What are the required mathematics for studying Quantum Field Theory?

Click For Summary

Discussion Overview

The discussion focuses on the mathematical prerequisites for studying Quantum Field Theory (QFT). Participants explore various mathematical concepts and their relevance to QFT, including foundational topics and more advanced theories.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant lists foundational topics in quantum mechanics, such as wave functions, the Schrödinger equation, and perturbation theory, as a starting point for studying QFT.
  • Another participant suggests additional mathematical topics including Fourier analysis, Lagrangian and Hamiltonian mechanics, Lie groups and algebras, and tensors.
  • A different participant highlights renormalization as a complex aspect of QFT that is often poorly treated mathematically.
  • One participant expresses the need to start with Lie algebras, indicating a progression in their mathematical understanding.
  • Another participant qualifies the necessity of certain mathematical concepts, stating that while some theories may not require Lie groups or tensors, Lagrangian mechanics, Fourier analysis, and renormalization are essential for perturbation theory.
  • It is mentioned that knowledge of the wave equation and its solutions is also important for QFT.

Areas of Agreement / Disagreement

Participants express a range of views on the required mathematics for QFT, with some suggesting that certain topics are essential while others argue that they may not be strictly necessary depending on the specific theory being studied. The discussion remains unresolved regarding the prioritization of these mathematical areas.

Contextual Notes

Some participants note that the importance of certain mathematical topics may depend on the specific approach to QFT being taken, and that the context of learning (e.g., physics vs. mathematics department) may influence the relevance of mathematical proofs.

Black Integra
Messages
56
Reaction score
0
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.
 
Physics news on Phys.org
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.
 
Black Integra said:
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
 
Haha
I might have to start from Lie algebras as expected. That renormalization may be the very last one.

Thank you people. :)
 
torquil said:
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.
 

Similar threads

High School Quantum Field Theory As Simple As Possible
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad What are anomalies in quantum field theory?
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Confusion about creation/annihilation operators with interaction
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Quantum Field theory vs. many-body Quantum Mechanics
  • · Replies 36 ·
2
Replies
36
Views
7K
Graduate Why is Quantum Field Theory Local?
  • · Replies 182 ·
7
Replies
182
Views
16K
Graduate Multiparticle Relativistic Quantum Mechanics in an external potential
  • · Replies 22 ·
Replies
22
Views
3K
Graduate Measurement in QFT: Mapping Fields to Theory's Math Formalism
  • · Replies 31 ·
2
Replies
31
Views
3K
Graduate General Covariance in Quantum Field Theory
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Classical field in quantum field theory?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Quantum field theory: an informative approach
  • · Replies 5 ·
Replies
5
Views
2K