Would a theory with ONLY logarithmic divergencies be renormalizable ?

In summary, the conversation discusses the concept of renormalizability in a well-behaved theory where only logarithmic divergences occur. It is suggested that QED works because only logarithmic divergencies appear and the integral can be regularized using Hadamard finite part integral. The discussion also mentions that renormalizability refers to the possibility of absorbing the divergences into redefinitions of constants, and the nature of the divergence does not affect this concept.
  • #1
zetafunction
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the idea is let us supppose we have a well behaved theory where only logarithmic divergences of the form

[tex] \int_{0}^{\infty} \frac{dx}{x+a}=I(a) [/tex] for several values of 'a' occur ,

then would this theory be renormalizable ?? , i think QED works because only logarithmic divergencies appear , in fact the integral above I(a) can be regularized in the sense of Hadamard (or either differentiating respect to 'a' ) in the form

[tex] I(a)= -log(a)+c_a [/tex] here c_a is a free parameter to be fixed by experiments... Hadamard finite part integral says that


[tex] \int_{0}^{\infty} \frac{dx}{x+a}=I(a)= \int_{-\infty}^{\infty}dx \frac{H(x-a)}{x} [/tex]

so in the sense of distribution theory the integral I(a) exits and is equal to log(a)
 
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  • #2
Yes, but not because of the nature of the divergence. Renormalisability refers to whether it's possible to absorb the divergences into some redefinitions of constants. If this can be done with a finite number of constants (which would then be determined by experiment), then it's called renormalisable.

The log divergence just means that we can be happy about the accuracy of the theory, since it means that the difference between the bare value and the measured value is fairly small. (Remember that usually, the bare value is not the value at infinity energy, but something large, like Plank scale.) This produces confidence in the convergence rate of perturbation expansions.
 
  • #3
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Yes, a theory with only logarithmic divergences would be renormalizable. This is because logarithmic divergences are considered to be mild divergences and can be renormalized using the process of dimensional regularization. In fact, QED (Quantum Electrodynamics) is an example of a theory that only has logarithmic divergences and is renormalizable.

The process of dimensional regularization involves extending the number of dimensions in the theory from 4 to d and then taking the limit as d approaches 4. This allows for the cancellation of the logarithmic divergences, making the theory finite.

In the case of the integral \int_{0}^{\infty} \frac{dx}{x+a}, the Hadamard finite part integral provides a way to regularize the divergence and obtain a finite result. This is because the Hadamard finite part integral can be interpreted as the limit of a sequence of integrals with a cutoff parameter, which can then be renormalized using dimensional regularization.

Therefore, in summary, a theory with only logarithmic divergences can be renormalizable using the process of dimensional regularization and the Hadamard finite part integral, making it a well-behaved and consistent theory.
 

1. What is a logarithmic divergence?

A logarithmic divergence is a type of divergence that occurs in quantum field theory calculations when the energy or momentum of a particle approaches infinity. It is characterized by a logarithmic dependence on the energy or momentum in the calculation, resulting in an infinite value.

2. How does a theory with only logarithmic divergences differ from other theories?

A theory with only logarithmic divergences is considered to be less divergent than other theories, such as those with power-law divergences. This is because the logarithmic divergence is slower and does not grow as quickly as power-law divergences, making it easier to handle in calculations.

3. Can a theory with only logarithmic divergences be renormalizable?

Yes, a theory with only logarithmic divergences can be renormalizable. Renormalizability is determined by the behavior of the divergences in a theory, not the type of divergence. If the logarithmic divergences can be absorbed into the parameters of the theory, it can still be renormalized.

4. What are the implications of a theory with only logarithmic divergences being renormalizable?

If a theory with only logarithmic divergences is renormalizable, it means that it can be used to make meaningful and accurate predictions about physical phenomena. It also suggests that the theory is self-consistent and does not require any additional parameters or corrections to produce finite results.

5. Are there any real-life examples of theories with only logarithmic divergences being renormalizable?

Yes, there are several examples of renormalizable theories with only logarithmic divergences, such as the Gross-Neveu model in 2 dimensions and the Thirring model in 2 dimensions. These theories have been successfully used in particle physics and condensed matter physics to make predictions and explain experimental results.

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