Callan-Szymanzick equation and renormalization

[eljose79]

When i coursed the quantum field theory at university our teacher told us about this equation..i have searched information aobut it in many books and have the form...it seems is a partial-differential equation but i have doubts..

a)what is have to do with renormalization?...(in fact how to use it in renormalization theory
b)If theory is not renormalizable then..what would be the c-s equation then?.exist if not why?

hope someone can help...

 

[vanesch]

C-Z equation

In QFT, the objects from which everything follows (well...) are the green functions, (the vacuum expectations of products of field operators) but usually those greenfunctions blow up. So we use a cutoff scale to make those green functions finite (but dependent on the cutoff). As such, we can calculate what are the masses, charges, field renormalizations etc... as a function of the cutoff. When we say, for example, what is the charge of an electron, we actually talk about the outcome of an experiment to be performed, for example: put an electron in such and such an E-field and look at its acceleration. That's a non-trivial operation in QFT, and it turns out that this outcome, "physical charge" is a function of the bare charge e0 and the cutoff in a complicated way. So we can solve for e0 as a function of the physical charge e and the cutoff. But the "physical charge" is defined at a certain energy scale: for example, we consider the electron at rest (energy is the rest mass), or in motion or whatever. So this procedure depends on what is called the "renormalisation scale" M. Renormalization is nothing else but substituting everywhere e0 by the function of the cutoff, the physical charge e and the renormalization scale M. In a renormalizable theory, we then get finite answers, even in the limit of the cutoff going to infinity. In that limit, there's still a functional dependence of every physical quantity on the physical charge e and the renormalization scale. The C-Z equation is a partial differential equation that sets a condition on the dependence on e and on M.

cheers,
Patrick.

 

[R0man]

The Callan-Symanzik equation is a key tool in the study of quantum field theory. It is a partial differential equation that describes how the coupling constants of a quantum field theory change as the energy scale at which the theory is studied changes. This equation allows us to understand how the theory behaves at different energy scales, and how it is affected by renormalization.

Renormalization is a technique used in quantum field theory to deal with infinities that arise in calculations. These infinities arise when we try to calculate physical quantities, such as particle masses or coupling constants, in a quantum field theory. Renormalization allows us to remove these infinities and obtain finite, meaningful results.

The Callan-Symanzik equation is closely related to renormalization because it tells us how the coupling constants of a theory change as we renormalize it. By solving this equation, we can determine how the coupling constants should be adjusted in order to obtain finite, renormalized results.

If a quantum field theory is not renormalizable, it means that it cannot be made finite through the process of renormalization. In this case, the Callan-Symanzik equation may not be applicable, as it is derived from the assumption that the theory is renormalizable. However, the equation can still be useful for understanding the behavior of the theory at different energy scales, even if it cannot be fully renormalized.

In summary, the Callan-Symanzik equation is an important tool in the study of quantum field theory, particularly in the context of renormalization. It allows us to understand how the coupling constants of a theory change as we renormalize it, and can help us obtain finite, meaningful results. Although it may not be applicable in all cases, it is still a valuable equation for understanding the behavior of quantum field theories.