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By C. Piron
'Foundations of Quantum Physics' offers a unified formula of 'quantum' physics, fending off the widely permitted department among classical and quantum mechanics. The formalism takes heavily Einstein's standpoint and describes a actual method when it comes to 'elements of reality.' all of the easy notions are explicitly outlined, and just about all theorems, propositions and lemmas are proved intimately.
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This condition is difficult, if not impossible, to realize perfectly. For this reason the precise definition of a particular physical system always depends somewhat on the point of view and on the degree of idealization considered. Thus a free proton is a simple example of a physical system. But if one wants to bring into evidence a difference of mass due to the gravitational field, or some other, more subtle difference due 19 20 Foundations of Quantum Physics to the surroundings of the laboratory, it will doubtless be necessary to make explicit whether one is concerned with a proton from the CERN, BNL, or any other accelerator.
If(a, a ' ) induces the identity on Gp (V, K), this means that for every f E V there exists tjJ(j) E K such that af= tjJ(f)f; we may therefore write: a(f + g) = tjJ(f + g) (f + g) = af + ag = tjJ(f)f + tjJ(g)g. If f and g are linearly independent, we deduce from this that tjJ (f) = tjJ(f + g) = tjJ(g). If f and g are not linearly idependent, but are both different from the null element, and since V is of dimension at least 2, there exists h E V such that the pairs f, hand h, g are each linearly independent, and by the same argument tjJ(f) = tjJ(h) = tjJ (g).
This is the reason why, for these elements, we shall modify f so as to map bijectively the countable set of numbers of the form m2- n onto the set of exceptional sequences. The mapping j thus modified is certainly a bijection of C onto I, and it is even a homeomorphism if one excludes from it numbers ofthe form m2- n • Therefore, according to the preceding argument, j defines an iso- 1-3: 17 A New Language morphism of B(COl) onto B(I). Similarly one would show that B(COlN) and B(IN) are isomorphic.