Aug 9, 2006

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.

7 comments:

Anonymous said...

Here's the website: http://m0squit0.com/delta/

If you want to hear some of their music check out http://www.isound.com/delta_de_dirac

Roberto Alamino said...

Thanks! I will surely listen them.

Anonymous said...

The official web site is www.deltadedirac.com i am the bass player jussef...nice to meet you :) you can reach our mail to jussef@deltadedirac.com or info@deltadedirac.com please write me and tell me what ou think

Roberto Alamino said...

I'm downloading the demo file right now and will listen it in my house. :)

But I'm curious. With this name, you should have a mathematics background, don't you?

Anonymous said...

HEHEHE WELL KIND OFF, BASICALLY OUR GUITAR PLAYER HAS LOTS OF MATH INFLUENCES SINCE HE STUDIED FOR A DEGREE IN ELECTRONIC ENGINEER OR SOMETHING LIKE THAT AND HE EXPLAINED THE DELTA FUNCTION TO ALL OF US AND WE LOVED AND DECIDED INCLUSIVE TO NAME ONE OF THE EIGHT SONGS AFTER THE DELTA FUNCTION.

Roberto Alamino said...

I tried to send you an email in the address you gave, but I always get a failure. Well, I was going to say that I really liked your music. Your very skilled and talented. Do you know if I can buy your CD here in the UK?

Jussef Sota said...

Hey there, yes you can buy the cd from the uk but you will have to pay the shipping and handling and the overall price will go up, the way we do it is send cash by western union and then about 2 or 3 weeks after the money arrives you will get the cd. Any questions i will drop you 3 e-mail directions for you to contact me.

jussef@deltadedirac.com
jussef@yahoo.com
jussef@puebla.megared.net.mx

i check the last one every day and the other two like three times a week, hope to recive an answer. Bye