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

Alexey Kryukov (Department of Mathematics, University of Wisconsin)

**Abstract**

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.