- #1

MathematicalPhysicist

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(2.6)[tex]e^{i\alpha_a T^a_R} e^{i\beta_a T^a_R}=e^{i\delta_a T^a_R}[/tex]

where [tex]T^a_R[/tex] is the generator of the group represented by R.

Now in equation (2.7) he take the logarithm:

(2.7)[tex] i\delta_a T^a_R=log{[1+i\alpha_aT^a_R+0.5(i\alpha_aT^a_R)^2][1+i\beta_a T^a_R+0.5(i\beta_a T^a_R)^2]}=log[1+i(\alpha_a+\beta_a)T^a_R-0.5(\alpha_a T^a_R)^2-0.5(\beta_a T^a_R)^2-\alpha_a \beta_b T^a_R T^b_R][/tex]

and I don't understand from where did he get the term with the b's, I guess it should a's instead of b's, but then again he writes that he uses the taylor expansion of log(1+x) upto second order to get to equation (2.8)[tex]\alpha_a \beta_b [T^a_R,T^b_R]=i\gamma_c(\alpha,\beta)T^c_R[/tex], I don't understnad why did he change indexes in equation 2.7, can anyone enlighten me with this?

Thanks.