Understanding Complex Group Parameters for SO(4) Lorentz Algebra

[geoduck]

When you rewrite the angular momentum generators J_i and boost generators K_j in terms of the linear combinations N^±_i=J_i±iK_i, does this mean that your group parameters can now be complex? So for example a group element R can be written as:

$$R(z_1,z_2)=\exp[i(z_1 N^+ +z_2 N^-)]$$

where the z's are complex? z1 and z2 must be complex conjugates in order to get something of the form:

$$R(z_1,z_2)=R(x,y)=\exp[i(xJ+yK)]$$

where x and y are real group parameters instead of complex ones.

So is there an implicit rule that whatever the coefficient of N⁺, the coefficient of N^- must be the complex conjugate? So in order to specify a group element of SO(4), you have to give 3 complex numbers (one for i=1,2,3), and the coefficients in front of the N^- generators would just be the complex conjugate of those numbers?

 

[tom.stoer]

When you rewrite the angular momentum generators J_i and boost generators K_j in terms of the linear combinations N^±_i=J_i±iK_i, does this mean that your group parameters can now be complex?

Yes; this step is called complexification and what you get is

so(3,1; R) → so(3,1; C) ~ so(4; C) = sl(2; C) + sl(2; C)

So is there an implicit rule that whatever the coefficient of N⁺, the coefficient of N^- must be the complex conjugate? So in order to specify a group element of SO(4), you have to give 3 complex numbers (one for i=1,2,3), and the coefficients in front of the N^- generators would just be the complex conjugate of those numbers?

No; the complexification deals with arbitrary complex coefficients. That does not mean that complex coefficients or the whole complex algebra represent physically meaningful entities. It's an extension of a mathematical structure G(R) → G(C) in order to study the properties of G(R) in terms of the larger structure G(C); in order to do physics you may restrict yourself again to G(R).