Discussion Overview
The discussion revolves around the application of Fourier transforms in the context of critical phenomena, specifically addressing the transition between equations 7.37 and 7.38. Participants are exploring the mathematical intricacies involved in this transformation and the implications of convolution in Fourier space.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Homework-related
Main Points Raised
- One participant expresses difficulty in understanding how to transition from equation 7.37 to 7.38 using Fourier transforms.
- Another participant suggests multiplying equation 7.37 by the denominator of the right-hand side (RHS) and applying the convolution theorem, indicating that a product in Fourier space becomes a convolution in direct space.
- A participant questions whether the Fourier transform of C (with hat) is simply C without the hat, expressing confusion about the arguments of the functions in equation 7.38.
- One participant clarifies that the difference in variables (r1-r2) is valid in a homogeneous system, where translation invariant quantities depend on their difference rather than individual positions.
- Another participant emphasizes the need to understand that the integral represents a convolution.
- A participant mentions that despite using the convolution theorem, their results differ from the author's, particularly in the arguments of the functions, which adds to their confusion.
- A suggestion is made to rename variables to clarify the relationships in the equations.
Areas of Agreement / Disagreement
Participants express varying levels of understanding regarding the application of the convolution theorem and the interpretation of function arguments. There is no consensus on the resolution of the confusion surrounding the differences in arguments of functions under the integral versus those outside of it.
Contextual Notes
Participants reference earlier discussions in the text regarding the properties of Fourier transforms and convolution, but the specific assumptions and definitions that lead to their confusion remain unresolved.
