page 27 of this paperI don't understand the relation between the Fourier transform and translational invariance.Let’s take advantage of translational invariance in d dimensions, ##x_μ → x_μ + a_μ## , to Fourier decompose the scalar field:
## \phi(z,x^\mu)=e^{ik_\mu x^\mu} f_k(z) ##
What does that have to do with Fourier transforming it?The Bill said:I think the author is referring to the translational invariance of the equation of motion for ##ϕ##.
He Fourier transforms the x coordinates and because the derivatives in the equation are only w.r.t. z, you can write the equation for each mode of the transformed field separately.The Bill said:The author mentioned Fourier decomposition. This is not the same as a Fourier transform.
A Fourier transform is a mathematical operation that decomposes a function or signal into its constituent frequencies. It is used in signal processing, image processing, and other fields to analyze and manipulate data.
Translational invariance refers to the property of a system or function remaining unchanged under translations, or shifts, in space or time. The Fourier transform is also invariant under translations, meaning that shifting a function in one domain will result in a corresponding shift in the transformed domain.
The translational invariance of Fourier transform allows for the separation of translation from other operations, making it a powerful tool in analyzing and manipulating data. It also allows for the efficient computation of the Fourier transform using fast Fourier transform algorithms.
Yes, Fourier transform can be applied to both continuous and discrete data. In the case of discrete data, the Fourier transform is called the discrete Fourier transform (DFT) and is commonly used in digital signal processing.
Fourier transform and translational invariance have a wide range of practical applications, including image and audio compression, signal filtering and analysis, pattern recognition, and solving differential equations. They are also used in fields such as physics, engineering, and economics.