DIRAC:

When I first got that equation, of course I was very anxious to know whether it would work for the hydrogen atom, and I just tried it by an approximation method. I thought that if I got it anywhere near right with an approximation method, I would be very happy about that. It needed someone else, namely Darwin (and Gordon), to tackle that equation as an exact equation and see what the exact solutions were; I think I would have been too scared myself to consider it exactly. I would be too scared that il would get unfortunate results which would compel the whole theory to be abandoned.T. s. KUHN:

That"s fascinating; does this mean that you had yourself not tried to handle exactly before going to an approximation method?DIRAC:

That is so, yes.KUHN:

You looked for the approximation method from the start?DIRAC:

Yes, yes. Of course 1 had the fear that the whole theory was nowhere near right, and if 1 could get it approximately right, well my confidence would already be substantially increased in that way. It"s just that one has a lack of confidence when one introduces something quite new.

Dirac, P.A.M. 1963a. Interview with T. S. Kuhn 1963. Archives for the History of Quantum

Physics, Niels Bohr Library, AIP, New York.

Aún el gran Dirac necesitaba ganar confianza antes de encarar todas las consecuencias de su famosa ecuación.

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The M.V. was an excellent school for science and modem languages. There was no Latin or Greek, something of which I was rather glad, because I did not appreciate the value of old cultures. I consider myself very lucky in having been able to attend the School. I was at the M.V. during the period 1914-18, just the period of the First World War. Many of the boys then left the School for National Service. As a result, the upper classes were rather empty; and to fill the gaps the younger boys were pressed ahead, as far as they were able to follow the more advanced work. This was very beneficial to me: I was rushed through the lower forms, and was introduced at an especially early age to the basis of mathematics, physics and chemistry in the higher forms. In mathematics I was studying from books which mostly were ahead of the class. This rapid advancement was a great help to me in my later career.

The rapid pushing-ahead was a disadvantage from the point of view of Games—which we had on Wednesday afternoons. I played soccer anti cricket, mostly with boys older and bigger than myself, and never had much success. But all through my schooldays, my interest in science was encouraged and stimulated.

It was a great advantage, that the School was situated in the same building as the Merchant Venturers’ Technical College. The College “took over” in the evenings, after the School was finished. The College had excellent laboratories, which were available to the School during the daytime. Furthermore, some of the staff combined teaching in the School in the daytime with teaching in the College in the evenings. (Dirac 1980, p. 9)

Se refiere Dirac, P.A.M. 1980. A little ‘prehistory.’ The Old Cothamian 1980, p. 9. Lo encuentro en el excelente "QED and The Men Who Made It", de Silvan Schweber.

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Dirac, Heisenberg, Paili y la religión

Hoy leo una versión más completa de la opinión de Dirac, en ese congreso Solvay de 1927:

It we are honest -and scientist have lo be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.... I can't for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much miscry and injustice, the exploitation of the poor by the rich and all other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but siinply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also easier to exploit... Hence the close alliance between those two great public forces, the State and the Church.

Lo encuentro en el excelente "QED and The Men Who Made It" de Silvan Schweber. Asombra un poco la postura "dura" de un Dirac circumspecto, casi tímido en otros aspectos.

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Dirac siempre comentó que su relación con su padre Charles nunca fue fácil. A partir de 1925, año del suicidio del hermano de Dirac, Reginald, padre e hijo se distanciaron aún más. Leo a Schwber, QED and the Men Who Made It:

