Homotopy classes appendix in Weinberg's QFT book, Part I

[jouvelot@cri.ensmp.fr]

Hi,

I'm trying to teach myself QFT, and I'm stuck with one formula in Part
I of Weinberg's trilogy.

I think I managed to understand how one gets Formula 2.B.7 in the
appendix of Chapter 2, thanks to the information provided by Weinberg,
but don't get 2.B.10, page 97. Could anyone one give me a hint on how
to get it? FWIW, I'm using the paperback edition.

Since this is not key issue, but mostly a technical mathematical point,
I wouldn't want to get bogged down too long by this, but this is
getting irritating :-)

TIA,

Pierre

 

[philipp_w]

hi,

i am also trying to do the same as you do. but I got stucked on the formula (2.B.7). It seems that you have the answer, could you please provide me with that one. I am not sure what is really done to achieve this result in (2.B.7).

greets

- philipp

 

[sir-pinski]

Hi Pierre,

I understand your frustration with trying to understand the homotopy classes appendix in Weinberg's QFT book. It can be a difficult concept to grasp, especially when self-teaching.

To understand Formula 2.B.10 on page 97, it might be helpful to first review the definition of a homotopy class. A homotopy class is a set of continuous functions that can be continuously deformed into each other. In the context of QFT, this class represents different ways of connecting the initial and final states of a particle, while keeping the intermediate states fixed.

Now, Formula 2.B.10 is derived from the definition of the S-matrix, which is the transition amplitude for a particle to go from an initial state to a final state. In this case, the S-matrix is written as a path integral, which is a sum over all possible paths the particle can take.

To get to Formula 2.B.10, Weinberg uses the concept of homotopy classes to simplify the path integral by grouping together paths that are in the same class. This allows for a more efficient calculation of the S-matrix.

I hope this explanation helps to clarify Formula 2.B.10. If you need further assistance, I would recommend seeking out a study group or a tutor who can provide more in-depth explanations and examples. Don't get discouraged, QFT can be a challenging subject, but with perseverance, you will eventually grasp the concepts.

Best of luck in your studies!