A question about the derivation of Fermion Quantization in QFT

In summary, Fermion quantization is a crucial aspect of Quantum Field Theory (QFT) that allows for the description of fundamental particles with half-integer spin, such as electrons, protons, and neutrons. It involves converting classical fields into quantum fields and using creation and annihilation operators to create and destroy fermion particles. This process is derived by applying quantum mechanics principles to classical fields and provides a framework for understanding fermion behavior in a quantum field. While specific to fermions, the principles of QFT can be applied to all types of particles, making it a valuable tool in various fields such as particle physics, condensed matter physics, and nuclear physics. Additionally, fermion quantization has practical applications in predicting and calculating particle interactions
  • #1
Andrew3
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Homework Statement
I am studying the part of Fermion Quantization in QFT. However, I am puzzled about the derivation in one class in youtube.
https://www.youtube.com/watch?v=FNxespJgDDE&list=PLbMVogVj5nJQ3slQodXQ5cSEtcp4HbNFc&t=826s
At 13:46, why does (sigma^3)_a^c (sigma^1)_{cb}=(sigma^3 sigma^1)_{ab}?
Relevant Equations
sigma^0 is the unit. sigma^k is the pauli matrices (k=1,2,3). 'a','b' and 'c' denotes the indexes of matrices elements.
fermionqm1.png
fermionq0.png
fermionq1.png
fermionq2.jpg
 

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  • #2
Because contraction of the ##c## index is just an index form of noting matrix multiplication. What he wrote is just matrix multiplication law for those matrices. For example, the element of product of two matrices is:
$$(AB)_{mn} = \sum_{k}A_{mk}B_{kn}$$
In the notation on the board, the sum is implied over indices that appear twice(that's Einstein summation convention), and also lifting and lowering indices is probably trivial since metric is trivial.
 

FAQ: A question about the derivation of Fermion Quantization in QFT

1. What is Fermion Quantization in QFT?

Fermion quantization in QFT refers to the process of quantizing fermionic fields, which are fields associated with particles that follow the Fermi-Dirac statistics. This is an essential step in the development of quantum field theory, as it allows for the description of fermionic particles in terms of quantum fields.

2. Why is Fermion Quantization important in QFT?

Fermion quantization is important in QFT because it allows for the description of fermionic particles, such as electrons and protons, in terms of quantum fields. This is crucial for understanding the behavior of these particles at a quantum level and for predicting their interactions with other particles.

3. What is the derivation of Fermion Quantization in QFT?

The derivation of Fermion Quantization in QFT involves applying the principles of quantum mechanics and special relativity to fermionic fields. This results in the quantization of these fields, meaning that their values are restricted to discrete levels rather than continuous values.

4. What are the implications of Fermion Quantization in QFT?

The implications of Fermion Quantization in QFT are far-reaching. It allows for the development of a consistent and mathematically rigorous theory of fermions at a quantum level, which has been successfully applied in many areas of physics, including particle physics and condensed matter physics.

5. Are there any challenges or limitations to Fermion Quantization in QFT?

While Fermion Quantization in QFT has been successful in describing the behavior of fermionic particles, it does face some challenges and limitations. For example, it does not fully account for the effects of gravity, and there are still ongoing efforts to unify quantum mechanics and general relativity. Additionally, some aspects of fermionic particles, such as their mass and charge, are still not fully understood within the context of QFT.

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