Alternate Representations of Lorentz Generators Mμν

[sshaep]

I know the representations of Lorentz generators M\mu\nu as 4X4 matrices.

This matrices satisfy the commutation relation(Lie algebra of O(3,1))

However I think these 4X4 matrix representations are not unique.

Is there any other representations satisfying the commutation relation? 2X2 matrix or

another?

 

[Ben Niehoff]

Yes, there are infinitely many representations of the Lorentz algebra. One trivial example is to form a set of 8x8 matrices by taking the Kronecker product of M with the 2x2 identity.

 

[George Jones]

Is there any other representations satisfying the commutation relation? 2X2 matrix or another?

Because SL \left(2 , \mathbb{C} \right) is the universal cover of the restricted Lorentz group, they have isomorphic Lie algebras.

 

[sshaep]

Because SL \left(2 , \mathbb{C} \right) is the universal cover of the restricted Lorentz group, they have isomorphic Lie algebras.

This means can I get the 2x2 representation of M\mu\nu from SL(2,C)?

But how? What I know is \sigma_{\mu}A^{\mu} _{\phantom{\mu}\nu}}=L\sigma_{\nu}L^{\dagger}

 

[samalkhaiat]

Take

<br /> M^{0i} = \frac{i}{2} \sigma^{i}, \ \ M^{ij} = \frac{1}{2} \epsilon^{ijk}\sigma^{k}<br />

Also, if you define

\omega_{i0} = 2 Re(\alpha_{i}), \ \ \omega_{ij} = 2 \epsilon^{ijk} Im(\alpha_{k})

then

<br /> 1 + \frac{i}{2}\omega_{\mu\nu}M^{\mu\nu} = \left( \begin {array}{rr} (1 + \alpha_{3}) &amp; (\alpha_{1} - i \alpha_{2}) \\ (\alpha_{1} + i \alpha_{2}) &amp; (1 - \alpha_{3}) \end {array} \right)<br />

represents elements of SL(2,C) infinitesimally close to the identity element.

regards

sam