Reginald, Paul"s older brother, had wanted to become a physician, but Charles forced him to study mechanical engineering at Bristol. He obtained only a third-class degree upon graduating and accepted a position as a draftsman with an engineering firm in Wolverhampton. He committed suicide when he was twentyfour years old. The death of his oldest son deeply disturbed Charles, and for a while Paul feared that his father might lose his sanity—and he resolved that he would never take his own life no matter what the circumstances. Thereafter Paul"s relationship with his father became chilled and they had very little interaction with one another. One manifestation of Paul"s feeling toward his father was that throughout his life he avoided going to Switzerland, a country he associated with his father (Mehra and Rechenberg 1982). Paul invited only his mother to attend the ceremonies in Stockholm honoring him with the Nobel Prize in 1933. Dalitz and Peierls report that when Professor Tyndall, who had headed the physics department at Bristol University for three decades, gave a set of public evening lectures on modem physics in the early 1930s, he noticed a regular listener in the front row, a man much older than the others there, who was taking careful note of all that he said. At the end of the last lecture of the series, this old man came up to him to thank him, saying: "I am glad to have heard all this. My son does physics but he never tells me anything about it." The old man was Charles Dirac. Charles Dirac died in 1935. Paul was in Russia at the time to watch an eclipse of the sun; when infomied of the seriousness of his father"s illness, Paul flew back to England, but it was too late. The first letter he wrote his wife after his father"s death was to say, "1 feel much freer now" (Margit Dirac 1987, p. 5).

No conocía la anécdota del padre tratando de conocer física. Debió ser duro para él encontrarse en la madurez, sin sus hijos. Dirac siempre tuvo una conducta lejana, sin grandes relaciones, con excepción de su esposa (recordemos que era la hermana de Wigner).

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Es notable todo los temas en los que se involucró Jordan, estando cerca de Born:

For his dissertation under Born, Jordan worked on a problem in the theory of light quanta (Jordan 1925) that dealt with the interaction of electrons and radiation. In it he tried to disprove Einstein"s hypothesis that in the process of absorption or emission of a photon of energy hv by an atom an amount of momentum hi"/c is transferred by or to the photon. Einstein, in a brief note to the Zeitschrift für Physik, took exception with Jordan"s work. He pointed out that Jordan"s paper was based on a hypothesis that implied "that the amounts ol radiation taken (by an atom exposed to blackbody radiation) from rays of different directions were treated as not being independent of each other" and that this would result in consequences contrary to observation (Einstein 1925). After finishing his thesis in the fall of 1924, Jordan worked with James Franck on problems connected with spectroscopy. He helped him write volume 3 of the series Struktur der Materie which Bom and Franck edited. The book was published with Franck and Jordan as coauthors, with the title Anregung von Quantensprüngen der Stosse. During that same year Jordan wrote several papers dealing with problems in atomic structure and spectroscopy. He also collaborated with Born on a paper in the quantum theory of aperiodic processes. By generalizing Kramers and Heisenberg"s dispersion-theoretic approach, they calculated the effect on an atom of an electric field whose time dependence is arbitrary. The arbitrary time dependence was to allow them to simulate the effect of a charged projectile particle during an atomic collision.

Es interesante notar que se había familiarizado con las ideas y formalismo matemático de Kramers y Heisenberg sobre la dispersión de radiación por un átomo. Parte de ese formulismo será usado por Heisenberg en su gran "paper" de 1925.

Con este post termino la serie. Luego, la historia de Jordan ya cae en un periodo más allá de sus primeros años, donde comienza a aportar nuevas ideas a la mecánica cuántica y la teoría de campos.

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Veamos como Jordan va apareciendo en escena, a partir de 1922:

The Bohr Festspiel of June 1922 gave Jordan a taste of the drama of physics. Heisenberg, Fermi, Pauli, and Hund were also in Göttingen at the time. Fermi was evidently left out of the Göttingen intellectual community (Segre 1970, p.32) but Jordan got to know the other three well, particularly Heisenberg and Pauli, and came to appreciate the company of these brilliant young men. But he was overshadowed by their brash and confident ways. Jordan was rather short, and his presentation of self reflected his physical stature. He gave the impression of being insecure, an impression that was reinforced by his stuttering (he in fact suffered a breakdown in the early 1930s).

