# Matrix Element of (1/2, 0) Representation of the Lorentz Group

+ 1 like - 0 dislike
165 views

Let us consider the Lorentz group $SO(1, 3)$. There are two copies of $SU(2)$ in it. We specify a matrix representation of $SO(1, 3)$ by a doublet $(j, j')$; where $j$ corresponds to the SU(2) generated by $N_i^+$ and $j'$ corresponds to the SU(2) generated by $N_i^-$; $N_i^\pm$ are generators of the $SU(2)$s.

Say, I want to write a matrix element of the $(0, \frac{1}{2})$ representation of $SO(1, 3)$. How can I mathematically construct such an element? A detailed discussion would be highly appreciated.

asked Oct 22, 2016
recategorized Oct 22, 2016

Represent the first SU(2) trivially and the second in its fundamental representation. Represent the desired infinitesimal  generator of SO(1,3) as a linear combination of the generators of the two SU(2)s. Then you can take the matrix elements without problems.

@ArnoldNeumaier  The problem is that the generators and the matrix elements of the $j = 0$ representation of the $SU(2)$ are $1 \times 1$ matrices. But the generators and the matrix elements of the $j' = 1$ representation of the $SU(2)$ are $2 \times 2$ matrices. How can I take their linear combination, when we can't add $1 \times 1$ matrices with the $2 \times 2$ matrices?

You need to take the tensor product of the representations, not (as your comment implies that you currently do) the direct sum. Note that $1 \otimes 2 = 2$. This means that your Hilbert space is $1\times 2=2$-dimensional and the action of the generators of the 0-representation is identically zero.

@ArnoldNeumaier: Would you mind if I request you write an equation?

A general element of $so(1,3)$ is $X= a\cdot N_1+b\cdot N_2$ with 3-vectors $a,b$, and $X(1\otimes \psi_2)=1\otimes b\cdot N_2 \psi_2$.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.