What are the required mathematics for studying Quantum Field Theory?

[Black Integra]

Hello everyone!
I want to study QFT and I want to know all of the required mathematics for it.
I know most of quantum mechanics topics
-wave function
-Schrödinger equation
-state, operator
-perturbation theory
I know some of special relativity. And I almost familiar with einstein's notation.
I don't know much about symmetries, group theory, and other abstract algebras.

Please list me some of the required maths for QFT.
Thanks in advance.

 

[torquil]

Some additions to those you have mentioned:

Fourier analysis
Langrangian, Hamiltonian mechanics, Principle of least action.
Lie groups and algebras
Tensors

Whether you learn it at the physics of math department can also be important. Many might not have any use for all the mathematical proofs that are involved.

 

[A. Neumaier]

Hello everyone!
I want to study QFT and I want to know all of the required mathematics for it.

Renormalization is the ugliest part of QFT, and usually not treated well mathematically.
But see http://arnold-neumaier.at/ms/ren.pdf

 

[Black Integra]

Haha
I might have to start from Lie algebras as expected. That renormalization may be the very last one.

Thank you people. :)

 

[torquil]

Some additions to those you have mentioned:

Fourier analysis
Langrangian, Hamiltonian mechanics, Principle of least action.
Lie groups and algebras
Tensors

Whether you learn it at the physics of math department can also be important. Many might not have any use for all the mathematical proofs that are involved.

I want to qualify my answer just a bit. It depends on what you mean by "required". It is possibly to study quantum field theories that do not involve e.g. Lie groups or tensors. E.g. the quantum theory of a real scalar field avoids some of the mathematical preliminaries.

However, it still involves Lagrangian/Hamiltonian mechanics, Fourier analysis and renormalisation, which is at the heart of quantum field theory if you want to consider perturbation theory beyond the lowest order approximation.

So Lie groups is not really the most important mathematical discpline involved in QFT, although it is used extensively in more complicated theories, e.g. gauge theories like QED.

Also, you need to know about the wave equation and its solutions.