The Quantum Structure of Space and Time (293 pages)

98 The Quantum Structure of Space and Time

a’ large

a‘ M 0

described in terms of loop spaces and sums over surfaces.

In theories of four-dimensional gravity, the Planck scale is determined as

conformal field theory M-theory

strings string fields, branes

quantum mechanics quantum field theory

particles fields

ep = gses.

This is the scale at which we expect to find the effects of quantum geometry, such

as non-commutativity and space-time foam. So, in a perturbative regime, where g,

is by definition small, the Planck scale will be much smaller than the string scale

ep << e, and we will typical have 3 regimes of geometry, depending on which length

scale we will probe the space-time: a classical regime at large scales, a “stringy”

regime where we study the loop space for scales around e,, finally and a truly

quantum regime for scales around ep.

qz-larztum “stringy ” classical

geometry geometry geornet y

4 es smooth

For large values of g, this picture changes drastically. In the case of particles
we know that for large Ti it is better to think in terms of waves, or more precisely
quantum fields. So one could expect that for large gs and e, the right framework is
string field theory [MI. This is partly true, but it is in general difficult to analyze
(closed) string field theory in all its generality.
Summarizing we can distinguish two kinds of deformations: strzngy effects pa-
rameterized by l, or a’, and quantum effects parameterized by gs. This situation
can be described with the following diagram