How to show gamma matrices are unique?

In summary, the conversation discusses the uniqueness of gamma matrices in physics textbooks, and how they can be proven using the Clifford algebra. There is also mention of similarity transformations and the concept of basis in this context. The main point is that any two sets of matrices satisfying the Clifford algebra can be linked by a similarity transformation.
  • #1
kof9595995
679
2
In most of the physics textbooks I read they only give one or two representations of gamma matrices, but none gives a proof, so how can I prove it from the Clifford algebra?
 
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  • #2
kof9595995 said:
In most of the physics textbooks I read they only give one or two representations of gamma matrices, but none gives a proof, so how can I prove it from the Clifford algebra?

? The form of gamma matrices is not unique in general.
Consider similarity transformations like:

[tex]
\gamma' ~=~ W \, \gamma \, W^{-1}
[/tex]

Such new gammas still satisfy the defining Clifford algebra relations.

(Or did I misunderstand your question?)
 
  • #3
I mean, given a basis, gamma matrices must be unique, aren't they?
 
  • #4
kof9595995 said:
I mean, given a basis, gamma matrices must be unique, aren't they?

Er, what precisely do you mean by "basis" in this context?

(I'm still having trouble understanding what point is bugging you... :-)
 
  • #5
Maybe I should rephrase it: Any two sets of matrices satisfying Clifford algebra, can be linked by a similar transformation.
I think this should be a quite famous theorem, but I don't know where to find it.
 

1. What are gamma matrices and why are they important in physics?

Gamma matrices are mathematical objects used in the study of quantum mechanics and special relativity. They represent the generators of rotations and boosts in the Dirac equation, which describes the behavior of spin-1/2 particles. Gamma matrices are important because they provide a way to mathematically describe the behavior of fundamental particles in the universe.

2. How do we know that gamma matrices are unique?

The uniqueness of gamma matrices can be proven mathematically using the properties and relationships between the matrices. One way to show this is by using the properties of the Pauli matrices and the relationship between the gamma matrices and the Pauli matrices.

3. Can gamma matrices be derived from other mathematical concepts?

Yes, gamma matrices can be derived from the Clifford algebra, which is a mathematical concept used to study geometric transformations. The gamma matrices are a representation of the Clifford algebra in the Dirac basis.

4. Are there different ways to represent gamma matrices?

Yes, there are different ways to represent gamma matrices, including the Dirac, Weyl, and Majorana representations. Each representation has its own unique properties and can be used to solve different problems in physics.

5. How are gamma matrices used in experiments and real-world applications?

Gamma matrices are used in a wide range of experiments and applications, including particle physics, quantum computing, and cosmology. They play a crucial role in understanding the behavior of particles and predicting their interactions in various physical systems.

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