Angel "Java" Lopez en Blog

Noviembre del 2016


Publicado el 30 de Noviembre, 2016, 15:01

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Este es un tema que amerita mayor consideración y detalle técnico. Por ahora, sigo citando y comentando brevemente a Dirac:

Let us see what can be done with putting the present quantum electrodynamics on a logical footing. We must keep to the standard practice of neglecting only quantities which one can believe to be small, even though the grounds for this belief may be rather shaky.

In order to handle infinities, we must refer to a process of cut-off. We must do this in mathematics whenever we have a series or an integral which is not absolutely convergent. When we have introduced a cut-off, we may proceed to make it more and more remote and go to a limit, which then depends on the method of cut-off. Alternatively, we may keep the cut-off finite. In the latter case, we must find quantities that are insensitive to the cutoff.

The divergencies in quantum electrodynamics come from the high-energy terms in the energy of interaction between the particles and the field. The cut-off thus involves introducing an energy, g say, beyond which the interaction energy terms are omitted. It is found that we cannot make g tend to infinity without destroying the possibility of solving the equations logically. We have to keep a finite cutoff.

Dirac prefiere perder la invariancia relativista que seguir en un problema de base:

The relativistic invariance of the theory is then destroyed. This is a pity, but it is a lesser evil than a departure from logic would be. It results in a theory which cannot be valid for high-energy processes, processes involving energies comparable with g, but we may still hope that it will be a good  approximation for low-energy processes.

On physical grounds we should expect to have to take g to be of the order of a few hundred Mev, as this is the region where quantum electrodynamics ceases to be a self-contained subject and the other particles of physics begin to play a role. This value for g is satisfactory for the theory.

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Angel "Java" Lopez
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Por ajlopez, en: Ciencia

Publicado el 25 de Noviembre, 2016, 15:51

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Por ajlopez, en: Emprender

Publicado el 23 de Noviembre, 2016, 16:38

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Por ajlopez, en: Emprender

Publicado el 21 de Noviembre, 2016, 17:51

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A Dirac le preocupa que en la explicación con modelo matemático de la interacción entre electrón y campo electromagnético, hay divergencias, integrales cuyas sumas divergen. Las divergencias surgen de tener que lidiar con singularidades en el campo, o en interacciones cada vez más cercanas, alrededor de un punto. El tema de llegar a divergencias (o a sumas infinitas) al disminuir las distancias es una señal que nos hace la naturaleza: para mí, nos está diciendo "el modelo que adoptaron es incorrecto, en la realidad no hay singularidades". Veamos que piensa Dirac:

If one deals classically with point electrons interacting with the electromagnetic field, one finds difficulties connected with the singularities in the field. People have been aware of these difficulties from the time of Lorentz, who first worked out the equations of motion for an electron. In the early days of the quantum mechanics of Heisenberg and Schrodinger, people thought these difficulties would be swept away by the new mechanics. It now became clear that these hopes would not be fulfilled. The difficulties reappear in the divergencies of quantum electrodynamics, the quantum theory of the interaction of electrons and the electromagnetic field. They are modified somewhat by the infinities associated with the sea of negative-energy electrons, but they stand out as the dominant problem.

De nuevo es un tema técnico, pero vemos que para Dirac es importante. Los físicos no siempre se preocupan a tal grado por alguna imperfección en la teoría, pero Dirac ha sido consecuente con sus ideas, y durante décadas pregonó sobre este problema "básico".

The difficulty of the divergencies proved to be a very bad one. No progress was made for twenty years. Then a development came, initiated by Lamb's discovery and explanation of the Lamb shift, which fundamentally changed the character of theoretical physics. It involved setting up rules for discarding the infinities, rules which are precise, so as to leave well-defined residues that can be compared with experiment. But still one is using working rules and not regular mathematics. Most theoretical physicists nowadays appear to be satisfied with this situation, but I am not. I believe that theoretical physics has gone on the wrong track with such developments and one should not be complacent about it. There is some similarity between this situation and the one in 1927, when most physicists were satisfied with the Klein-Gordon equation and did not let themselves be bothered by the negative probabilities that it entailed.

