A possibility of unifying gravity with electromagnetism in the Kaluza-Klein sense is an "affine theory" in the principal bundle $P(M,G)$. This theory is based on a class of $G$-invariant linear connections in $P(M,G)$ that induce the gauge potentials. The class of totally vertical connections is defined and studied. First, the case of $\operatorname{GL}(n,\bold R)$-invariant connections is considered. If $\nabla$ is a totally vertical connection and satisfies a regularity condition, then it generates uniquely a principal connection in $P$, whose gauge potentials satisfy a given system. Next, when $G$ is a reductive subgroup of $\operatorname{GL}(n,\bold R)$ (this covers all cases of physical interest in gauge theories), the principal connection $\omega$ in $P(M,G)$ and a dimensionally reducible linear connection $\nabla$ in $P$ are defined. It is proved that $\nabla$ generates a unique principal connection $\omega$ in $P(M,G)$ and that the local gauge potentials of $\omega$ can be calculated by projecting the solutions of the system onto the corresponding Lie subalgebra $\germ g$ of $\operatorname{gl}(n,\bold R)$.
Principal connections from G-invariant linear connections
FERRARIS, Marco;FRANCAVIGLIA, Mauro
1990-01-01
Abstract
A possibility of unifying gravity with electromagnetism in the Kaluza-Klein sense is an "affine theory" in the principal bundle $P(M,G)$. This theory is based on a class of $G$-invariant linear connections in $P(M,G)$ that induce the gauge potentials. The class of totally vertical connections is defined and studied. First, the case of $\operatorname{GL}(n,\bold R)$-invariant connections is considered. If $\nabla$ is a totally vertical connection and satisfies a regularity condition, then it generates uniquely a principal connection in $P$, whose gauge potentials satisfy a given system. Next, when $G$ is a reductive subgroup of $\operatorname{GL}(n,\bold R)$ (this covers all cases of physical interest in gauge theories), the principal connection $\omega$ in $P(M,G)$ and a dimensionally reducible linear connection $\nabla$ in $P$ are defined. It is proved that $\nabla$ generates a unique principal connection $\omega$ in $P(M,G)$ and that the local gauge potentials of $\omega$ can be calculated by projecting the solutions of the system onto the corresponding Lie subalgebra $\germ g$ of $\operatorname{gl}(n,\bold R)$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.