Although Jordan did not enjoy his courses in experimental physics—he actually stopped attending them—he found the laboratory course in zoology in which he had enrolled very satisfying; he also faithfully attended Alfred Kuhn"s lectures on heredity. In fact, he chose zoology as one of the minor subjects for his doctorate. Most of his energies, however, were spent on theoretical physics and mathematics. He helped Courant in the preparation of the book he was then writing with Hilbert, the famous Mathematische Methoden der Physik (Jordan 1963, p. 12). He also assisted Born with an article on crystal dynamics and became quite close to him.

El "festspiel" de Bohr fue el primer encuentro de Heisenberg y Bohr, y al parecer de Pauli con Bohr también. No sabía que había acudido Fermi también. Vean que Jordan era tartamudo, lo que influyó en su relación con los demás. No sabía que había ayudado en el famoso libro de Hilbert y Courant.

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Veamos como llega Jordan a conocer a Born:

Jordan entered the Technische Hochschule of the University of Hannover in 1921 intending to study physics. He had by that time learned some special and general relativity from Moritz Schlick"s Raum und Zeit in der gegenwärtigen Physik, had mastered electromagnetic theory, and had carefully studied Sommerfeld's Atombau und Spektrallinien. Jordan found that physics was not taught well at the Technische Hochschule and he transferred to Göttingen in 1922. However, he had made good use of his year at the Hochschule taking courses in mathematics, electrical engineering, and physical chemistry. In Göttingen he attended Courant"s course on mathematical methods for physicists and became the official note taker for the course. For a while he toyed with the idea of becoming a mathematician. But he came into Bom"s orbit, and under his influence and with his guidance became more and more committed to physics. When Born died, Jordan, in a brief eulogy for him, wrote: "He was not only my teacher who in my student days introduced me to the wide world of physics—his lectures were a wonderful combination of intellectual clarity and horizon widening overview. But he was also, I want to assert, the person who next to my parents, exerted the deepest, longest lasting influence on my life"....

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Ya saben que me interesa la historia de la ciencia, y en especial, he escrito bastante sobre el "mágico" año 1925 y alrededores, con el nacimiento de la mecánica cuántica. Uno de los personajes que aparecen una y otra vez en cualquier historia sobre el periodo, pero que tal vez no es tan conocido, es Pascual Jordan. Nunca consiguió el premio Nobel, aunque colegas como Born y Heisenberg lo consiguieron, por trabajos que de alguna manera también compartieron con Jordan. Como ayudante de Born, estuvo con él cuando llego el tiempo de escribir y expandir las ideas de Heisenberg de 1925. Luego involucrado con la actividad nazi en Alemania en los treintas, se dedicó a otros temas, además de la física. Encuentro hoy un texto, relato, de sus primeros años, en el excelente "QED and the Men Who Made It", de Silvan Schweber:

Jordan was bom in Hannover. Germany, in 1902. He was the younger of the two children in the family; a sister some ten years older than Pascual was the older sibling. Both his parents were well read in the natural sciences. His father was a painter and he got the young Pascual interested in the geometrical concepts involved in the "perspective" of drawing at an early age. In his interview with T. S. Kuhn, Jordan recalled that as a young boy his father read him books from the Kosmos series that acquainted him with the writings of Darwin and Haeckel. His mother introduced him to the world of plants, animals, and stars. "From her I... learned that... light has to go eight minutes from the sun to here. She was also very interested in calculation, in numbers and so on and from her 1 learned the first steps in arithmetic and so on" (Jordan 1963, p. 1). She often took him to visit the local zoo and he remembered collecting pictures of extinct animals, particularly those of dinosaurs. In his early teens he thought of becoming a painter or an architect, but gradually his interests shifted to natural history and biology, and eventually to physics and mathematics. He was clearly quite gifted and ambitious: "At fourteen, I had a plan of writing a big book on all the fields of science linking them all together" (Jordan 1963, p. 5). He had by then read and absorbed such books as Pauly"s Darwinismus und Lamarkismus and F. A. Lange"s Geschichte des Materialismus. He had also studied by himself classical physics and a great deal of mathematics. While in Gymnasium he taught himself the differential and integral calculus from Nernst and Schoenfiiess"s Kurzgefasstes Lehrbuch der Differentialuiid Integralrechnung, and the theory of complex variables from Knoff"s Funktionentheorie. During his last year in the Gymnasium he began to study physics in depth and carefully read Mach"s Mechanik and Prinzipien der Wärmelehre. Mach"s views influenced Jordan deeply and he became an ardent positivist. He later declared that he took up physics in order to help resolve the discrepancy he felt existed between Mach"s teachings and the old quantum theory (Jordan 1936). He adopted as the central tenet of his philosophical outlook what he considered to be the essential and decisive principle of the positivistic theory of knowledge: that scientifically sound proposilions arc limiled lo those that can be proved experimentally.

