Pero Dirac tenía más que aportar a la nueva mecánica cuántica que los corchetes de Poisson. Estos ponían de manifiesto la no conmutatividad, algo que NO es clásico:
With the development of quantum mechanics one had a new situation in theoretical physics. The basic equations, Heisenberg's equations of motion, the commutation relations and Schrodinger's wave equation were discovered without their physical interpretation being known. With noncommutation of the dynamical variables, the direct interpretation that one was used to in classical mechanics was not possible, and it became a problem to find the precise meaning and mode of application of the new equations.
Había que aprender a manejar correctamente el nuevo aparato matemático:
This problem was not solved by a direct attack. People first studied examples, such as the nonrelativistic hydrogen atom and Compton scattering, and found special methods that worked for these examples. One gradually generalized, and after a few years the complete understanding of the theory was evolved as we know it today, with Heisenberg's principle of uncertainty and the general statistical interpretation of the wave function.
Eso se hizo, pero en general, dentro del régimen no relativístico. Había habido algún progreso, pero no satisfacía a Dirac:
The early rapid progress of quantum mechanics was made in a nonrelativistic setting, but of course people were not happy with this situation. A relativistic theory for a single electron was set up, namely Schrodinger's original equation, which was rediscovered by Klein and Gordon and is known by their name, but its interpretation was not consistent with the general statistical interpretation of quantum mechanics.
Para Dirac, la ecuación de Klein y Gordon no era adecuada, porque al aplicarla daba probabilidades tantos positivas como negativas. Pasaron algunos años hasta que Pauli la volvió a aplicar, esta vez sobre probabilidades de carga eléctrica.
Había un aparato matemático, los tensores, que se habían aplicado hasta entonces en toda teoría relativista, pero que se "quedaban cortos" en cuanto se los aplicaba en cuántica, como en la ecuación de Klein-Gordon.
As relativity was then understood, all relativistic theories had to be expressible in tensor form. On
this basis one could not do better than the Klein-Gordon theory. Most physicists were content with the Klein-Gordon theory as the best possible relativistic quantum theory for an electron, but I was always dissatisfied with the discrepancy between it and general principles, and continually worried over it till I found the solution.
Parece que Pauli fue el primero en usar espinores, luego adoptados entusiastamente por Dirac:
Tensors are inadequate and one has to get away from them, introducing two-valued quantities, now called spinors. Those people who were too familiar with tensors were not fitted to get away from them and think up something more general, and I was able to do so only because I was more attached to the general principles of quantum mechanics than to tensors. Eddington was very surprised when he saw the possibility of departing from tensors. One should always guard against getting too attached to one particular line of thought.
Y hasta trajeron un regalo inesperado: la explicación del spin del electrón:
The introduction of spinors provided a relativistic theory in agreement with the general principles of quantum mechanics, and also accounted for the spin of the electron, although this was not the original intention of the work. But then a new problem appeared, that of negative energies. The theory gives symmetry between positive and negative energies, while only positive energies occur in nature.
En el próximo post, veremos cómo aún esta dificultad fue fructífera, de una manera que aún Dirac no supo prever.
Angel "Java" Lopez