# What does it mean for a theory to be renormalizable

• Lemaho
In summary, renormalizable means that a theory can either have only finite expressions or can absorb infinities by redefining a finite number of quantities. It is usually referred to as "perturbatively renormalizable", but this can be misleading as a theory may appear unrenormalizable when treated perturbatively but may behave differently when treated non-perturbatively. The concept is closely related to the scaling dimension of a theory.
Lemaho
Dear all,

I have a simple question with a possibly long answer.
What does it mean for a theory to be renormalizable. And how is it related to the scaling dimension?

Renormalizable means that either the theory contains only finite expressions or that infinities can be absorbed by redefining finitly many quantities. A theory is not renormalizable if you need infinitly many quantities to absorb the infinities.

Usually renormalizable is used in the sense of "perturbatively renormalizable", but this is misleading. It can very well be that a theory seems to be unrenormalizable when treated perturbatively, but this behavior can simply be due to the (unjustified) perturbation expansion and may vanish when treated non-perturbatively.

## 1. What is the definition of a renormalizable theory?

A renormalizable theory is a theory that can be used to describe physical phenomena at different scales without encountering infinite or non-physical results. It is a theoretical framework that allows for the mathematical treatment of divergent quantities in a way that still produces meaningful and accurate results.

## 2. How is a theory determined to be renormalizable?

A theory is considered renormalizable if the divergent quantities in its mathematical equations can be absorbed into a finite number of parameters. This process is known as renormalization and involves making changes to the theory's parameters in order to account for the effects of very small or very large scales.

## 3. What is the significance of a renormalizable theory?

A renormalizable theory is significant because it allows for the prediction of physical phenomena at different scales, from the very small to the very large. It also allows for the calculation of physical quantities with high precision, making it a useful tool in theoretical physics and particle physics.

## 4. Can all theories be renormalized?

No, not all theories can be renormalized. Some theories, such as those that involve gravity, cannot be renormalized because they lead to infinite or non-physical results that cannot be absorbed into a finite number of parameters.

## 5. How does renormalizability relate to the search for a theory of everything?

The concept of renormalizability is important in the search for a theory of everything because it allows for the merging of different theories at different scales. A theory of everything must be able to accurately describe physical phenomena at all scales, and the ability to renormalize a theory is a key aspect in achieving this goal.

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