Interesante la influencia de su madre, y su inclinación a la ciencia, y varias ramas a la vez, por ejemplo, su interés en la biología evolutiva.

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Grupos y Física, por Dirac

Teoría de Grupos y Partículas Elementales

Hermann Weyl, Teoría de Grupos y Teoría Cuántica

Encuentro hoy unos párrafos de Abraham País, en su excelente "Inward bound, of matter and forces in the physical world", en un capítulo dedicado a grupos y la "clásica" mecánica cuántica:

Nearly a year after Heisenberg had considered the theory of one linear oscillator and so discovered quantum mechanics, he had something interesting to say about two identical oscillators symmetrically coupled to each other. The quantum states of this system, he found, separate into two sets, one symmetric, the other anti-symmetric under exchange of the oscillator coordinates. Assuming further that the oscillators carry electric charge, he noted that radiative transitions can occur between states within each set, never between one set and the other. He further conjetural that non-combining sets should likewise exist if the number n of identical particles is larger than two, but had not yet found a proof. He left this problem aside; another question was on his mind. Six weeks later he gave the theory of the hellium spectrum, that bane of the old quantum theory. To Pauli he complianed that his calculations were 'imprecise and incomplete'. It is true that others were able to refine his answer in later years. Nevertheless, his outstanding paper contains all the basic ingredients used today. The first quantum mechanical application of the Pauli principle is given: two-electron wave functions are antisymmetric for simultaneous exchange of space and spin coordinates. That principle is the subject of the following section...

Meanwhile in Berlin a young Hungarian chemistry engineer, Jenö Pál (better known since as Eugene Paul) Wigner, had become interested in the n > 2 identical particle problem. He rapidly mastered the case n = 3 (without spin). His method were rather laborious; for example, he had to solve a (reducible) equation of degree six. It would be pretty awful to go on this way to higher n. So, Wigner told me, he went to consult his friend the mathematician Johnny von Neumann. Johnny thought a few moments then told him that he should read certain papers by Frobenius and by Schur which he promised to bring the next day. As a result Wigner's paper on the case n (no spin), was ready soon and was submitted in November 1926. It contains an acknowledment to von Neumann, and also the following phrase: 'There is a well-developed mathematical theory which one can use here: the theory of transformation groups which are isomorphic with the symmetric group (the group of permutations)'.

Thus did group theory enter quantum mechanics.

El propio Wigner escribió un libro "clásico" "Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra". Y ya mencioné en un post de arriba, al trabajo de Weyl.

Vean cómo el trabajo de Heisenberg fue precursor de la aplicación de grupos. Luego, él mismo se encargaría de introducir las simetrías internas, en el caso protón-neutrón, que inauguraría otra rama de aplicación, la de teoría de grupos en partículas elementales.

