A DC potential difference applied at the boundary
between two superconductors can produce an AC Josephson current
whose frequency is precisely related to the size of the applied
potential and the electron’s charge. Precision measurements of
frequency and voltage are in this way converted into a precise
measurement of , and so of
. But use of this
effect to determine
only makes sense if the predicted
relationship between frequency and voltage is also known to an
accuracy which is better than the uncertainty in
.
It is, at first sight, puzzling how such an accurate prediction for this effect can be possible. After all, the prediction is made within the BCS theory of superconductivity (see, for example, [139]), which ignores most of the mutual interactions of electrons, focussing instead on a particular pairing interaction due to phonon exchange. Radical though this approximation might appear to be, the theory works rather well (in fact, surprisingly well), with its predictions often agreeing with experiment to within several percent. But expecting successful predictions with an accuracy of parts per million or better would appear to be optimistic indeed!
The astounding theoretical accuracy required to
successfully predict the Josephson frequency may be understood at
another level, however. The key observation is that this prediction
does not rely at all on the details of the BCS theory, depending
instead only on the symmetry-breaking pattern which it predicts.
Once it is known that a superconductor spontaneously breaks the
gauge symmetry of electromagnetism, the Josephson
prediction follows on general grounds in the low-energy limit (for
a discussion of superconductors in an effective-Lagrangian spirit
aimed at a particle-physics audience see [151]). The
validity of the prediction is therefore not controlled by the
approximations made in the BCS theory, since any theory with the same low-energy
symmetry-breaking pattern shares the same predictions.
The accuracy of the predictions for the Josephson
effect are therefore founded on symmetry arguments, and on the
validity of a low-energy approximation. Quantitatively, the
low-energy approximation involves the neglect of powers of the
ratio of two scales, , where
is the low energy scale of the observable under consideration -
like the applied voltage in the Josephson effect - and
is the higher energy scale - such as the
superconducting gap energy - which is intrinsic to the system under
study.
Indeed, arguments based on a similar low-energy approximation may also be used to explain the surprising accuracy of many other successful models throughout physics, including the BCS theory itself [130, 135, 136, 35]. This is accomplished by showing that only the specific interactions used by the BCS theory are relevant at low energies, with all others being suppressed in their effects by powers of a small energy ratio.
Although many of these arguments were undoubtedly known in various forms by the experts in various fields since very early days, the systematic development of these arguments into precision calculational techniques has happened more recently. With this development has come considerable cross-fertilization of techniques between disciplines, with the realization that the same methods play a role across diverse disciplines within physics.
The remainder of this article briefly summarizes the techniques which have been developed to exploit low-energy approximations. These are most efficiently expressed using effective-Lagrangian methods, which are designed to take advantage of the simplicity of the low-energy limit as early as possible within a calculation. The gain in simplicity so obtained can be the decisive difference between a calculation’s being feasible rather than being too difficult to entertain.
Besides providing this kind of practical
advantage, effective-Lagrangian techniques also bring real
conceptual benefits because of the clear separation they permit
between of the effects of different scales. Both of these kinds of
advantages are illustrated here using explicit examples. First
Section 2.2 presents a toy model
involving two spinless particles to illustrate the general method,
as well as some of its calculational advantages. This is followed
by a short discussion of the conceptual advantages, with quantum
corrections to classical general relativity, and the associated
problem of the non-renormalizability of gravity, taken as the
illustrative example.