The Fourier-Minkowski transform?

[jason12345]

Can anyone suggest any references explaining the motive behind it's definition?

I'm unfortunately too thick to see the necessity of the sign of the time part of the exponentials being opposite to that of the space part. It seems that the transform must preserve some property of the invariance of the space time interval, but i don't see what.

Thanks

 

[Fredrik]

Isn't it just a convention to ensure that the 4-momentum operator can be represented as $i\partial_\mu$ (when we Fourier transform a wave function)?

This question has been asked a couple of times recently, but I don't think I read the answers. Try searching for it.

 

[Entropia]

for your question about the Fourier-Minkowski transform. The Fourier-Minkowski transform is a mathematical tool that is used to analyze signals and functions in both the time and frequency domains. It is a generalization of the traditional Fourier transform, which only operates in the frequency domain. The motivation behind its definition lies in the fact that it allows for the analysis of signals that vary in both space and time, such as electromagnetic waves.

The reason for the opposite signs in the time and space components of the exponentials is related to the concept of space-time invariance. This is a fundamental principle in physics that states that the laws of physics should be the same for all observers, regardless of their relative motion. The Fourier-Minkowski transform takes into account this principle by preserving the space-time interval, which is a measure of the distance between two events in space and time. By using opposite signs in the time and space components, the transform ensures that the space-time interval remains invariant.

If you are looking for references to better understand the motivation behind the Fourier-Minkowski transform, I would recommend looking into books or articles on mathematical physics or signal processing. Some examples include "Mathematical Methods in the Physical Sciences" by Mary L. Boas and "Introduction to Fourier Analysis and Wavelets" by Mark A. Pinsky. These resources can provide a deeper understanding of the concept and its applications. Additionally, there are many online resources and tutorials available that can help explain the concept in a more approachable way. I hope this helps clarify the motivation behind the Fourier-Minkowski transform.