304
10. Loops and Renormalization I: The ABC Theory
procedure known as renormalization, to which we shall provide an introduction
in this and the following chapter.
The physical ideas behind renormalization are, however, just as relevant
in cases – such as condensed matter physics – where the analogous higherorder (loop) corrections are not infinite, though possibly large. In quantum
mechanics, infinite momentum corresponds to zero distance, and our fields
are certainly ‘point-like’. But in condensed matter physics there is generally a
natural non-zero smallest distance – the lattice size, or an atomic diameter, for
example. In quantum field theory, such a ‘shortest distance’ would correspond
to a ‘highest momentum’, meaning that the magnitudes of loop momenta
would run from zero up to some finite limit Λ, say, rather than infinity. Such
a Λ is called a (momentum) ‘cut-off’. With such a cut-off in place, our loop
integrals are of course finite – but it would seem that we have then maltreated
our field theory in some way. However, we might well ask whether we seriously
believe that any of our quantum field theories is literally valid for arbitrarily
high energies (or arbitrarily small distances). The answer is surely no: we are
virtually certain that ‘new physics’ will come into play at some stage, which is
not contained in – say – the QED, or even the Standard Model, Lagrangian.
At what scale this new physics will enter (the Planck energy? 1 TeV?) we
do not know, but surely the current models will break down at some point.
We should not be too alarmed, therefore, by formal divergences as Λ → ∞.
Rather, it may be sensible to regard a cut-off Λ as standing for some ‘new
physics’ scale, accepting some such manoeuvre as physically realistic as well
as mathematically prudent.
At the same time, however, we would not want our physical predictions,
made using quantum field theories, to depend sensitively on Λ – i.e. on the
unknown short-distance physics, in this interpretation. Indeed, theories exist
(for example, those in the Standard Model and the ABC theory) which can be
reformulated in such a way that all dependence on Λ disappears, as Λ → ∞;
these are, precisely, renormalizable quantum field theories. Roughly speaking,
a renormalizable quantum field theory is one such that, when formulae are
expressed in terms of certain ‘physical’ parameters taken from experiment,
rather than in terms of the original parameters appearing in the Lagrangian,
calculated quantities will be finite and independent of Λ as Λ → ∞.
Solid state physics provides a close analogy. There, the usefulness of a
description of, say, electrons in a metal in terms of their ‘effective charge’ and
‘effective mass’, rather than their free-space values, is well established. In this
analogy, the free-space quantities correspond to our Lagrangian values, while
the effective parameters correspond to our ‘physical’ ones. In both cases, the
interactions are causing changes to the parameters.
It is clear that we need to understand more precisely just what our ‘physical parameters’ might be and how they might be defined. This is what we aim
to do in the remainder of the present section, and in the next one, before returning in section 10.3 to the mathematical details associated with evaluating
(10.7), and indicating how renormalization works for the self-energy. Having
10. Loops and Renormalization I: The ABC Theory
procedure known as renormalization, to which we shall provide an introduction
in this and the following chapter.
The physical ideas behind renormalization are, however, just as relevant
in cases – such as condensed matter physics – where the analogous higherorder (loop) corrections are not infinite, though possibly large. In quantum
mechanics, infinite momentum corresponds to zero distance, and our fields
are certainly ‘point-like’. But in condensed matter physics there is generally a
natural non-zero smallest distance – the lattice size, or an atomic diameter, for
example. In quantum field theory, such a ‘shortest distance’ would correspond
to a ‘highest momentum’, meaning that the magnitudes of loop momenta
would run from zero up to some finite limit Λ, say, rather than infinity. Such
a Λ is called a (momentum) ‘cut-off’. With such a cut-off in place, our loop
integrals are of course finite – but it would seem that we have then maltreated
our field theory in some way. However, we might well ask whether we seriously
believe that any of our quantum field theories is literally valid for arbitrarily
high energies (or arbitrarily small distances). The answer is surely no: we are
virtually certain that ‘new physics’ will come into play at some stage, which is
not contained in – say – the QED, or even the Standard Model, Lagrangian.
At what scale this new physics will enter (the Planck energy? 1 TeV?) we
do not know, but surely the current models will break down at some point.
We should not be too alarmed, therefore, by formal divergences as Λ → ∞.
Rather, it may be sensible to regard a cut-off Λ as standing for some ‘new
physics’ scale, accepting some such manoeuvre as physically realistic as well
as mathematically prudent.
At the same time, however, we would not want our physical predictions,
made using quantum field theories, to depend sensitively on Λ – i.e. on the
unknown short-distance physics, in this interpretation. Indeed, theories exist
(for example, those in the Standard Model and the ABC theory) which can be
reformulated in such a way that all dependence on Λ disappears, as Λ → ∞;
these are, precisely, renormalizable quantum field theories. Roughly speaking,
a renormalizable quantum field theory is one such that, when formulae are
expressed in terms of certain ‘physical’ parameters taken from experiment,
rather than in terms of the original parameters appearing in the Lagrangian,
calculated quantities will be finite and independent of Λ as Λ → ∞.
Solid state physics provides a close analogy. There, the usefulness of a
description of, say, electrons in a metal in terms of their ‘effective charge’ and
‘effective mass’, rather than their free-space values, is well established. In this
analogy, the free-space quantities correspond to our Lagrangian values, while
the effective parameters correspond to our ‘physical’ ones. In both cases, the
interactions are causing changes to the parameters.
It is clear that we need to understand more precisely just what our ‘physical parameters’ might be and how they might be defined. This is what we aim
to do in the remainder of the present section, and in the next one, before returning in section 10.3 to the mathematical details associated with evaluating
(10.7), and indicating how renormalization works for the self-energy. Having