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Summary: REPRESENTATION OF STATE PROPERTY SYSTEMS
D. AERTS AND S. PULMANNOV´A
Abstract. A `state property system' is the mathematical structure which
models an arbitrary physical system by means of its set of states, its set of
properties, and a relation of `actuality of a certain property for a certain state'.
We work out a new axiomatization for standard quantum mechanics, starting
with the basic notion of state property system, and making use of a general-
ization of the standard quantum mechanical notion of `superposition' for state
property systems.
1. Introduction
In standard quantum mechanics a state p¯c of a quantum entity S is represented by
the one dimensional subspace or the ray ¯c of a separable complex Hilbert space H.
An experiment eA testing an observable A is represented by a self adjoint operator
A on H, and the set of outcomes of this experiment eA is the spectrum spec(A) of
this self-adjoint operator A. Measurable subsets B spec(A) represent the events
(in the sense of probability theory) of outcomes. The interaction of the experiment
eA with the physical entity being in state p¯c is described in the following way: (1)
the probability for a specific event B spec(A) to occur if the entity is in a specific
state p¯c is given by c, PB(c) , where PB is the spectral projection corresponding to
B, c is the unit vector in the ray ¯c representing state p¯c, and , is the inproduct
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