9780192806727.pdf

where Rw is the curvature scalar in /? 4. Thus /?(5) is the unified Lagrangian

for gravitation and electromagnetism! Equation 17.8 makes clear why the fac-

tor VZK was introduced in Eq. 17.6a and why it is important that 755 be taken

positive (and normalized to +1).

5. In 1926 Klein already believed that the fifth dimension might have something
   to do with quantization [K4], an idea that stayed with him for many years
   *In the same year, the five-dimensional unification was discovered independently by Mandel [Ml];
   see also [M2] and [Fl].

The relations 17.2 and 17.3 are invariant under G 5.

2. Define g^ by

(17.5)

(17.6)

UNIFIED FIELD THEORY 331

Kaluza proved his results only for the case where the fields are weak (i.e.,
g,u, = Vw + h^, | /z,J «1, Tj 55 = 1) and the velocity is small (y/c«l). An impor-
tant advance was made by Oskar Klein, who showed in 1926* that these two
constraints are irrelevant [K3]. Unification (at least this version) has nothing to
do with weak fields and low velocities. The resulting formulation has since become
known as the Kaluza-Klein theory. Its gist can be stated as follows.

1. Start with the quadratic form Eq. 17.1 and demand that it be invariant under
   a group G 5 of transformations that is the product of the familiar group of point
   transformation G 4 in R 4 and the group S,, defined by

which shows that S\ is a geometrized version of the local electromagnetic gauge

group.

4. Let /?<5) be the curvature scalar in five-space. A straightforward calculation
   shows that

They are a four-vector under G 4 and (since Eq. 17.1 is invariant under G 5 )

thev transform under S, as

(17.7)

(17.6a)

The g,k are symmetric; they are a tensor under G 4 and are invariant under 5,.

Thus we can define g^ dx'dxk as the standard line element in R 4.

3. Define the electromagnetic potentials $, by

(17.8)

(17.9)