Everything about Renormalisation totally explained
In
quantum field theory, the
statistical mechanics of fields, and the theory of
self-similar geometric structures,
renormalization refers to a collection of techniques used to take a continuum limit.
When describing space and time as a continuum, certain statistical and quantum mechanical constructions are ill defined. In order to define them, the continuum limit has to be taken carefully.
Renormalization determines the relationship between parameters in the theory, when the parameters describing large distance scales differ from the parameters describing small distances. Renormalization was first developed in
quantum electrodynamics to make sense of
infinite integrals in perturbation theory. Initially viewed as a suspect, provisional procedure by some of its originators, renormalization eventually was embraced as an important and
self-consistent tool in several fields of
physics and
mathematics.
Self-interactions in classical physics
The problem of infinities first arose in the
classical electrodynamics of
point particles in the
19th and early
20th century.
The mass of a charged particle should include the mass-energy in its electrostatic field. Assume that the particle is a charged spherical shell of radius
. The energy in the field is
»
Using Hurwitz zeta regularization plus rectangle method with step h (not to be confused with
Planck's constant)
Attitudes and interpretation
The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.
Dirac's criticism was the most persistent. As late as 1975, he was saying:
» Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we don't have to worry about it any more.' I must say that I'm very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it's small - not neglecting it just because it's infinitely great and you don't want it!
Another important critic was
Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote:
» The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it's still what I'd call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization isn't mathematically legitimate.
While Dirac's criticism was based on the procedure of renormalization itself, Feynman's criticism was very different. Feynman was concerned that all field theories known in the 1960s had the property that the interactions becomes infinitely strong at short enough distance scales. This property, called a
Landau pole, made it plausible that quantum field theories were all inconsistent. In 1974,
Gross,
Politzer and
Wilczek showed that another quantum field theory, Quantum Chromodynamics, doesn't have a landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.
The general unease was almost universal in texts up to the
1970s and
1980s. Beginning in the 1970s, however, inspired by work on the
renormalization group and
effective field theory, and despite the fact that Dirac and various others -- all of whom belonged to the older generation -- never withdrew their criticisms, attitudes began to change, especially among younger theorists.
Kenneth G. Wilson and others demonstrated that the renormalization group is useful in
statistical field theory applied to
condensed matter physics, where it provides important insights into the behavior of
phase transitions. In condensed matter physics, a
real short-distance regulator exists:
matter ceases to be continuous on the scale of
atoms. Short-distance divergences in condensed matter physics don't present a philosophical problem, since the field theory is only an effective, smoothed-out representation of the behavior of matter anyway; there are no infinities since the cutoff is actually always finite, and it makes perfect sense that the bare quantities are cutoff-dependent.
If
QFT holds all the way down past the
Planck length (where it might yield to
string theory,
causal set theory or something different), then there may be no real problem with short-distance divergences in
particle physics either;
all field theories could simply be effective field theories. In a sense, this approach echoes the older attitude that the divergences in QFT speak of human ignorance about the workings of nature, but also acknowledges that this ignorance can be quantified and that the resulting effective theories remain useful.
In QFT, the value of a physical constant, in general, depends on the scale that one chooses as the renormalization point, and it becomes very interesting to examine the renormalization group running of physical constants under changes in the energy scale. The coupling constants in the
Standard Model of particle physics vary in different ways with increasing energy scale: the coupling of
quantum chromodynamics and the weak isospin coupling of the
electroweak force tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase. At the colossal energy scale of 10
15 GeV (far beyond the reach of our civilization's
particle accelerators), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about
grand unified theory. Instead of a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.
Renormalizability
From this philosophical reassessment a new concept follows naturally: the notion of
renormalizability. Not all theories lend themselves to renormalization in the manner described above, with a finite supply of counterterms and all quantities becoming cutoff-independent at the end of the calculation. If the Lagrangian contains combinations of field operators of excessively high
dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number, and, at first glance, the theory would seem to gain an infinite number of free parameters and therefore lose all predictive power, becoming scientifically worthless. Such theories are called
nonrenormalizable.
The
Standard Model of particle physics contains only renormalizable operators, but the interactions of
general relativity become nonrenormalizable operators if one attempts to construct a field theory of
quantum gravity in the most straightforward manner, suggesting that
perturbation theory is useless in application to quantum gravity.
However, in an effective field theory, "renormalizability" is, strictly speaking, a
misnomer. In a nonrenormalizable effective field theory, terms in the Lagrangian do multiply to infinity, but have coefficients suppressed by ever-more-extreme inverse powers of the energy cutoff. If the cutoff is a real, physical quantity—if, that is, the theory is only an effective description of physics up to some maximum energy or minimum distance scale—then these extra terms could represent real physical interactions. Assuming that the dimensionless constants in the theory don't get too large, one can group calculations by inverse powers of the cutoff, and extract approximate predictions to finite order in the cutoff that still have a finite number of free parameters. It can even be useful to renormalize these "nonrenormalizable" interactions.
Nonrenormalizable interactions in effective field theories rapidly become weaker as the energy scale becomes much smaller than the cutoff. The classic example is the
Fermi theory of the
weak nuclear force, a nonrenormalizable effective theory whose cutoff is comparable to the mass of the
W particle. This fact may also provide a possible explanation for
why almost all of the particle interactions we see are describable by renormalizable theories. It may be that any others that may exist at the GUT or Planck scale simply become too weak to detect in the realm we can observe, with one exception:
gravity, whose exceedingly weak interaction is magnified by the presence of the enormous masses of
stars and
planets.
Renormalization schemes
In actual calculations, the counterterms introduced to cancel the divergences in Feynman diagram calculations beyond tree level must be
fixed using a set of
renormalization conditions. The common renormalization schemes in use include:
Minimal subtraction (MS) scheme and the related modified minimal subtraction (MS-bar) scheme
On-shell scheme
Application in Statistical Physics
As mentioned in the introduction, the methods of renormalization have been applied to Statistical Physics, namely to the problems of the critical behaviour near second-order phase transitions, in particular at fictitious spatial dimensions just below the number of 4, where the above-mentioned methods could even be sharpened (for example, instead of "renormalizability" one gets "super-renormalizability"), which allowed extrapolation to the real spatial dimensionality for phase transitions, 3. Details can be found in the book of Zinn-Justin, mentioned below.
For the discovery of these unexpected applications, and working out the details, in 1982 the physics Noble prize was given to Kenneth G. Wilson.
Further Information
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