Canonical Transformation and renormalization

• eljose79
In summary, the conversation discusses the use of Canonical Transformation and renormalization in field theory. It is proposed that by taking the Hamiltonian and applying a Canonical Transformation, a renormalizable Hamiltonian can be found and solved. However, there are concerns about the validity and complexity of this approach, and some argue that the perturbative approach is more systematic. There is also mention of using "perfect actions" to map into themselves under the renormalization group. Overall, the preference is to see the scaling procedure at work instead of hiding it through calculations.
eljose79
Canonical Transformation and renormalization...

Let be L a lagrangian of a Non-Renormalizable theory..then we could take its hamiltonian.

Then after taking Hamiltonian you could take a Canonical Transformation to find another (renormalizable) Hamiltonian..and solve it..¿why this trick is not valid?...

Hmm, I can not see why not. Perhaps it is just complicated. First it is field theory, so the canonical transformations must be generated by functional derivatives from an action or so. Secondly, even in quantum mechanics one has that ordering problems are more evident in the transformed equation, for instance if one changes to action-angle variables. Third, can one grant that the new theory is going to be renormalizable? The perturbative approach at least is systematic, an universal recipe.

Still, some people uses "perfect actions", mapping into themselves under renormalization group. Perhaps this is close to the canonical transf approach, I do not know for sure.

In any case, I prefer to see the scaling procedure working, instead of hidding it under the carpet of calculus.

1. What is a canonical transformation?

A canonical transformation is a mathematical procedure in Hamiltonian mechanics that transforms the equations of motion from one set of generalized coordinates and momenta to another set. It preserves the form of Hamilton's equations and allows for a more convenient choice of coordinates that simplifies the equations of motion.

2. What is the purpose of renormalization?

Renormalization is a technique used in quantum field theory to remove infinities that arise in calculations of physical quantities, such as particle masses and coupling constants. It is necessary because the equations of quantum field theory often produce infinite results, making the theory mathematically inconsistent. Renormalization allows for the calculation of meaningful and finite physical quantities.

3. How do canonical transformations and renormalization relate to each other?

Canonical transformations and renormalization are both techniques used in theoretical physics to simplify and make sense of complex equations. Canonical transformations are used in classical mechanics, while renormalization is used in quantum mechanics. However, both techniques involve transforming equations in a way that preserves certain physical properties, such as symmetries and conservation laws.

4. What are the benefits of using canonical transformations and renormalization?

The main benefit of using canonical transformations and renormalization is that they allow for the simplification and understanding of complex physical theories. In classical mechanics, canonical transformations can simplify equations of motion and reveal underlying symmetries. In quantum field theory, renormalization allows for the calculation of finite physical quantities and the elimination of infinities.

5. Can canonical transformations and renormalization be applied to any physical system?

Yes, canonical transformations and renormalization can be applied to a wide range of physical systems, including classical and quantum mechanical systems. However, the specific techniques used may vary depending on the system and the desired outcome. These techniques are commonly used in the study of fundamental particles and their interactions, but they can also be applied to other areas of physics, such as condensed matter and statistical mechanics.

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