Solving Simple Matrix Algebra Homework Problem

In summary, Ryder thinks that the commutation relations are not satisfied by K = \pm i\frac{\sigma}{2}. He makes a substitution and gets the correct relation.
  • #1
Jimmy Snyder
1,127
21

Homework Statement


This is from Ryder's QFT book, second ed. page 37. At the bottom of the page it says that the commutation relations (eqn 2.68?) are satisfied by:
[itex]K = \pm i\frac{\sigma}{2}[/itex]
However, I do not find this to be so. What am I missing?

Homework Equations


Here is one of the commutation relations that I think he means.
[itex][K_x,K_y] = -iJ_z[/itex]

The Attempt at a Solution


Using [itex]K = i\frac{\sigma}{2}[/itex], I get:
[itex][K_x,K_y] = [i\frac{\sigma_x}{2},i\frac{\sigma_y}{2}] = \frac{-1}{4}[\sigma_x,\sigma_y] = -\frac{1}{2}\sigma_z = iK_z \neq -iJ_z[/itex]
 
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  • #2
Isn't there an i in the commutation relations of the Pauli matrices as well?
 
  • #3
yep,
[tex]
\left[\sigma_j,\sigma_k\right]=2i\epsilon_{jkl}\sigma_l
[/tex]
 
  • #4
Thanks Dick. Here is the corrected attempt. I still don't get the right commutation relation.

[itex][K_x,K_y] = [i\frac{\sigma_x}{2},i\frac{\sigma_y}{2}] = \frac{-1}{4}[\sigma_x,\sigma_y] = -i\frac{\sigma_z}{2} = -K_z \neq -iJ_z[/itex]
 
  • #5
yes, you do get the right relation. Ryder is talking about (2-component) Pauli spinors for which [tex]J_z=\frac{\sigma_z}{2}[/tex].

Look at equation (2.74). That is a boost and a rotation of a 2-component spinor where the rotation generator is
[tex]\frac{\vec \sigma}{2}[/tex] and the boost generator is [tex]i\frac{\vec \sigma}{2}[/tex].
 
  • #6
olgranpappy said:
[tex]J_z=\frac{\sigma_z}{2}[/tex].
Thanks olgranpappy, your reply is what I needed. If I make the substitutions [tex]K = i\frac{\sigma}{2}[/tex] and [tex]J = \frac{\sigma}{2}[/tex], then I get:

[tex][K_x,K_y] = [i\frac{\sigma_x}{2},i\frac{\sigma_y}{2}] = -i\frac{\sigma_z}{2} = -iJ_z[/tex] just as in (2.68)

I have also verified the other relations in (2.68). I came close to solving it this morning as I was driving to work. It occurred to me that there might be a typo in the book and that the author meant J instead of K in eqn (2.69). If I had followed that thought a while longer, I might have come up with the solution on my own. Thanks again for your help.
 
  • #7
no problem. cheers.
 

FAQ: Solving Simple Matrix Algebra Homework Problem

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Matrix algebra is a branch of mathematics that deals with the study and manipulation of matrices, which are rectangular arrays of numbers or symbols. It involves operations such as addition, subtraction, multiplication, and division of matrices to solve various problems.

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