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Quantum Field Theory

Renormalizable Theories

As mentioned earlier, it is very difficult to compute the viscosity from the Schrodinger equation, although the former must somehow follow from the latter. However, in relativistic theory of elementary particles, the form of the equations is highly restrictive by the requirement of relativistic invariance. For that reason it is actually not difficult to carry out the renormalization procedure in a relativistic theory, at least in the perturbation framework.

It is also mentioned that divergences (infinities) appear with certain loop diagrams in QED. The origin of these infinities can be traced back to the singular nature of the inverse-square law 1 / r2 as r 0. But there is no direct evidence that this law is still valid below a distance of some 10-16 cm. If QED is renormalizable, then the detail at the deeper scale is not important. The inverse-square law can be replace by a constant 1 / r02 for r r0 ~ 10-16 cm. It turns out that QED is indeed renormalizable. The schemes to make the force law less singular is called regularization. There are three popular schemes :
  1. Pauli-Villars regularization - A fictitious particle of mass M is introduced to modify the propagator such that it behaves now as 1/k4, which is usually enough to render all graphs finite in the off mass-shell integral. Then the limit of M2 is taken at the end of calculation to decouple the unphysical particle from the theory.
  2. Dimensional regularization - The action is generalized to arbitrary dimension d, where there are regions in complex d space in which the Feynman integrals are all finite. The Feynman graphs pick up poles in d space to absorb the divergences of the theory into the physical parameters as d is reduced to four.
  3. Lattice regularization - This is the most widely used in QCD for non-perturbative calculations. The lattice spacing serves as the cutoff for the space-time integrals.
In layman's language the renormalization procedure goes like this: one theoretical calculation produces a quantity mZ which involves an infinity. As the inherent mbare is another infinite, then the observed quantity could be in such form as mobs = mbare - mZ in which mbare cancels out most of the infinity in mZ leaving behind a finite mobs, which is to be determined by experiment. In other word, QED provides mZ to modify the original quantity mbare when the electromagnetic interaction is switched on.

Mathematically, the divergences come from the integration of the 4-momentum p (or sometimes denoted as k) for the virtual particles (both fermion and boson) to infinity. The integration "4-volume" element d4p contributes a p4 term, which can be "diluted" by a p-1 term from each fermion propagator (internal line), a p-2 term from each boson propagator, and in addition the coupling constant in each vertex will contribute a (-n) power of p if its unit has n-dimension of p (in order to keep the total dimension of the graph to be the same). The origin of the other negative power p terms can be traced ultimately to the number of spacetime derivatives (within individual term) in either the Lagrangian or the field equation. While the Dirac equation involves only first order derivatives, the field equations for most boson are written down in second order derivatives. Thus, if we denote the Divergence as D, then the degree of divergence can be written as :

D = 4 - F - 2B - nV ---------- (39a)

where F and B are respectively the number of fermion and boson propagators (internal lines), V is the number of vertices in the Feynman diagram, which is divergent if D 0. The coupling constants g for any gauge theory is dimensionless, so n = 0. Depending on whether n = 0, n > 0, or n < 0, the corresponding theories are renormalizable, super-renormalizable, or non-renormalizable.

According to the formula for D, only a small variety of graphs in a renormalizable theory are divergent. It is summerized in Table 04 below:

Graph Diagram F B D D' D'+1 Parameter(s)
Vacuum 2 1 0 0 1 (Ignored)
Photon Vacuum Polarization 2 0 2 0 1 Z3
Electron Self-energy 1 1 1 1 2 m and Z2
Electron-Photon Vertex 2 1 0 0 1 Z1
Photon-Photon Scattering 4 0 0 0 > 0 > Nil (Convergent)
Electron-Electron Scattering 2 2 -2 -2 -1 Nil (Convergent)

Table 04 Degree of Divergence

For a super-renormalizable theory, n > 0, so there are even fewer divergent diagrams than the renormalizable theories. Moreover, if V is large enough, Eq.(39) indicates that all diagrams would be primitively convergent.

A non-renormalizable theory has n < 0, so D' grows indefinitely with V. Since D' + 1 divergent parameters are required in each case, a non-renormalizable theory contains an infinite number of divergent parameters. Even if we had an infinite number of bare parameters to adjust, there would be an infinite number of parameters in the theory. It would take a prohibited long time to measure all of them. The gravitational constant G = 6.7087x10-39 c (Gev / c2)-2 in quantum gravity has a negative energy dimension of n = -2 and hence it is non-renormalizable. Presumably some new physics will happen at the scale of 1019 Gev to turn quantum gravity into a renormalizable or a finite theory.

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