Angel "Java" Lopez en Blog

Publicado el 15 de Mayo, 2014, 11:07

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Sigo compartiendo el prefacio de "Mathematical Foundations of Quantum Mechanics", de John Von Neumann. Viene una mención y crítica al trabajo de Dirac:

Dirac, in several papers, as well as in his recently published book, has given a representation of
quantum mechanics which is scarcely to be surpassed in brevity and elegance, and which is at the same time of invariant character. It is therefore perhaps fitting to advance a few arguments on behalf of our method, which deviates considerably from that of Dirac.

Von Neumann destaca el carácter invariante del trabajo de Dirac, es decir, independiente de las coordenadas.

The method of Dirac, mentioned above, (and this is overlooked today in a great part of quantum mechanical literature, because of the clarity and elegance of the theory) way satisfies the requirements of  mathematical rigor — not even if these are reduced in a natural and proper fashion to the extent common elsewhere in  theoretical physics. For example, the method adheres to the fiction that each self-adjoint operator can be put in diagonal form. In the case of those operators for which this is not actually the case, this requires the  introduction of "improper" functions with self-contradictory properties. The insertion of such a mathematical "fiction" is frequently necessary in Dirac's approach, even though
the problem at hand is merely one of calculating  numerically the result of a clearly defined experiment.

Supongo que se refiere a las delta de Dirac. Años después, éstas serían adoptadas en rigor gracias al trabajo de Schwartz. Ver

There would be no objection here if these concepts, which cannot be incorporated into the present day framework of analysis, were intrinsically necessary for the physical theory. Thus, as Newtonian mechanics first brought about the development of the infinitesimal calculus, which, in its original form, was undoubtedly not self-consistent, so quantum mechanics might suggest a new structure for our "analysis of infinitely many variables" — i.e., the  mathematical technique would have to be changed, and not the
physical theory.

Interesante que ponga a la física newtoniana y el cálculo (no riguroso al principio), como ejemplo. Pero los diferencia de lo que se necesita para la cuántica:

But this is by no means the case. It should rather be pointed out that : the quantum mechanical "Transformation theory" can be established in a manner which is just as clear and unified, but which is also  without mathematical objections. It should be emphasized that the correct structure need not consist in a  mathematical refinement and explanation of the Dirac method, but rather that it requires a procedure differing from the very beginning, namely, the reliance on the Hilbert theory of operators.

Veré qué puedo aprender de este "approach" distinto. Seguimos leyendo en el próximo post.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia