Multidimensional Unified Theories

An extended spacetime, M^4+N, is a Riemannian (4 + N)-dimensional manifold which admits an N-parameter group G of (spacelike) isometries and is such that ordinary spacetime M⁴ is the space M^4+N/G of the equivalence classes under G-transformations of M^4+N. A multidimensional unified theory (MUT) is a dynamical theory of the metric tensor on M^4+N, the metric being determined from the Einstein-Hilbert action principle: in absence of matter, the Lagrangian is (essentially) the total curvature scalar of M^4+N. A MUT is an extension of the Cho-Freund generalization of Jordan's five-dimensional theory. A MUT can be faithfully translated in four-dimensional language: as a theory on M⁴, a MUT is a gauge field theory with gauge group G. A unifying aspect of MUT's is that all fields occur as elements of the metric tensor on M^4+N. When the isometry generators are subjected to strongest constraints, a MUT becomes the De Witt-Trautman generalization of Kaluza's five-dimensional theory; in four-dimensional language, this is the theory of Yang-Mills gauge fields coupled to gravity. With weaker constraints, a MUT appears to be more natural than a Yang-Mills theory as a physical realization of the gauge principle for an exact symmetry of gauged confined color. Such weakly-constrained MUT leads to bag-type models without the need for ad hoc surgery on the basic. Lagrangian. The present paper provides a detailed introduction to the formalism of multidimensional unified gauge field theory.