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

• jouvelot@cri.ensmp.fr
In summary, the conversation is about two individuals trying to teach themselves QFT and getting stuck on a formula in Part I of Weinberg's trilogy. One of the individuals managed to understand Formula 2.B.7 with the help of Weinberg's information, but is struggling with Formula 2.B.10. They are both using the paperback edition and are asking for a hint on how to solve it. They acknowledge that it is a technical mathematical point and do not want to spend too much time on it.
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

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

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!

## 1. What is a homotopy class in Weinberg's QFT book?

A homotopy class in Weinberg's QFT book refers to a set of mathematical entities that are related to each other through continuous transformations. In the context of QFT, these transformations represent different ways of defining and describing the same physical phenomenon.

## 2. Why does Weinberg include an appendix on homotopy classes in his QFT book?

Weinberg includes an appendix on homotopy classes to provide a deeper understanding of the mathematical concepts and tools used in QFT. Specifically, these classes help to clarify the relationship between different mathematical formulations of QFT and how they can be interconnected.

## 3. How are homotopy classes relevant to QFT?

Homotopy classes are relevant to QFT because they allow for a more comprehensive understanding of the mathematical structures and symmetries underlying the theory. By considering different homotopy classes, we can gain insights into the physical implications of these mathematical structures and how they relate to observable phenomena.

## 4. Are homotopy classes unique in QFT?

No, homotopy classes are not unique in QFT. In fact, there can be an infinite number of homotopy classes for a given physical system. This reflects the fact that there can be multiple equivalent mathematical formulations for the same physical phenomenon in QFT.

## 5. How can I apply the concept of homotopy classes in my own research in QFT?

The concept of homotopy classes can be applied in various ways in QFT research. For example, it can be used to explore the relationship between different mathematical formulations of QFT, or to investigate the physical implications of different homotopy classes for a given system. Ultimately, understanding and utilizing homotopy classes can help to deepen our understanding of QFT and its applications.

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