On the problem of emergence of classical space-time: The quantum-mechanical approach

Alexey Kryukov (Department of Mathematics, University of Wisconsin)


We take for granted that physical events take place in space-time. Mathematically this is reflected in the realization of physical quantities as functions of space-time points. The shortcomings of such a realization are well known. In particular, the position of a particle in quantum field theory is only defined to energies less than the particle's mass and the concept of a quantum field at a point is ill-defined as well.

The interpretation of space-time as a Riemannian manifold so successfully used in classical physics is based on the possibility of physically (i.e. experimentally) identifying points of space-time. Most typically, the identification is done by scattering experiments. As a result of a high resolution scattering event needed to identify a point of space, the scattering center (i.e. a particle) assumes a localized quantum state. In the non-relativistic quantum mechanics such a state is described by the Dirac delta-function. The points of classical space are in one-to-one correspondence with the states of a particle localized at these points.

The resulting identification of points and point-supported states provides an unexpectedly fruitful interplay between the classical and the quantum worlds. It turns out that there exists an elegant mathematical framework that allows recovering of the usual Riemannian geometry on the classical space from the structure of a Hilbert space of states. Moreover, the specific functional realizations of the Hilbert space are capable of generating space-times of different geometry and topology. Finally, it turns out that the classical dynamics of macroscopic test-particles in a curved space-time and the Schroedinger evolution of microscopic particles can be both derived by variation of a single length-type action functional on the Hilbert space. More generally, the above approach leads one to an intriguing scenario of emergence of the space and time from a purely quantum reality.