Dirac spinors and commutation

In summary, the conversation is about Dirac spinors and the confusion surrounding the product u^r u^s\bar{u}^s and whether it is the same as u^s\bar{u}^s u^r. The conclusion is that they are not the same, as u^r is a vector while u^s\bar{u}^s is a matrix. The conversation also touches on the completeness relation for spinors and the concept of tensor products.
  • #1
earth2
86
0
Hey guys,

i'm stuck (yet again! :) )

I am somewhat confused by Dirac spinors [tex]u,\bar{u}[/tex]. Take the product (where Einstein summation convention is assumed):

[tex]u^r u^s\bar{u}^s[/tex] Is this the same as [tex]u^s\bar{u}^s u^r[/tex]? Probably not because u^r is a vector while the other thing is a matrix, right?

Cheers,
earth2
 
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  • #2
So it's a sum after the spinor index s ? Well, then [itex]u^r u^s\bar{u}^s = u^s\bar{u}^s u^r [/itex], because the sum of products is a scalar wrt the Lorentz transformations and is a bosonic variable, as it has Grassmann parity 0.
 
  • #3
Thanks! But i don't get it if i look at it in terms of vectors and matrices...

So, [tex]u^s\bar{u}^s[/tex]=4x4 matrix where [tex]u^r[/tex] is a 1x4vector. How can i then have vector times matrix = matrix times vector?
 
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  • #4
Well, actually, No, actually the [itex]u^s\bar{u}^s[/itex] is not well defined, the barred spinor should always be put on the left, so that [tex]\bar{u}^s u^s[/tex] becomes just an ordinary complex number which commutes with everything, that's why you can switch it around.

EDIT: It is well defined, as a tensor product. See the below comments.
 
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  • #5
But look for instance at the completeness relation for spinors. It is nothing but

\slashed{P}= [tex]u^s\bar{u}^s[/tex] with a sum over s. I.e. it is a matrix :) See Peskin Schröder in the beginning... :)
 
  • #6
Hmm, you're right, I guess. It's a tensor product. Why didn't I realize that ? :)) So in that case, the answer to your initial question is NO, you can't switch them around. A line is multiplied by a sq. matrix and not viceversa.
 
  • #7
:) Thanks!
 

1. What are Dirac spinors?

Dirac spinors are mathematical objects used in quantum field theory to describe the spin (intrinsic angular momentum) of particles. They are represented by four-component vectors that obey certain mathematical properties.

2. How do Dirac spinors commute?

Dirac spinors commute by following the rules of Clifford algebra. This means that the order in which the spinors are multiplied does not affect the final result, as long as the order of the individual components within each spinor is maintained.

3. What is the significance of commutation in Dirac spinors?

Commutation in Dirac spinors is significant because it allows us to describe the behavior of fermions (particles with half-integer spin) in a consistent and mathematically rigorous way. It also helps us understand the symmetries and conservation laws in quantum field theory.

4. How are Dirac spinors used in particle physics?

Dirac spinors are used in particle physics to describe the spin states of fundamental particles, such as electrons, quarks, and neutrinos. They are also used in the calculations of scattering amplitudes and cross sections, which are important in understanding the interactions between particles.

5. Can Dirac spinors be applied to other areas of physics?

Yes, Dirac spinors have applications beyond particle physics. They have been used in condensed matter physics to describe the behavior of electrons in a crystal lattice. They also have applications in general relativity, where they are used to describe the spin of particles in curved spacetime.

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