The range of elementary interactions: a criticism of the standard classroom approach

Abstract

The finite scope of some elementary interactions is usually presented to physics students as a natural consequence of the time - energy uncertainty relation. It is demonstrated that this heuristic derivation is not a priori valid. Accordingly, the doubtful usefulness of this `proof' in the classroom is argued, for it may lead to misconceptions.Resumen. Usualmente se presenta a los estudiantes de Física el alcance finito de ciertas interacciones fundamentales como una consecuencia natural de la relación de incertidumbre energía-tiempo. Se demuestra que esta derivación heurística no es a priori válida. Por tanto, se sostiene que es dudosa la utilidad de una tal `demostración' en el aula, dado que puede conducir a concepciones erróneas. In many textbooks, both introductory (see for example [1]) and advanced (such as [2,3], to cite a couple), the finite range of an elementary interaction with massive field quanta is explained as follows. Assume that the intermediary particle of a force possesses mass m. Then the energy required to produce it is . Thus, from Heisenberg's principle the life of the particle will be of order Since an upper bound for the speed of the virtual particle is c, an upper bound for the range l of the interaction will be Sometimes [3] this line of reasoning is extended to also include the electromagnetic interaction: in this case is the energy of the exchanged virtual photon, and from a straightforward repetition of the previous argument we obtain In either case should be an estimation of the maximum distance that the virtual particle can travel before being reabsorbed. However, this is a misleading argument. To see this, it should be noticed that the time - energy uncertainty relation is [4,5] Therefore, from this equation it is only possible to deduce a lower bound for . Then, if v < c is the average speed of the virtual particle, we arrive at Hence, the time - energy uncertainty relation implies the existence of a core of radius such that the virtual particle cannot be reabsorbed inside it. In other words, equation (5) establishes a lower bound for the range of the interaction. To retrieve a formula similar to (2) or (3), it is first necessary to assume that the actual value of l is of the order of the lower bound in (5): Note that this hypothesis is extraneous to the uncertainty principle itself. Then, since v < c, we obviously have This statement is apparently similar to that expressed by equation (3) (or by (2), if we substitute by ). However, the meaning of (7) is entirely different to this one, since in (7) we have an upper bound for a lower bound, that is, an upper bound for the radius of the core. Clearly, the obtainment of the standard heuristic formula requires an awkward argument, and the interpretation of the result looks too involved and confusing for the student. Consequently, the previous development seems to imply that the usual heuristic derivation is undesirable from a pedagogic point of view. As we have seen, it does not provide a complete description of the process. Moreover, to explain the finite range of the interaction we are in fact making use of an argument leading to the opposite result: the existence of a lower bound for the range or, in other words, the existence of a core where the interaction is forbidden. Conversely, as the author has had the opportunity to observe, the well known fact that the life of virtual particles is finite may lead to the student to deduce that the uncertainty principle is instead of the correct expression (4), because the wrong formula leads immediately to the expected result in which l has an upper bound given by It seems therefore reasonable to conclude that the standard trick used to establish the finite range of those interactions with massive field quanta should be carefully employed, or perhaps avoided, in the classroom.