- #1

Kane O'Donnell

Science Advisor

- 124

- 0

Hi everyone,

From the condition:

how does one formally proceed to show that the objects [tex]\gamma_{\mu}[/tex] must be 4x4 matrices? I unfortunately know very little about Clifford algebras, and for this special relativity project of mine I'd much rather not need brute force!

Cheerio!

Kane

PS: I'm using the signature (+---) for the metric tensor, although this should only change the content of the matrices, not the proof itself, I suspect.

PPS: I quite realise that it probably cannot be shown that the gamma matrices *must* be 4x4 matrices, what I want to know is if the anticommutator conditions are precisely the defining relations for a R(1,3) Clifford algebra or something like that, and how we eliminate the possibility of lower-dimensional 'isomorphisms' (don't know the correct algebra mapping term) existing.

From the condition:

[tex] \gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu} = 2g_{\mu\nu} [/tex]

how does one formally proceed to show that the objects [tex]\gamma_{\mu}[/tex] must be 4x4 matrices? I unfortunately know very little about Clifford algebras, and for this special relativity project of mine I'd much rather not need brute force!

Cheerio!

Kane

PS: I'm using the signature (+---) for the metric tensor, although this should only change the content of the matrices, not the proof itself, I suspect.

PPS: I quite realise that it probably cannot be shown that the gamma matrices *must* be 4x4 matrices, what I want to know is if the anticommutator conditions are precisely the defining relations for a R(1,3) Clifford algebra or something like that, and how we eliminate the possibility of lower-dimensional 'isomorphisms' (don't know the correct algebra mapping term) existing.

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