FIZIKA B 10 (2001)  3, 113-138

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Institute Rudjer BoskoviŠ, 10000 Zagreb, Croatia

Dedicated to Professor Kseno Ilakovac on the occasion of his 70th birthday

Received 10 May 2001; Accepted 22 October 2001
Online 30 December 2001

The present article is the first in a program that aims at generalizing quantum mechanics by keeping its structure essentially intact, but constructing the Hilbert space over a new number system much richer than the field of complex numbers. We call this number system ``the Quantionic Algebra''. It is eight dimensional like the algebra of octonions, but, unlike the latter, it is associative. It is not a division algebra, but "almost" one (in a sense that will be evident when we come to it). It enjoys the minimum of properties needed to construct a Hilbert space that admits quantum-mechanical interpretations (like transition probabilities), and, moreover, it contains the local Minkowski structure of space-time. Hence, a quantum theory built over the quantions is inherently relativistic. The algebra of quantions has been discovered in two steps. The first is a careful analysis of the abstract structure of quantum mechanics (the first part of the present work), the second is the classification of all concrete realizations of this abstract structure (several additional articles). The classification shows that there are only two realizations. One is standard quantum mechanics, the other its inherently relativistic generalization. The present article develops the abstract algebra of observables.

PACS numbers: 87.15.Rn, 87.50.-a
UDC 535.217, 539.21
Keywords: quaternion, octonion, division algebra, quantion, quantum, quantal, quantization, unification.
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