Dirac Matrices

  • #1
311
1

Main Question or Discussion Point

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

"Now let us find Dirac Matrices [tex]\gamma^\mu[/tex] 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.
 

Answers and Replies

  • #2
tom.stoer
Science Advisor
5,766
161
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.
 
  • #3
311
1
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.
 
  • #6
dextercioby
Science Advisor
Homework Helper
Insights Author
12,985
540
I am currently reading Dirac Equation from Peskin-Schroeder. In a particular para it says,

"Now let us find Dirac Matrices [tex]\gamma^\mu[/tex] 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
 
  • #8
dextercioby
Science Advisor
Homework Helper
Insights Author
12,985
540
In 4 space-time dimensions ? How ?
 
  • #9
A. Neumaier
Science Advisor
Insights Author
2019 Award
7,269
3,155
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.
 

Related Threads on Dirac Matrices

  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
4
Views
392
  • Last Post
Replies
14
Views
13K
  • Last Post
Replies
5
Views
8K
  • Last Post
Replies
6
Views
927
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
6
Views
1K
Top