order $v^2/c^2$ that account for the fine structure of the hydrogen atom.

Of course one should note that it is possible to do quantum mechanics -- actually, quantum field theory -- in curved spacetime backgrounds. Here background means that one neglects the gravitational field produced by the matter that is modeled quantum mechanically. E.g., in Hawking's famous calculation, the black hole and the rest of spadetime is entirely classical, and the gravitational field produced by the radiation is neglected, and only the radiation is quantum mechanical. For a general formalism see the recent review by Fredenhagen and Rejzner.

If we take general relativity to be "matter tells spacetime how to curve, curvature tells matter how to move", then one could say that we know how to treat the latter quantum mechanically, but not the former.

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