Como había yo mencionado, a Dirac le preocupaba que la Klein-Gordon pudiera producir probabilidades negativas. Notablemente, el efecto Lamb fue explicado exitosamente por la QED aún usando los trucos de eliminar los infinitos, siendo uno de los valores mejor deducidos de la historia de la física.

We must realize that there is something radically wrong when we have to discard infinities from our equations, and we must hang on to the basic ideas of logic at all costs. Worrying over this point may lead to an important advance. Quantum electrodynamics is the domain of physics that we know most about, and presumably it will have to be put in order before we can hope to make any fundamental progress with the other field theories, although these will continue to develop on the experimental basis.

Y no sólo aparecen divergencias en electrodinámica cuántica.

Nos leemos!

Angel "Java" Lopez
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Por ajlopez, en: Ciencia

Publicado el 20 de Noviembre, 2016, 14:13

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¿Por qué encuentra energía negativa? Porque al pasar a relatividad, la expresión del operador energía se ve transformado por una raíz cuadrada, que en mecánica cuántica se traduce en que los dos valores son posible. Es algo técnico para discutirlo ahora en detalle, pero el resultado es que la solución de Dirac hace aparece no SOLO un spinor sino DOS spinores. Y el tema de las energías negativas da para la aparición de uno de los temas más interesantes del siglo pasado: el descubrimiento de las antipartículas.

As frequently happens with the mathematical procedure in research, the solving of one difficulty leads to another. You may think that no real progress is then made, but this is not so, because the second difficulty is more remote than the first. It may be that the second difficulty was really there all the time, and was only brought into prominence by the removal of the first.

This was the case with the negative energy difficulty. All relativistic theories give symmetry between positive and negative energies, but previously this difficulty had been overshadowed by more crude imperfections in the theory.

The difficulty is removed by the assumption that in the vacuum all the negative energy states are filled. One is then led to a theory of positrons together with electrons. Our knowledge is thereby advanced one stage, but again a new difficulty appears, this time connected with the interaction between an electron and the electromagnetic field.

Eso de tener todos los estados negativos llenos solo sirve cuando manipulamos fermiones, partículas que obedecen al principio de exclusión de Pauli.

When one writes down the equations that one believes should describe this interaction accurately and tries to solve them, one gets divergent integrals for quantities that ought to be finite. Again this difficulty was really present all the time, lying dormant in the theory, and only now becoming the dominant one.

Esta divergencia es la que realmente preocupa a Dirac, como veremos en los próximos posts.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
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Por ajlopez, en: Ciencia

Publicado el 19 de Noviembre, 2016, 14:30

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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.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez

Por ajlopez, en: Ciencia

Publicado el 18 de Noviembre, 2016, 5:46

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Angel "Java" Lopez
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Por ajlopez, en: China

Publicado el 4 de Noviembre, 2016, 18:36

Se acerca fin de año, yo sigo con resoluciones personales por mes. Primero, repaso de las octubre:

- Continuar mi serie sobre el último teorema de Fermat [pendiente]
- Continuar mi serie sobre matemáticas y física cuántica [pendiente]
- Continuar mi serie sobre la teoría de la transformación de Dirac [pendiente]
- Continuar mi serie sobre teoría de Galois [completo] ver post
- Comenzar una serie sobre los azulejos de Wang [pendiente]
- Estudiar blues en guitarra [completo]

Tuve bastante trabajo profesional. Igual me las arreglé para estudiar y escribir de otros temas:

La Influencia de Lagrange en Galois, por Edwards

Notas sobre Segunda Cuantización (1)
Notas sobre Segunda Cuantización (2)
Notas sobre Segunda Cuantización (3)

El camino a la unificación en física (1)
El camino a la unificación en física (2)
El camino a la unificación en física (3)
El camino a la unificación en física (4)

Métodos en Física Teórica, por Dirac (5)
Métodos en Física Teórica, por Dirac (6)

Sigo con las resoluciones para noviembre

- Continuar mi serie sobre el último teorema de Fermat
- Continuar mi serie sobre matemáticas y física cuántica
- Continuar mi serie sobre la teoría de la transformación de Dirac
- Continuar mi serie sobre teoría de Galois
- Comenzar una serie sobre los azulejos de Wang
- Estudiar blues en guitarra

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez