Renormalization

The quantization prescription is based on the existence of a Hamiltonian, and since ${\cal{H}}$ generates infinitesimal time displacements, this corresponds to a differential development in time. Lorentz invariance then demands a differential development of space as well. The assumption that the field description holds down to arbitrarily small space-time intervals however leads to divergencies in the perturbative expressions. No matter how small a volume of space one considers, there are always some very short wavelengths present, giving rise to infinities in the calculations.

In a special class of field theories this is cured by the renormalization procedure. By redefining certain quantities like masses and charges, by subtraction of some infinitely big numbers, the masses and charges are renormalized, and finite predictions are obtained.
In the renormalization scheme, the parameters of a particle depend on the scale at which the particle is examined. That is, the mass, charge and coupling constants change according to the distance scale at which they are perceived. If a process is calculated beyond leading order, divergences can arise in the integrations over momenta in closed loops of Feynman diagrams. These divergences get contributions from all levels of energy, i.e. from all distances, down to zero.

One can introduce a regulator or UV cutoff that limits the maximal energy up to which one integrates the loop integrals. The cutoff scale can be absorbed at all levels of energy into an effective cutoff dependent bare coupling constant. The cutoff disappears when renormalizing (reparametrizing) theory in terms of a new coupling constant that is normalized with some physical input at some energy scale. Cutoff independent predictions can then be made for all processes; renormalization thus removes the dependence on the ultraviolet cutoff. The fact that the same physics can be described at different renormalization scales $\Lambda$, gives rise to the renormalization group equation, the solution of which is the running coupling constant. In QCD one obtains

$$\frac{d\alpha_{s}(Q^{2})}{d lnQ^{2}}=\beta (\alpha_{s}(Q^{2}))$$ (32)

where $\beta$ can be determined perturbatively, $\beta(\alpha)$ = $-b\alpha^{2}+O(\alpha^{3})$, and $b$ is a constant. This is at best good for large $Q^2$. Thus

$$\alpha(Q^{2})\approx \frac{1}{b{\hspace{2mm}} ln(Q^{2}/\Lambda^{2})}$$ (33)

for large values of $Q^2$.
The important role of a renormalization scheme is to establish a quantitative connection between the constants in the Lagrangian and some measurable quantities.

In conclusion, the present theories of elementary particle interactions can be understood deductively as a consequence of symmetry principles and renormalizability. Different interactions are apparently governed by different symmetries, but according to the unification philosophy, these different symmetries may be the remnants of a symmetry that appear as broken in our low energy world.
The symmetry principle at the heart of the matter, is gauge symmetry. Gauge symmetries are believed to govern electroweak, strong and gravitational interactions. However, in the electroweak model not all forces are gauge forces. Due to the phenomenon of spontaneous symmetry breaking there are, in the electroweak model, non-gauge interactions mediated by the "Higgs particle".