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FIZIKA B 10 (2001) 3, 113-138
 
INHERENTLY RELATIVISTIC QUANTUM THEORY
Part I. THE ALGEBRA OF OBSERVABLES
EMILE GRGIN
Institute Rudjer Boskovi�, 10000 Zagreb, Croatia
E-mail: [email protected]
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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