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Encontré datos adicionales sobre la historia de la aparición de la "segunda cuantización". Leo:

In the midst of their efforts to cobble together a new quantum mechanics to treat the physics of atoms during the mid-1920s, several theorists—Werner Heisenberg, Pascual Jordan, Wolfgang Pauli, Paul Dirac, and others—began trying to quantize Maxwell"s electromagnetic field as well. Jordan began the process in the midst of his work on Heisenberg"s matrix mechanics, composing a long and difficult section of the famed Dreim¨annerarbeit, or "three-man paper" by Heisenberg, Jordan, and Max Born of 1926, on suggestive ways to quantize vibrating strings (seen as a first step toward treating waves and fields). The following year, Paul Dirac demonstrated that the electromagnetic field"s infinite number of degrees of freedom could be decomposed as a sum over quasi-particulate oscillators, each corresponding to a specific frequency or energy—a representation, soon dubbed "second quantization," that Jordan quickly extended tomatter fields aswell.By 1927, Jordan had become convinced that all physical quantities—everything from the electrons and protons of ordinary matter to the electromagnetic fields that bound them together into atoms—arose ultimately from quantum fields. Although resisting some of Jordan"s maneuvers at first, other theorists, including Heisenberg and Pauli, soon came to share Jordan"s view.

Lo encuentro en el libro de David Kaiser, Drawing theories apart, the dispersion of Feynman Diagrams in Postwar Physics, que trata sobre cómo los diagramas de Feynman se fueron popularizando luego de la segunda guerra mundial. Menciona en una nota, estas fuentes:

Dirac, "Quantum theory of radiation" (1959 [1927]); Born, Heisenberg, and Jordan, "Quantenmechanik" (1926); Jordan, "Quantenmechanik des Gasentartung" (1927); Heisenberg and Pauli, "Quantenelektrodynamik" (1929–30); Darrigol, "Origin of quantized matter waves" (1986); Pais, Inward Bound (1986), 334–40; Schweber,QED(1994), 23–56; andMiller, "Frame-setting essay" (1994), 18–28.

El "paper" de Dirac se puede encontrar en:

http://isites.harvard.edu/fs/docs/icb.topic474774.files/dirac.pdf

El "paper" de los "tres autores" lo tengo en el libro de van der Waerden, Sources of Quantum Mechanics, pero no sé si completo. Y es muy interesante la descripción del trabajo de Dirac que hace el libro de País, Inward bound.

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It is worth mentioning, in passing, that in 1900, the same year in which Planck"s paper on blackbody radiation appeared, Lord Kelvin gave a lecture that drew attention to another difficulty with the classical theory of statistical mechanics. Kelvin described two "clouds" over nineteenth century physics at the dawn of the twentieth century. The first of these clouds concerned aether—a hypothetical medium through which electromagnetic radiation propagates—and the failure of Michelson and Morley to observe the motion of earth relative to the aether. Under this cloud lurked the theory of special relativity. The second of Kelvin"s clouds concerned heat capacities in gases. The equipartition theorem of classical statistical mechanics made predictions for the ratio of heat capacity at constant pressure (cp) and the heat capacity at constant volume (cv). These predictions deviated substantially from the experimentally measured ratios. Under the second cloud lurked the theory of quantum mechanics, because the resolution of this discrepancy is similar to Planck"s resolution of the blackbody problem. As in the case of blackbody radiation, quantum mechanics gives rise to a correction to the equipartition theorem, thus resulting in different predictions for the ratio of cp to cv, predictions that can be reconciled with the observed ratios.

Lo encuentro en el excelente libro de Brian C. Hall, "Quantum Theory for Mathematicians".

¡Qué puntería en señalar "nubes" que tenía Lord Kelvin! Justo apuntó a dos temas que terminarían provocando gran parte del desarrollo de la nueva física del siglo que estaba por comenzar. ¿Tenemos nubes así, ahora que estamos en el siglo XXI? Debe haber varias, pero a mí me llaman la atención dos en particular: las llamadas "materia oscura" y la "energía oscura" que aparecieron con distinto sustrato experimental, pero que no encuentran una explicación evidente en estos días. A veces pienso que la respuesta a estas "nubes" realmente será insólita, y veremos el nacer de una novísima física.

Vivimos tiempos interesantes...

