Kiefer C. Quantum gravity

302 STRING THEORY
the metric is flat, the dynamical equation for a massless scalar field Φ in five
dimensions is given by the wave equation
✷ 5 Φ(x µ ,y)=0, (9.73)
where ✷ 5 is the five-dimensional d’Alembert operator. Making a Fourier expansion
with respect to the fifth dimension,
Φ(x µ ,y)= ∑ n
ϕ n (x µ )e iny/R ,n∈ Z , (9.74)
one obtains for the ϕ n (x µ ) an effective equation of the form
)
(✷ 4 − n2
R 2 ϕ n (x µ )=0, (9.75)
where ✷ 4 is the four-dimensional d’Alembert operator. Equation (9.75) is nothing
but the four-dimensional Klein–Gordon equation for a massive scalar field ϕ n (x µ )
with mass
m n = |n|
R . (9.76)
From the four-dimensional point of view one thus has a whole ‘Kaluza–Klein
tower’ of particles with increasing masses. For low energies E 1/R, the massive
Kaluza–Klein modes remain unexcited and only the massless mode for n =0
remains. The higher dimensions only show up for energies beyond 1/R. Sinceno
evidence has been seen yet at accelerators for the massive modes, the size of the
fifth dimension must be very small, definitely smaller than about 10 −17 cm.
The Kaluza–Klein scenario has been generalized in various directions. In Section
9.2.3, we have seen that the notion of T-duality in string theory gives rise
to the concept of D-branes. Gauge and matter fields are localized on the brane,
whereas gravity can propagate freely through the higher dimensions (the bulk).
One can, therefore, assume that our observed four-dimensional world is, in fact,
such a brane being embedded in higher dimensions. This gives rise to various
‘brane-world scenarios’ which often are very loosely related to string theory itself,
taking from there only the idea of a brane without giving necessarily a dynamical
justification from string theory. A general review is Rubakov (2001).
In one scenario, the so-called ‘ADD’ approach, the brane tension (energy per
unit three-volume of the brane) and therefore its gravitational field is neglected;
see Arkani-Hamed et al. (1998) and Antoniadis et al. (1998). A key ingredient
in this approach is to take the extra dimensions compact (as in the standard
Kaluza–Klein approach) but not microscopically small. The reason for this possibility
is the fact that only gravity can probe the extra dimensions, and the
gravitational attraction has only been tested down to distances of about 0.2
mm. Any value for R with R 0.1 mm would thus be allowed. This could
give a clue for understanding the ‘hierarchy problem’ in particles physics—the
problem why the electroweak scale (at about 1 TeV) is so much smaller than