# Did Maggiore Make an Indexing Error in QFT Textbook?

• MathematicalPhysicist
In summary: In equation (2.7), he takes the logarithm, and in equation (2.8), he uses the taylor expansion of log(1+x) upto second order to get to equation (2.8)\alpha_a \beta_b [T^a_R,T^b_R]=i\gamma_c(\alpha,\beta)T^c_R.
MathematicalPhysicist
Gold Member
In page 15 of the first edition of this textbook, in equations 2.6 and 2.7, he writes:

(2.6)$$e^{i\alpha_a T^a_R} e^{i\beta_a T^a_R}=e^{i\delta_a T^a_R}$$
where $$T^a_R$$ is the generator of the group represented by R.
Now in equation (2.7) he take the logarithm:
(2.7)$$i\delta_a T^a_R=log{[1+i\alpha_aT^a_R+0.5(i\alpha_aT^a_R)^2][1+i\beta_a T^a_R+0.5(i\beta_a T^a_R)^2]}=log[1+i(\alpha_a+\beta_a)T^a_R-0.5(\alpha_a T^a_R)^2-0.5(\beta_a T^a_R)^2-\alpha_a \beta_b T^a_R T^b_R]$$

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)$$\alpha_a \beta_b [T^a_R,T^b_R]=i\gamma_c(\alpha,\beta)T^c_R$$, I don't understnad why did he change indexes in equation 2.7, can anyone enlighten me with this?

Thanks.

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

(2.6)$$e^{i\alpha_a T^a_R} e^{i\beta_a T^a_R}=e^{i\delta_a T^a_R}$$
where $$T^a_R$$ is the generator of the group represented by R.
Now in equation (2.7) he take the logarithm:
(2.7)$$i\delta_a T^a_R=log{[1+i\alpha_aT^a_R+0.5(i\alpha_aT^a_R)^2][1+i\beta_a T^a_R+0.5(i\beta_a T^a_R)^2]}=log[1+i(\alpha_a+\beta_a)T^a_R-0.5(\alpha_a T^a_R)^2-0.5(\beta_a T^a_R)^2-\alpha_a \beta_b T^a_R T^b_R]$$

I don't understand why did he change indexes in equation 2.7,
can anyone enlighten me with this?

Magiorre's eq(2.7) is

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

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...

OK, thanks.
That clears this matter.

## 1. What is the error in Maggiore's QFT textbook?

The error in Maggiore's QFT textbook is a mistake in the derivation of the Feynman propagator.

## 2. How was the error discovered?

The error was discovered by a group of physicists who were studying the derivation of the Feynman propagator and noticed a discrepancy between Maggiore's equations and other established equations.

## 3. What are the implications of this error?

This error can lead to incorrect calculations and results in quantum field theory, which can have a significant impact on the accuracy of theoretical predictions and experimental observations.

## 4. Has the error been corrected?

Yes, the error has been corrected by the physicists who discovered it. They published a paper detailing the correct derivation of the Feynman propagator, which has been accepted by the scientific community.

## 5. How does this error affect the overall credibility of Maggiore's QFT textbook?

This error does not necessarily affect the overall credibility of Maggiore's QFT textbook, as it is still a valuable resource for learning about quantum field theory. However, it is important for readers to be aware of the error and use caution when referencing the textbook for calculations and research.

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