The Dirac Delta

I've been doing replica calculations since I arrived in UK. Appart from the replica trick itself, these calculations involve a lot of interesting and useful mathematical tricks and techniques. One of them, which I particularly like, is related to the Dirac delta function.

The Dirac delta is not a function in the correct sense of the word, it is a distribution and is a very odd object. It is defined rigorously as a limit of functions. There are different functions that gives the Dirac delta in some limit, but (in my opinion) the most elegant is the Gaussian distribution. It goes like this:


The main characteristics of the delta is that its integral over the entire real line is one and it has the filtering property:


The legend (as I couldn't find more accurate historical details) says that Dirac introduced the delta to study point charge distributions in electromagnetism. Soon, a lot of other applications were found for it. I stumbled with one in the replica papers. It is usual in replica calculations to do the following transformation:


Let me explain. In the above equation, s is a random vector with probability distribution P(s) and x a constant vector. What I'm doing is a change of variables under the integral. Now, the new one-dimensional (and this is important!) variable v is a random variable and its distribution will be


The average of the delta over s. If you look more closely, there is a very interesting thing ocurring here: if you consider the equations above without the probability distributions like


The integral of the delta is nothing more than the Jacobian of the transformation. The interesting is that it is a Jacobian of a transformation between variables with different dimensionality. I'll leave to the reader the pleasure to see that all the Jacobian properties and delta function properties match beautifully indeed. I also bet that this can be related with Gauss theorem (or the generalized Stokes Theorem) although I haven't done the detailed calculations.

Now, changing the topic a little, I found this nice site about Singular Value Decomposition, a kind of matrix decomposition which is important in numerical calculation, but not only there. I remember that once I entered the office of (again) Prof. Caticha and he was playing with SVD. He transformed a picture in a matrix, calculated the singular values and took the resulting matrix as the original ignorating the less important singular values. The result was an image almost equal to the original one: a kind of compression technique.

Picture: I also discovered that 'Delta de Dirac' is the name of a Mexican band of progressive rock and the picture is of one of its CDs. Unfortunately, I didn't find their website. If someone knows, please tell me.