Exploring Dirac Matrices and Their Dimensions

In summary: And this is the point.In summary, The Dirac Equation from Peskin-Schroeder states that the Dirac matrices for four-dimensional Minkowski Space must be at least 4X4. This is proven by a general theorem for Clifford algebras, which determines the size of the matrices for every spacetime dimension. The original article by Pauli also gives an argument for why this is the case. While the matrices can be of higher dimensions, this does not change the fact that there are only four degrees of freedom.
  • #1
praharmitra
311
1
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.
 
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  • #2
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
tom.stoer said:
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
praharmitra said:
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
 
  • #9
bigubau said:
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.
 

1. What are Dirac matrices and what are their dimensions?

Dirac matrices are a set of mathematical tools used in quantum mechanics to represent the spin and angular momentum of particles. They are 4x4 matrices, meaning they have 4 rows and 4 columns.

2. How are Dirac matrices used in physics?

Dirac matrices are used to describe the spin and angular momentum of particles, which are essential concepts in quantum mechanics. They are also used in the Dirac equation, which describes the behavior of fermions, such as electrons and quarks.

3. Can Dirac matrices be represented by other dimensions?

No, Dirac matrices are unique in their 4x4 dimensionality. However, they can be combined with other matrices to create larger matrices with higher dimensions.

4. What is the significance of the 4x4 dimension in Dirac matrices?

The 4x4 dimension of Dirac matrices is significant because it corresponds to the four components of the Dirac spinor, which represents the quantum state of a fermion. These components include the particle's spin and momentum in the x, y, and z directions.

5. How are Dirac matrices related to special relativity?

Dirac matrices are used to incorporate special relativity into quantum mechanics. They allow for the description of relativistic effects, such as time dilation and length contraction, in the behavior of particles.

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