Confused About Lorentz Generators: A Guide

[nigelscott]

I am looking at the generators of the Lorentz group. The literature commonly refers to the generators as
M_ij, J_i and K_i and defines:

J_i = (1/2)∈_ijkM_jk

I am confused about the factor of (1/2) in this equation as I thought that M_ij is essentially the same as J_i

This also shows up in

Λ= exp((1/2)Ω_ρσM^ρσ) ≡ 1 + (1/2)Ω_ρσM^ρσ

see http://www.phys.ufl.edu/~fry/6607/lorentz.pdf

 

[dextercioby]

The 1/2 is needed not to count the same term of the sum twice, because the epsilon and M have the same index symmetry, i.e. they are both antisymmetric over the two indices summed over.

 

[nigelscott]

OK. Just to clarify. For M₂₃ this would look like:

J₁ = (1/2)[∈₁₂₃M₂₃ + ∈₁₃₂M₃₂]

Since M₂₃ = -M₃₂ and ∈₁₂₃ = -∈₁₃₂ the second term is additive.

Correct?

 

[dextercioby]

Yes, that is correct.