Post relacionado: Lord Kelvin y Rutherford

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Llego al final de esta conferencia de Dirac. Luego de observar algunos problemas, menciona la renormalización de los valores del electrón:

A feature of the calculations leading to the Lamb shift and anomalous magnetic moment should be noted. One finds that the parameters m and e denoting the mass and charge of the electron in the starting equations are not the same as the observed values for these quantities. If we keep the symbols m and e to denote the observed values, we have to replace the m and e in the starting equations by m + dm and e + be, where dm and be are small corrections which can be calculated. This procedure is known as renormalization.

Dirac ve que este "truco" funciona porque es un cambio global:

Such a change in the starting equations is permitted. We can take any starting equations we like, and then develop the theory by making deductions from them. You might think that the work of the theoretical physicist is easy if he can make any starting assumptions he likes, but the difficulty arises because he needs the same starting assumptions for all the applications of the theory. This very strongly restricts his freedom. Renormalization is permitted because it is a simple change which can be applied universally whenever one has charged particles interacting with the electromagnetic field.

Yo no sabía que hay otros problemas asociados con la autoenergía del fotón, también:

There is a serious difficulty still remaining in quantum electrodynamics, connected with the self-energy of the photon. It will have to be dealt with by some further change in the starting equations, of a more complicated kind than renormalization.

Finalmente, el objetivo final según Dirac:

The ultimate goal is to obtain suitable starting equations from which the whole of atomic physics can be deduced. We are still far from it. One way of proceeding towards it is first to perfect the theory of low-energy physics, which is quantum electrodynamics, and then try to extend it to higher and higher energies. However, the present quantum electrodynamics does not conform to the high standard of mathematical beauty that one would expect for a fundamental physical theory, and leads one to suspect that a drastic alteration of basic ideas is still needed.

Fue una larga serie post, con varios puntos que tengo que estudiar y comentar con mejor detalle.

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No sabía que el tema del "corte" obtiene distintos resultados, según se use una imagen u otra:

Working with a finite cut-off, we have to search for quantities which are not sensitive to the precise mode and value of the cut-off. We then find that the Schrodinger picture is not a suitable one. Solutions of the Schrodinger equation, even the one describing the vacuum state, are very sensitive to the cutoff. But there are some calculations that one can carry out in the Heisenberg picture that lead to results insensitive to the cut-off.

Notablemente, Dirac afirma que por este camino puede obtener el valor del corrimiento Lamb:

One can deduce in this way the Lamb-shift and the anomalous magnetic moment of the electron. The results are the same as those obtained some twenty years ago by the method of working rules with discard of infinities. But now the result can be obtained by a logical process, following standard mathematics in which only small quantities are neglected.

Pero perdemos los avances de la imagen Schrodinger:

As we cannot now use the Schrodinger picture, we cannot use the regular physical interpretation of quantum mechanics involving the square of the modulus of the wave function. We have to feel our way towards a new physical interpretation which can be used with the Heisenberg picture. The situation for quantum electrodynamics is rather like that for elementary quantum mechanics in the early days when we had the equations of motions but no general physical interpretation.

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Veamos que hay casos de operadores donde las autofunciones forman base. Sea un operador en un espacio vectorial de dimensión finita N. Entonces, ese operador puede expresarse como una matriz N x N. Para encontrar sus autovalores:

Donde M es una matriz cuadrada, lambda un número variable, rho es un vector columna. Si calculamos el determinante igualado a cero:

Obtenemos un polinomio de grado N, con incógnita lambda. Tiene que tener N raíces, tal vez algunas repetidas. Cada raíz es un autovalor, y por cada autovalor, tenemos un vector columna que es un autovector. Ya vimos que las autovectores de autovalores diferentes son ortogonales. Si algún autovalor se repite, igual podemos formar un subespacio con todos sus autovectores, y elegir una base ortonormal (siguiendo procedimientos conocidos de espacios vectoriales).

En definitiva, para estos operadores en dimensión finita N, tenemos autovectores que forman base. Como en general en los libros de divulgación se tratan ejemplos de este tipo, de dimensión finita, no tenemos mayor problema.

Pero hay casos donde no hay base de autovectores de manera tan simple. Veremos un caso sobre dimensión infinita no numerable en el próximo post.

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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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