Mass of the photon

Με αφορμή την κουβέντα στο forum.atheia.gr για το Big Bang και όχι μόνο, θυμήθηκα ένα πολύ όμορφο ανέκδοτο του Feynman όπου συζητάει με κάποιον καθηγητή για την μάζα του φωτονίου.

Πέρα από την ουσία της συζήτησης για την μάζα του φωτονίου και πέρα από το ότι ο Feynman ξεδιπλώνει το θεϊκό ταλέντο του, το όλο ανέκδοτο αναδεικνύει κάποια ενδιαφέροντα σημεία που υπάρχουν ή ακόμα και προβλήματα που μπορεί να προκύψουν στην όποια συζήτηση μπορεί να κάνει κάποιος για ένα θέμα Φυσικής (ή οποιοδήποτε άλλο επιστημονικό θέμα). 2 είναι τα μεγάλα προβλήματα σε τέτοιες συζητήσεις. Το πρώτο πρόβλημα είναι το τι θεωρεί ο καθένας ως δεδομένη γνώση και το δεύτερο είναι κατά πόσο "παίζει" με τους κανόνες του "παιχνιδιού".

In this connection I would like to relate an anecdote, something from a conversation after a cocktail party in Paris some years ago. There was a time at which all the ladies mysteriously disappeared, and I was left facing a famous professor, solemnly seated in an armchair, surrounded by his students. He said, “Tell me, Professor Feynman, how sure are you that the photon has no rest mass?” I answered “Well, it depends on the mass; evidently if the mass is infinitesimally small, so that it would have no effect whatsoever, I could not disprove its existence, but I would be glad to discuss the possibility that the mass is not of a certain definite size. The condition is that after I give you arguments against such mass, it should be against the rules to change the mass.” The professor then chose a mass of 10^-5 of an electron mass.
My answer was that, if we agreed that the mass of the photon was related to the frequency as \omega=\sqrt{k^2+m^2}, photons of different wavelengths would travel with different velocities. Then in observing an eclipsing double star, which was sufficiently far away, we would observe the eclipse in blue light and red light at different times. Since nothing like this is observed, we can put an upper limit on the mass, which, if you do the numbers, turns out to be of the order of 10^-9 electron masses. The answer was translated to the professor. Then he wanted to know what I would have said if he had said 10^-12 electron masses. The translating student was embarrassed by the question, and I protested that this was against the rules, but I agreed to try again.
If the photons have a small mass, equal for all photons, larger fractional differences from the massless behavior are expected as the wavelength gets longer. So from the sharpness of the known reflection of pulses in radar, we can put an upper limit to the photon mass which is somewhat better than from the eclipsing double star argument. It turns out that the mass had to be smaller than 10^-15 electron masses.
After this, the professor wanted to change the mass again, and make it 10^-18 electron masses. The student all became rather uneasy at this question, and I protested that, if he kept breaking the rules, and making the mass smaller and smaller, evidently I would be unable to make an argument at some point. Nevertheless, I tried again. I asked him whether he agreed that if the photon had a small mass, then from field theory arguments the potential should go as \exp{-mr}/r. He agreed. Then, the earth has a static magnetic field, which is known to extend out into space for some distance, from the behavior of the cosmic rays, a distance at least of the order of a few earth radii. But this means that the photon mass must be of a size smaller than that corresponding to a decay length of the order of 8000 miles, or some 10^-20 electron masses. At this point, the conversation ended, to my grate relief.

Richard Feynman
Feynman Lectures on Gravitation
pp. 22-23

Υπάρχουν αρκετά ενδιαφέροντα σημεία για να επισημάνει κανείς.
Το αφήνω ως "άσκηση".
:P

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Update:

Υπάρχει ένα πρόσφατο άρθρο στο Reviews of Modern Physics που παρουσιάζει τα όρια που έχουμε αυτή τη στιγμή για την μάζα του φωτονίου και του βαρυτονίου.

Το άρθρο είναι το, Rev. Mod. Phys. 82, 939–979 (2010), "Photon and graviton mass limits", των Alfred Scharff Goldhaber και Michael Martin Nieto. Για όποιον δεν έχει πρόσβαση στο περιοδικό, υπάρχει και στο arxiv.org και είναι το arXiv:0809.1003v4 [hep-ph].

Efforts to place limits on deviations from canonical formulations of electromagnetism and gravity have probed length scales increasing dramatically over time. Historically, these studies have passed through three stages: (1) testing the power in the inverse-square laws of Newton and Coulomb, (2) seeking a nonzero value for the rest mass of photon or graviton, and (3) considering more degrees of freedom, allowing mass while preserving explicit gauge or general-coordinate invariance. Since the previous review the lower limit on the photon Compton wavelength has improved by four orders of magnitude, to about one astronomical unit, and rapid current progress in astronomy makes further advance likely. For gravity there have been vigorous debates about even the concept of graviton rest mass. Meanwhile there are striking observations of astronomical motions that do not fit Einstein gravity with visible sources. “Cold dark matter” (slow, invisible classical particles) fits well at large scales. “Modified Newtonian dynamics” provides the best phenomenology at galactic scales. Satisfying this phenomenology is a requirement if dark matter, perhaps as invisible classical fields, could be correct here too. “Dark energy” might be explained by a graviton-mass-like effect, with associated Compton wavelength comparable to the radius of the visible universe. Significant mass limits are summarized in a table.

Στο άρθρο αναφέρονται και οι μέθοδοι που επικαλείται ο Feynman παραπάνω.