How to show gamma matrices are unique?

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Discussion Overview

The discussion revolves around the uniqueness of gamma matrices within the context of Clifford algebra. Participants explore the implications of different representations and transformations of gamma matrices, questioning the conditions under which they can be considered unique.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant notes that many physics textbooks provide limited representations of gamma matrices without proof, prompting a request for a demonstration from the Clifford algebra.
  • Another participant argues that the form of gamma matrices is not unique due to the possibility of similarity transformations, which still satisfy the defining Clifford algebra relations.
  • A participant questions whether gamma matrices must be unique given a specific basis, seeking clarification on the definition of "basis" in this context.
  • One participant suggests that any two sets of matrices satisfying the Clifford algebra can be connected through a similarity transformation, indicating this may be a well-known theorem but expressing uncertainty about its source.

Areas of Agreement / Disagreement

Participants express differing views on the uniqueness of gamma matrices, with some asserting that they can be transformed into one another while others seek clarification on the conditions that define uniqueness.

Contextual Notes

There is ambiguity regarding the definition of "basis" and the implications of similarity transformations on the uniqueness of gamma matrices. The discussion does not resolve these uncertainties.

kof9595995
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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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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:

<br /> \gamma&#039; ~=~ W \, \gamma \, W^{-1}<br />

Such new gammas still satisfy the defining Clifford algebra relations.

(Or did I misunderstand your question?)
 
I mean, given a basis, gamma matrices must be unique, aren't they?
 
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... :-)
 
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.
 

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