Showing posts with label General HEP. Show all posts

Random Matrices

I have been lately studying the random matrices and their application that widely defaults for the JT gravity. Though, random matrices need to be started with Wigner's idea of the random matrix in nuclear theory. Right now, random matrix theory can be considered an important subject, at least from my learning view. In the following (short), I present my rough ideas of random matrix theory, extracted from here.


Random matrix theory (RMT) is a classic example of statistical group theory in general physics. The most recent development for RMT is the equivalence of JT gravity with RMT, see here. From the correspondence of AdS/CFT, one learns that a bulk theory with gravity lives on the boundary of a quantum system. However, the equivalence of JT gravity is not given to a boundary theory (or a bulk theory). In fact, RMT shares the correspondence with JT gravity; hence JT gravity is dual to random matrix integral of Hamiltonian $ H $, where $ H $ is a random matrix.

(RMT is mainly concerned about groups' statistics, at least for us, whose applications are wide in physics, as indicated.)

Consider a matrix $ \sf M $, from linear algebra we know that $ \sf M$ holds eigenvalues $ {\sf {m}}_{ij} $ (for $ 2 \times 2 $ matrix). Suppose the elements of matrix $ \sf M $ are random variables, to which we say to obey some specific outlined properties. In that case, the study of those ($\mathsf{m}_{ij} $) eigenvalues are called the "random matrix theory'' problem. Now one can ask what the practical application of RMT is. Actually, there are many practical applications. Consider the well-studied example of the nucleus using these random matrices, which Wigner (and Dyson) developed. And the recent example of the success of RMT is JT gravity. I suggest the reader to read the most interesting book on this subject by  Madan Lal Mehta.


From a mathematical perspective, random matrices serve great as well. However, it tends that it is currently revolutionizing the physics. Nuclear physics had first exploitation of random matrices. And now, it is used as a tool in black hole information problems, JT gravity (see the Saad, Shenker, Stanford), and other statistics problems (include the pedastriation and all that researches).  The deformations of these matrices were done by Witten, and he also did the volume problems for the subject. 

Radom matrices can be classified into classes. See sec. 4 in this and Witten's paper on JT gravity. For a full study, consider the Mehta's book.

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Unparticles and Unnuclear

In this short paper, I sum up the ideas of unparticle field theory (UFT). UFT is a scale-invariant theory. Georgi proposed it in 2007. Some slides (of earlier times) by him are available here on the subject. Despite being a wonderful field theory, it lacks experimental evidence. However, indirect shreds of evidence are possible through the following channels


.

There are more channels other than these. (For technicalities of these interactions, refer to papers.) UFT has a parameter  in the field equations. UFT is unlikely (for now) to be observed because it can be integral values, for instance, 1/2. How would the field theory look? 

In the case of  , UFT is just standard model field theory (these scalar theories are scale-invariant). For instance a UFT with the propagator 


 is phase-space (refer to any paper on UFT).  When , the propagator becomes


which is a familiar propagator in SM. There are many more interesting things about UFT that I cover in the paper with results. You should check first (if new to UFT) papers by Georgi, Tzu-Yiang Yuan, and Kingman Cheung. Georgi is also carrying interesting results in Schwinger's problems. 


Recently, a new term was dropped in this area by Hammer and Son. Unnuclear physics is a non-relativistic theory of unparticles. This EFT claims much more experiments than unparticle physics (which is a relativistic version). A recent talk by Dam Son emphasized the phenomenology of unnuclear physics, which is his next paper (to be released this month). A lot is going in this field. However, it is a newborn field, so a lot can't be said. 

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Particle Physics for Begineers!

Recently, Prof. David Tong has released a set of lecture notes on particle physics. In the introduction, it has been claimed that notes cover things in high school mathematics, and it is true. However, a little more than high school, but a keen learner would find it suitable to learn through the process. 

You may check the notes and bookmark it if you are interested;

http://www.damtp.cam.ac.uk/user/tong/particle.html

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