The Quantum Structure of Space and Time (293 pages)

(^96) The Quantum Structure of Space and Time
theory.
Correspondence principle. Whatever quantum geometry is, it should reduce to
the classical space-times of general relativity in the limit lp t 0.
Space-time non-commutativity. The space-time coordinates xp are no longer
real numbers, but most likely should become eigenvalues of quantum operators.
These operators should no longer commute, but instead obey relations of the
form
[XP, xc”] - l;.
In particular, space-like and time-like coordinates should no longer commute.

There is a well-known simple physical argument for this: precise short-distance

spatial measurements Ax require such high energy waves compressed in such a

small volume, that a microscopic black hole can be formed. Due to Hawking

evaporation, such a black hole is only meta-stable, and it will have a typical

decay time At N e”,lax.

Quan,tum foam. In some sense one should be able to interpret quantum geom-

etry as a path-integral over fluctuating space-time histories. Short-distance

space-time geometries should be therefore be subject to quantum corrections

that have arbitrary complicated topologies. This induces some quantized, dis-

crete structures. The sizes of the topologically Ron-trivial cycles (handles, loops,

“holes”, ...) should be quantized in units of ep. Together with the idea of non-

commutativity of the space-time coordinates, this reminds one of semi-classical

Bohr-Sommerfeld quantization.

Holography. As is discussed in much greater details in the rapporteur talk of

Seiberg [13], the ideas of holography in black hole physics [14] suggest that

space-time geometry should be an emergent concept. It should arise in the

limit N + 00, where N is some measure of the total degrees of freedom of the

quantum system. In this context the analogy with the emergence of the laws of

thermodynamics out of the properties of a statistical mechanical system has of-

ten been mentioned. In fact, both thermodynamics and general relativity were

discover first as macroscopic theories, before the corresponding microscopic for-

mulations were found. They are also in a precise sense universal theories: in the

suitable macroscopic limit any system is subject to the laws of thermodynamics

and any gravity theory will produce Einsteinian gravity.

Probe dependence. Experience of string theory has taught us that the measured

geometry will depend on the object that one uses to probe the system. Roughly,

the metric gpv(x) will appear as an effective coupling constant in the world-

volume theory of a particle, string or brane that is used as probe. With C the

worldvolume of the probe, one has