Renormalization in Quantum Field Theory V
Algebra Number Theory 5 , no. Abstract Article info and citation First page References Abstract The aim of this paper is to describe how to use regularization and renormalization to construct a perturbative quantum field theory from a Lagrangian. Export citation. Export Cancel. References E. Epstein-Glaser normalization.
The construction of perturbative quantum field theories around a given gauge fixed relativistic free field vacuum is equivalently, by prop. By prop. By paying attention to the scaling degree def.
This conclusion is theorem Following this, statement and detailed proof appeared in Brunetti-Fredenhagen Then we show that these unique products on these special subsets do coincide on intersections. This yields the claim by a partition of unity. Moreover it is immediate that they form an open cover of the complement of the diagonal :. To see that this is the case, we have to consider any two such subsets. This means that there is a finite set. In terms of this, prop.
Therefore we now discuss extension of distributions def. Since the space of choices of such extensions turns out to depend on the scaling degree of distributions , we first discuss that def. Hence in this case. Then the degree of divergence def. Its scaling degree is. Brunetti-Fredenhagen 00, example 3 on p. The first three statements follow with manipulations as in example With the concept of scaling degree of distributions in hand, we may now discuss extension of distributions :.
This induces the operation of restriction of distributions. Brunetti-Fredenhagen 00, theorem 5. But by example This means that if the following limit exists. This is shown in Brunetti-Fredenhagen 00, p. Then u does admit at least one extension def. We follow Brunetti-Fredenhagen 00, theorem 5. Therefore every continuous linear projection. Observe that by prop. Therefore prop. By example In conclusion we obtain the central theorem of causal perturbation theory :.
Now if a polynomial local interaction is fixed, then via remark Such an interaction vertex redefinition def. A perturbative interaction vertex redefinition or just vertex redefinition , for short is an endofunction. The following proposition should be compared to the axiom of causal additivity of the S-matrix scheme :. If the theory is renormalizable see below for more on this , as it is in QED, the divergent parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term or by the similar counterterms that come from Z 0 and Z 3.
The diagram with the Z 1 counterterm's interaction vertex placed as in Figure 3 cancels out the divergence from the loop in Figure 2. Historically, the splitting of the "bare terms" into the original terms and counterterms came before the renormalization group insight due to Kenneth Wilson.
To minimize the contribution of loop diagrams to a given calculation and therefore make it easier to extract results , one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be independent of the choice of renormalization point, as long as it is within the domain of application of the theory.
Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of physical constants with changes in scale.
Wilsonian Renormalization and Effective Field Theories
This variation is encoded by beta-functions , and the general theory of this kind of scale-dependence is known as the renormalization group. Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity. This running does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction.
For example, since the coupling in quantum chromodynamics becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes largea phenomenon known as asymptotic freedom. Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.
An essentially arbitrary modification to the loop integrands, or regulator , can make them drop off faster at high energies and momenta, in such a manner that the integrals converge. With the regulator in place, and a finite value for the cutoff, divergent terms in the integrals then turn into finite but cutoff-dependent terms. After canceling out these terms with the contributions from cutoff-dependent counterterms, the cutoff is taken to infinity and finite physical results recovered.
If physics on scales we can measure is independent of what happens at the very shortest distance and time scales, then it should be possible to get cutoff-independent results for calculations. Many different types of regulator are used in quantum field theory calculations, each with its advantages and disadvantages.
One of the most popular in modern use is dimensional regularization , invented by Gerardus 't Hooft and Martinus J.
Renormalization and Invariance in Quantum Field Theory | Eduardo Caianiello | Springer
Veltman ,  which tames the integrals by carrying them into a space with a fictitious fractional number of dimensions. Another is Pauli—Villars regularization , which adds fictitious particles to the theory with very large masses, such that loop integrands involving the massive particles cancel out the existing loops at large momenta.
Yet another regularization scheme is the lattice regularization , introduced by Kenneth Wilson , which pretends that hyper-cubical lattice constructs our space-time with fixed grid size.
This size is a natural cutoff for the maximal momentum that a particle could possess when propagating on the lattice. And after doing a calculation on several lattices with different grid size, the physical result is extrapolated to grid size 0, or our natural universe. This presupposes the existence of a scaling limit. A rigorous mathematical approach to renormalization theory is the so-called causal perturbation theory , where ultraviolet divergences are avoided from the start in calculations by performing well-defined mathematical operations only within the framework of distribution theory.
In this approach, divergences are replaced by ambiguity: corresponding to a divergent diagram is a term which now has a finite, but undetermined, coefficient. Other principles, such as gauge symmetry, must then be used to reduce or eliminate the ambiguity.
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Julian Schwinger discovered a relationship [ citation needed ] between zeta function regularization and renormalization, using the asymptotic relation:. Although he reached inconsistent results, an improved formula studied by Hartle , J. Garcia, and based on the works by E. Elizalde includes the technique of the zeta regularization algorithm.
Or simply using Abel—Plana formula we have for every divergent integral:. The "geometric" analogy is given by, if we use rectangle method to evaluate the integral so:. Using Hurwitz zeta regularization plus the rectangle method with step h not to be confused with Planck's constant. In order to regularize these integrals one needs a regulator, for the case of multi-loop integrals, these regulator can be taken as. 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. Freeman Dyson argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method. Dirac 's criticism was the most persistent. Another important critic was Feynman. Despite his crucial role in the development of quantum electrodynamics, he wrote the following in . 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 s had the property that the interactions become infinitely strong at short enough distance scales. This property called a Landau pole , made it plausible that quantum field theories were all inconsistent. In , Gross , Politzer and Wilczek showed that another quantum field theory, quantum chromodynamics , does not 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 s and s. Beginning in the s, 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 physical short-distance regulator exists: matter ceases to be continuous on the scale of atoms. Short-distance divergences in condensed matter physics do not 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 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. Be that as it may, Salam 's remark  in seems still relevant.
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 current particle accelerators , they all become approximately the same size Grotz and Klapdor , p. Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.