# Dirac Matrices

I am currently reading Dirac Equation from Peskin-Schroeder. In a particular para it says,

"Now let us find Dirac Matrices $$\gamma^\mu$$ for four-dimensional Minkowski Space. It turns out that these matrices must be at least 4X4."

What is the proof of the above statement? I think (not sure) that it was once mentioned in class, that the above can be true only for a set of even dimensional matrices. Is that true? How?

And if yes, how do we know that the matrices can't be 2X2? Can someone show me a proof or guide me in the right direction.

Thanks.

tom.stoer
Of course there are Dirac matrices for 2-dim spacetime, too.

As far as I remember the statement is that if the index runs from 0 to 3 then one can show that the matrices must be at least 4*4. There is a general theorem (for Clifford algebras) that determines the size of the matrices for every spacetime dimension.

Of course there are Dirac matrices for 2-dim spacetime, too.

As far as I remember the statement is that if the index runs from 0 to 3 then one can show that the matrices must be at least 4*4. There is a general theorem (for Clifford algebras) that determines the size of the matrices for every spacetime dimension.

That is what I meant. I have reduced the problem to showing that the Dirac Matrices are traceless.

dextercioby
Homework Helper
I am currently reading Dirac Equation from Peskin-Schroeder. In a particular para it says,

"Now let us find Dirac Matrices $$\gamma^\mu$$ for four-dimensional Minkowski Space. It turns out that these matrices must be at least 4X4."

What is the proof of the above statement? I think (not sure) that it was once mentioned in class, that the above can be true only for a set of even dimensional matrices. Is that true? How?

And if yes, how do we know that the matrices can't be 2X2? Can someone show me a proof or guide me in the right direction.

Thanks.

This is a good question. The original article of Pauli gives an argument on why, in 4D-spacetime, the Dirac's matrices must be 4 by 4 http://www.numdam.org/item?id=AIHP_1936__6_2_109_0.

The "translation" in modern notation can be found in B. Thaller's book: "Dirac's Equation".

See the discussion here

dextercioby
Homework Helper
In 4 space-time dimensions ? How ?

A. Neumaier