 
FIZIKA B 10 (2001) 3, 139160
INHERENTLY RELATIVISTIC QUANTUM THEORY
Part II. CLASSIFICATION OF SOLUTIONS
EMILE GRGIN
Institute Ruđer Bošković, 10000 Zagreb, Croatia
Email: eg137@ix.netcom.com
Dedicated to Professor Kseno Ilakovac on the occasion of his 70^{th}
birthday
Received 22 May 2001; Accepted 22 October 2001
Online 30 December 2001
The abstract quantal algebra developed in Part I of the present work describes the
common structure of the two known mechanics, classical and quantum. By itself, however, it
is not physics. It is a mathematical object, or, as some might say, it is only
mathematics, a valid objection if quantal algebra were meant to be an end in itself, for
physics is not in abstract theories, but in their concrete realizations. Hence, the
immediate question is whether at least one new concrete realization of the quantal algebra
exists, for it is among these that a physically valid generalization of quantum mechanics
might be found. The search for all realizations of an abstract theory is known in
mathematics as structure theory, or the classification problem. Usually
difficult, it is relatively easy in our case because the foundations have already been
laid in Cartan's classification of the semisimple Lie algebras. Since the quantal algebra
contains a Lie algebra, we only need to adapt the standard work to our case by imposing
some additional conditions. The result is that the semisimple quantal algebra has exactly
two realizations. Expressed in terms of groups, one is the infinite family of unitary
groups, SU( n) , (i.e., standard quantum mechanics), the other is an exceptional solution,
the group SO( 2,4). Classical mechanics does not appear as a solution because the
requirement of semisimplicity eliminates the canonical group. Thus, if quantum mechanics
can be generalized, the generalization is somehow related to the group SO( 2,4) , and as
this group contains the relativistic spacetime structure, it appears that an inherently
covariant generalization might be possible.
PACS numbers: 87.15.Rn, 87.50.a
UDC 535.217, 539.21
Keywords: quantal, quaternion, quantum, unification
