*Class. Quantum
Grav. 1(1984) 517-530. Printed in Great Britain*

**Colour and Higgs
charges in G/H Kaluza-Klein theory**

** **

A Z Jadczyk*

International Centre for Theoretical Physics, Trieste, Italy

** On leave of
absence from Institute of Theoretical Physics, University of Wroclaw,
Cybulskiego 36, 50-205 Wroclaw, Poland.*

Received 24 April 1984

**Abstract. **Geodesics
in a multidimensional Universe with G-invariant metric are studied
and differential equations describing their space-time projection are
derived. It is shown that, when internal spaces are cosets rather
than group manifolds, then, in addition to the colour charge through
which the particle interacts with the gauge field, a new charge
arises which couples to the scalar fields only. This new charge,
called the Higgs charge, is shown to be of a nonlinear nature. For
certain cosets it may contribute to the colour charge non-conservation.

**1. Introduction**

** **

Recently a
geometrical theory of dimensional reduction was developed based on
the concept of G-invariance of a Riemannian metric in a
multidimensional Universe [1]. In the present paper geodesics in a
multidimensional Universe are studied, and it is shown that the
components of the velocity pointing into the extra dimensions give
rise to the coloured and higgsonic charges. From the formal point of
view our equations (5.1)-(5.3) generalise the well known Kerner-Wong
equation (see [2-5], also [6] where motion of a charged string is
treated) in two ways: firstly, the internal space is taken to be a
homogeneous space G/H* *rather than a group manifold, and
secondly, interaction with the Jordan-Thierry scalars, originating
from the metric on G/H*, *is taken into account. Before going
into the details of this paper's results, it is worthwhile analysing
the relation between the geometrical framework used in [1], and the,
so-called, Kaluza-Klein theory. The Kaluza-Klein mechanism is
supposed to work as follows (see e.g. [7, 8], and references
therein): one starts with a generally covariant field theory in (4 + n)* *dimensions
(e.g. eleven-dimensional supergravity), containing among its fields
{phi} the metric field *g _{AB} (A, B = 1, 2,..., 4 + n)*.
Suppose the theory admits a ground state

With the above in
mind, let us make some more comments on the results of this paper. It
is important to observe that the Lie-algebra decomposition
corresponding to a homogeneous space *G/H *is not *Lie(G) = Lie(H)+S*,
where *S* can be interpreted as the space tangent to *G/H *at
the origin, but a more subtle one: Lie(G) = *Lie(H)+Lie(K)+L*,
where *Lie(K)* is the Lie algebra of K = *N(H)|H *(the
effective gauge group), *N(H) *being the normaliser of *H*
in *G*. Consequently the metric *g _{alpha beta}* on

**2.
Geometry of a G-Universe**

**3. Geodesics in G-Universe**

**4. Equations of
motion for a particle carrying coloured and higgsonic charges**

**5. Summary and example**

**Download
Paper in PDF format (580 kB)**

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