Jacobian, Lorentz and Fourier Transformation

Question

Jacobian, Lorentz and Fourier Transformation. I am confused with the physical interpretation/meaning of all these transformations.

1. As far as I understood, Jacobian transforms from one coordinate space to another (there are examples for Jacobian from cartesian to spherical so and so...). But Lorentz transform also do similar transformation from one coordinate space to another right? So, Jacobian is also a kind of Lorentz transform! Or else what is the physical difference?

2. Finally, I got confused with the Fourier transform as well! It also transform from one space to another!.

How can we explain the physical difference and physical usefulness of all these transforms? How we decide what transformation can be done in some physical situation? Can anyone please explain with physical example and comparing these three transformations?

Answer 1

It's sloppy language that is confusing you here.

1. A Jacobian is not a transformation. The Jacobian of a transformation measures by how much the transformation expands or shrinks volume(/area/length/hypervolume/whatever) elements. Example: let $x' = 2x$. Then $dx = dx'/2$. The Jacobian is the $1/2$, meaning nothing more than "a unit of the $x'$ scale has a length of 1/2 a unit on the $x$ scale." The formula involving partial derivatives and a determinant is the generalization of this to arbitrary dimensionality & coordinate transformations, but the meaning is exactly the same.

2. Lorentz transformations are the transformations that preserve intervals in Minkowski spacetime. In terms of cartesian coordinate systems ($x,y,z,t$) they are the linear transformations that preserve $-\mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$ and their Jacobian is $1$, i.e. they don't expand or shrink spacetime volume elements $\mathrm{d}t\mathrm{d}x\mathrm{d}y\mathrm{d}z$. In crazy weird coordinates Lorentz transformations would surely look awful, and you would have to do gymnastics to keep track of their Jacobian, but that just means that if you care about Lorentz invariance you shouldn't be using crazy weird coordinate systems. :)

3. Fourier transforms are a change of basis on function space. You have a function on configuration space $f(x)$, say, and you would like to describe it by its wavenumber spectrum $\tilde{f}(k)$ instead. Fourier transform gives you the connection between the two bases. The function hasn't changed, and the function space hasn't changed either, but you are using a different set of basis functions to "map out" the space. Functions which look local in configuration space, like $\delta(x-x_0)$ map to extended waves in wavenumber space $\exp(i x_0 k)$ and vice versa, so which basis is more convenient depends on the application. But try to keep clear that you are not mapping positions to wavenumbers or vice versa. You are mapping the description of the abstract function $f$ in terms of local functions of coordinates $x$ into the description of the same abstract function $f$ in terms of local functions of wavenumbers $k$, no different from going from an abstract vector $\vec{v}$ written in components in the $\hat{x},\hat{y},\hat{z}$ basis to the same vector written in components in the $\hat{x}',\hat{y}',\hat{z}'$ basis.

Answer 2

A Fourier transform is a linear transformation between two particular bases, the point functions and the periodic functions. The vector space we are talking about here is the space of functions.

A Jacobian matrix is a linear approximation for a general transformation. For example, you mention transforming from a Cartesian basis to a spherical basis. This isn't a linear transformation, but we can approximate it as one if we restrict our attention to small regions of space.

A Jacobian is the determinant of the Jacobian matrix.

A Lorentz transformation is a linear transformation of space-time that has several additional restrictions (eg preserves the space-time interval, preserves parity) that are physical in nature.

Answer 3

Actually, they are all transformations from one space to the other. In order to clearly distinguish them, the key is to understand what are those spaces.

1. Jacobbian. $J:T_pM\rightarrow T_qM$ establishes a mapping between tangent spaces of two points in space. In the other word, suppose $f(p)=q$ and $v=v^idx_i\in T_pM$, then $$u^i=(Jv)^i=\frac{\partial f^i}{\partial x^j}v^j$$

2. Lorentz. $\mathcal{L}:\mathbb{R}^{1,3}\rightarrow\mathbb{R}^{1,3}$ establishes a mapping between two spacetimes.

3. Fourier. $\mathcal F:L^2\rightarrow\ell^2$ establishes a mapping between two function spaces. It is often applied to solve differential equation since it transform differential operation to algebraic operation which would be easier to solve.