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Quantum Field Theory

Green's Function and Renormalization

The mathematics becomes more complicated when there is interaction with the field. The simplest case is to include the source of the field in the free field equation. An additional term is inserted to the right of Eq.(1l) :
---------- (7)
where the label "o" designates quantities associated with the "bare field", and .

Invoking the Green's function technique, the solution of this equation is given by:

---------- (8)
where the first term is the free field solution and the Green's function G(x - y) inside the integral is the solution of the equation with a point source at y in the form:
---------- (9a)
where (x - y) is the delta function.

In the original formulation, y lables the source point while x is the field point. In the parlance of Quantum Field Theory the Green's function is better known as propagator denoted by , which is visualized as the propagation of virtual particles from x to x' with an infinite range of momentum via the Fourier transform (Figure 01e, and see formula below). The particle at space-time point x has x = (it, x), x = (it, -x). Similarly for the source at y.
Fourier Transform Feynman Diagram The 4-momentum k is associated with k = (iE, k), k = (-iE, k). The two particles interact via the virtual particles with different values of k (see Feynman diagram in Figure 01f). For (x - y)2 o the separation is said to be time-like or light-like meaning that the two particles can connect, otherwise they are space-like and cannot communicate with each other. It is forward in time for t > t' ; the reversed is called backward, which is interpreted as forward for anti-particle. Other often used notations are kx = kx = Et - kx, and d4k = dE d3k (all in natural unit with c = = 1).

Figure 01e Fourier Transfrom

Figure 01f Feynman Diagram

Negecting the distinction on the bare mo and renormalized mass m for the moment, the solution for Eq.(9a) is (see "contour integral" for an important step in the derivation) :
Feynman Propagator

Figure 01g Feynman Propagator [view large image]

The Feynman propagator has the important feature that includes the anti-particle as going backward in time with negative energy (in the second term), while the first term is for particle going forward in time.
NB There's some confusion about the "-" and "+" sign in the formulation. It is the consequence of adapting the convention of either (ds2 = dt2 - dx2 - dy2 - dz2) or (ds2 = -dt2 + dx2 + dy2 + dz2), i.e., either (g00 = +1, g11 = g22 = g33 = -1) or (g00 = -1, g11 = g22 = g33 = +1). The formulas up to this point is based on the former convention.
Similarly, the propagator for the ferminon is as derived from the Green's function of the Dirac equation :
---------- (9b)
where m0 is the mass of the fermion, and are the gamma matrices associated with the fermion (see "Derivation of the Dirac Equation").

Now back to renormalization, it can be shown that the "bare field" can be expressed in terms of ck's similar to the case of the free field as shown in Eq.(54), but these coefficients are now modified by an additional term related to the structure of the source. As a result the norm (length) of the eigenvectors are no longer equal to 1. To recover this definition, they have to be "renormalized" by the renormalization constant Z, which has the values ; it is equal to 1 for free field, 0 for a point source, and depends on the structure of the source in general. The renormalized field, mass, and energy are and EnR = Z-1Eno respectively. The physical mass mR is the experimentally observed mass, mo is an unspecified parameter (called "bare mass") which together with Z-1 determine a value in agreement with experiment. Since Z-1 is infinite for a point source, mo has to be slightly more or slightly less infinite to yield a finite value for mR. This technique of replacing the ignorance in detailed structure of the source by measurement is the more general definition of renormalization, although it is now more often referred to as the method to cancel the infinities in quantum field theory. For example, it involves a many-body Schrodinger equation to compute the viscosity in the Navier-Stokes equations. Since it is impossible to obtain the solution for such a complicated system, its value is determined by experiment instead. A renormalizable theory is one in which the details of a deeper scale are not needed to describe the physics at the present scale, save for a few experimentally measurable parameters (see more in the section about "Renormalizable Theories").

The problem with point mass was already apparent in the 19th century with the energy W stored in the electric field E surrounding a point charge :

Our ignorance stems from the fact that we are unable to formulate a relativistically invariant theory for extended (3-D) particles. The Superstring Theory with 1-dimensional object has already had some success in removing the infinities in Quantum Field Theory by spreading out the interacting point to an extended area to the Planckian length scale of 10-33 cm (see above insert).

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