Why Must Dirac Matrices in Minkowski Space Be at Least 4x4?

[praharmitra]

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.

 

[praharmitra]

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.

 

[tom.stoer]

Please have a look at http://en.wikipedia.org/wiki/Clifford_algebra#Basis_and_dimension

 

[praharmitra]

Please have a look at http://en.wikipedia.org/wiki/Clifford_algebra#Basis_and_dimension

Thank You. I was just typing that I managed to prove it. Thanks for the help.

 

[dextercioby]

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

https://www.physicsforums.com/showthread.php?t=49915&highlight=Pauli+Dirac+matrices

 

[A. Neumaier]

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.

They can be of higher dimension, though...

 

[dextercioby]

In 4 space-time dimensions ? How ?

 

[A. Neumaier]

In 4 space-time dimensions ? How ?

This has nothing to do with dimensions. You may for example add m zero rows and columns to your gamma matrices without altering their anticommutation relations.
And then you may apply an arbitrary unitary similarity transform to get rid of the zeros altogether.

Of course, this doesn't alter the fact that you only get four degrees of freedoms.