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2.1 The utility of low-energy approximations

It sometimes happens that predictions are much more accurate than would be expected based on an assessment of the approximations on which they appear to be based. A famous example of this is encountered in the precision tests of quantum electrodynamics, where the value of the fine-structure constant, α, was until recently obtained using the Josephson effect in superconductivity.

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 eāˆ• ā„, and so of α. But use of this effect to determine α only makes sense if the predicted relationship between frequency and voltage is also known to an accuracy which is better than the uncertainty in α.

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, [138]), 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 U (1 ) 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 [150]). 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, ωāˆ•Ω, where ω is the low energy scale of the observable under consideration – like the applied voltage in the Josephson effect – and Ω 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 [12913513435]. 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.

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