Lorentz Invariance and Internal Symmetry

One of the outstanding problems of elementary-particle physics is the problem of combining Lorentz invariance and internal symmetry in a nontrivial way. In this paper we attack this problem by making a general investigation of the possibilities of finding a Lie algebra of finite order $E$ such that the Lie algebras of the Lorentz and internal symmetry groups appear in it as subalgebras. We carry out this investigation by using some powerful standard results from the general theory of Lie algebras. The most relevant of these is Levi's result that any Lie algebra $E$ is the semidirect product of a semisimple algebra $G$ and an invariant solvable subalgebra $S$ (called the radical). Using this result we show that if $L$, the algebra of the inhomogeneous Lorentz group, is a subalgebra of $E$, and $M$ and $P$ are the homogeneous and translation parts of $L$ respectively, then either (a) $M$ lies completely in $G$ and $P$ lies completely in $S$, or (b) $L$ has no intersection with $S$. The relevance of this result is that it enables us to classify the ways in which $L$ can be a subalgebra of $E$ in a very simple way. The classification is carried out by subdividing case (a) into the three cases: (i) $S=P$, (ii) $S$ Abelian but larger than, and containing $P$, and (iii) $S$ solvable but not $A$ belian, and containing $P$; and by regarding case (b) as case (iv) $S\bigcap P=0\mathrm{}$. Each of these four cases is considered in detail. It turns out that case (i) is essentially a direct sum of $L$ and a semisimple Lie algebra; case (ii) is possible, but has the disadvantage of introducing a translation group of more than four dimensions; and case (iii) seems to be rather unphysical. Case (iv) is possible but is equivalent to imbedding $L$ in a simple Lie algebra. The over-all picture which emerges is that while there are a number of ways in which $L$ can be imbedded in an $E$, none of these (except the direct sum) seems to be particularly attractive from the physical point of view. In particular, it seems that, while it may be possible to make $\mathrm{SU}\left(6\right)\mathrm{}$ theory fully relativistic, it is probably not possible to do so within the context of a Lie algebra of finite order. [This does not contradict the $\tilde{U}\mathrm{}\left(12\right)\mathrm{}$ theory.] The question of explaining mass splitting within the context of a Lie algebra of finite order is considered, and it is shown that this cannot be done. The various negative-type theorems obtained by previous authors for special cases of $E$ are rederived here within the general framework, most of them being derived from much weaker assumptions.