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Maggiore's QFT textbook error?

  1. Sep 28, 2010 #1

    MathematicalPhysicist

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    Gold Member

    In page 15 of the first edition of this textbook, in equations 2.6 and 2.7, he writes:

    (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.
     
  2. jcsd
  3. Sep 28, 2010 #2

    strangerep

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    Your latex's not quite right...

    Magiorre's eq(2.7) is

    [tex]
    i\delta_a T^a_R ~=~ \log\big\{[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]\big\}
    ~=~ \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]

    which involves abuses of the summation convention. (Actually, even (2.6) should use another
    dummy index like b in the second exponential.)

    Basically, he uses the b dummy index so that you can correctly keep track of what's
    being summed with what...
     
  4. Sep 28, 2010 #3

    MathematicalPhysicist

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    Gold Member

    OK, thanks.
    That clears this matter.
     
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