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 M 4 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 M 4 , 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.