FIZIKA B 10 (2001)  3, 139-160

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Institute Ruđer Bošković, 10000 Zagreb, Croatia

Dedicated to Professor Kseno Ilakovac on the occasion of his 70th 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 semi-simple 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 semi-simple 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 semi-simplicity 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 space-time 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
Copyright by The Croatian Physical Society
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