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Renormalization and Quantum Field Theory

In physics when fields interact certain of their characteristics such as their field strengths or the masses of their quanta get modified (renormalized) from their free field values. For example, an electromagnetic field of electric and magnetic field strengths of E and B, respectively, in the presence of a permeable medium get modified to D=ε0(1+χe)E and H=μ0(1+χm)B. It is the values of D and H which are observed. In solid state an electron moving in a conduction band appears to have an effective (renormalized) mass greater than its actual mass.

An observed or renormalized characteristic can be considered the sum of the bare value and the effect induced by interaction. The induced effect is the result of nonlinearities in the equations describing the fields. The induced effects are finite when there is a lower bound to the size of the configurations involved in interactions. This is the case for atomic and particle phenomena. When vacuum units of space are involved there is no such lower bound and the induced effect is divergent; i.e., infinite. This has led to the notion that renormalization has solely to do with the removal of infinities. This not the case.

There are two approaches to renormalization: the Subtractive approach and the Counter Term approach. In the Subtractive approach the induced effect is analyzed into components and those that lead to infinities are identified and subtracted from the analysis. It has been found that this effectively reduces to renaormalizing coupling constants and the masses of field quanta. The proof of renormalizability in this approach uses a set of equations involving the Green functions known as the Dyson equations. This is a set of coupled integral equations. (The term Green function refers to the mathematician George Green.)

The Counter Term approach starts with the Lagrangian for the fields and their interaction. The Lagrangian consists of terms for the bare (noninteracting) fields and terms involving the fields interaction. The Counter Term approach involves transferring specific terms from free fields part to the interaction part. Such transfers are limited by the requirement that the single-particle states must remain stable.

The Lagrangian of the Interaction of Fields

Let ψ(x) be the electron-positron field strength and aμ(x) for (μ=1,…,4) be the vector meson fields strengths. The total action L is given by ∫dx4L(x), where L(x) is the Lagrangian density, which is given by three components: One due to the electric field Lel, one due to the radiation field of the vector mesons Lrad, and one due to their interaction Lint. These are given by:

Lel = −ψμ + m)ψ
Lrad = −½(∂νμνμ + κ²aμaμ)
Lint = −ie(ψγμψaμ)

where the Einstein summation convention applies; i.e, a repeated index in a terms means summation over that index. The expression ∂μ stands for ∂/∂xμ. The parameters m and e are the mass and charge (e>0) of the electron, respectively. The symbol i stands for the square root of −1 and ψ stands for the conjugate of ψ. The symbol γ stands for characteristics of space and time; i.e., γμ=1 for μ=1 to 3 and γ4=i=√(−1). It is assumed that h and c, Planck's constant and the speed of light, are both equal to unity.

(To be continued.)


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