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 :

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.
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.
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 m_Z which involves an infinity. As the inherent m_bare is another infinite, then the observed quantity could be in such form as m_obs = m_bare - m_Z in which m_bare cancels out most of the infinity in m_Z leaving behind a finite m_obs, which is to be determined by experiment. In other word, QED provides m_Z to modify the original quantity m_bare 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:

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

The time axis in the Feynman diagrams runs horizontally from left to right.
The processes are related to quantum electrodynamics (QED).
The number of Ferimon and boson internal lines (propagator) F and B can be counted from each diagram.
The degree of divergence D is computed from Eq.(39a) with n = 0 for gauge theories. D is often referred as "primitive" divergence. For example, the electron-electron scattering diagram has no primitive divergence since D = -2. However, there is a subdiagram in its upper corner that looks like the vertex diagram, which is know to have D = 0 producing logarithmic divergence. In other words, divergences in subdiagrams are not counted as primitive divergences for the whole diagram.
A gauge theory has a high degree of symmetry, which allows some primitive divergences to be cancelled and reduces the degree of divergence to D'. This is the case for the photon self-energy and the photon-photon scattering diagrams, which are actually primitively convergent.
It is known that the divergences of a diagram are summarized in D' + 1 divergent parameters. The vacuum diagram and its divergence is usually ignored, on the grounds that it is not related to any external particles and cannot be measured experimentally. If we assume that the bare mass m₀, and the bare charge e₀ are adjustable infinite quantities, then they can be combined with the divergent parameters to re-formulate the theory into one that does not involve infinities. The mass m_R, and charge e_R in this new form will be "dressed" and agree with experimental measurements. The bare and renormalized quantities are related by the following formulas:

m_R = m₀ + m
e_R = [(Z₁Z₂Z₃^1/2)]e₀ = (Z₃^1/2)e₀

It also requires that the fields should be transformed like:

_R = Z₂^-1/2₀
A_R = Z₃^-1/2A₀
The vacuum diagram in which virtual e^-e⁺ pairs and virtual photons are created and annihilated together, give terms that modify the vacuum energy. Energy shifts in perturbation theory are to be expected, but since there is no unperturbed vacuum with which to compare, such shifts are not measurable. The cosmological constant of general relativity gives a measure of the vacuum energy density that is very small. Thus, the vacuum energy density, whatever its origin, is taken to be zero.

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.