The Way Theorized

Winter is here and so the holidays and breaks. Here, I include few papers that might be interested to you to read in this winter break. In no specific order. Theme- Random Matrices and JT gravity.

•   JT gravity as a matrix integral by Phil Saad, Stephen Shenker and Douglas Stanford - https://arxiv.org/abs/1903.11115.
  

•  Nonperturbative Jackiw-Teitelboim gravity by Clifford V. Johnson and his other papers on this subject as well- https://arxiv.org/abs/1912.03637. 
  

•  Lower dimensional gravity by Roman Jackiw- https://doi.org/10.1016/0550-3213(85)90448-1.

•  Gravitation and hamiltonian structure in two spacetime dimensions by Claudio Teitelboim- https://doi.org/10.1016/0370-2693(83)90012-6.
  

• Comments on the Sachdev-Ye-Kitaev model by Juan Maldacena, Douglas Stanford- https://arxiv.org/abs/1604.07818.
  

• JT Gravity and the Ensembles of Random Matrix Theory by Douglas Stanford, Edward Witten- https://arxiv.org/abs/1907.03363.

Some bonus recommendations ;)

• Newly appeared paper by Ed Witten on Quantum field theory in curved spacetime (https://arxiv.org/abs/2112.11614). There are also lectured by him on the same on Youtube.
  

• Anna Karenina by Leo Tolstoy.
  

I simulated Albert Einstein with GPT-3. The best that I could. Here is the conversation we had. I have edited some stuff, I did not include anything, but all quite was fun.

Here are a few hot topics in high-energy physics. Not in any order. Moreover, of course, they are biased from my eye-view of hep-th.

Some basics of complex analysis. We will follow L. Ahlfors's Complex Analysis book. An analytic function is a complex-valued function with derivatives everywhere where function $f(x)$ is defined with an appropriate power series. Holomorphic functions are the same, with a different meaning. For an analytic function $f(z) = u+iv$, we write

It is a common misconception that renormalization is needed only when infinities are coming up. But R. Shankar beautifully tackles this in his textbook on Quantum Field Theory and Condensed Matter in chapter 11th. 

We will take a $4 \times 4$ random unitary matrix of Gaussian nature (Hermitian ensemble) -  generated from Mathematica.

One may now do many things with this matrix, such as generating its Gaussian distribution. 

There are many common misconceptions (or carelessness) that amateur string readers have. One of those is that bosonic string theory on worldsheet is for bosons, and superstring theory on superspace is for fermions (well, only fermions). This is technically wrong. Superstring theory has both bosonic sector (with Neveu-Schwarz - NS- boundary condition) and fermionic sector (with Ramond boundary condition). However, the NS bosonic sector (which uses the same $X^\mu$ worldsheet of the free bosonic theory of D=26) of Superstring is different from the bosonic theory in D=26. One of the things that differentiate the two is the presence of an extra oscillator in the former.  NS bosonic sector does not have the critical dimension $26$ but $10$.

In superstring theory, we add an extra wave-function $\psi^\mu$ which is related to $X^\mu$ by world-sheet supersymmetry (space-time SUSY is used in GS formalism). The fermionic sector (of course of Superstring) is also ghosts-free at $D=10$. And the Virasoro algebra of the free bosonic theory is replaced by the Super-Virasoro algebra.

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.

Let us catch with some updates. 

• Susskind has a good paper on de-Sitter space here. He has been pretty active in de-Sitter lately. One should first start with his and colleagues' paper on the causal patch and then follow up on his recent writings. (I recently saw a video of Susskind highly appraising Juan Maldacena here.)
• I wrote a fine and short piece of introduction (and basically mixed up) article on field theory and beyond, which can be found here and here.
• Subir Sachdev, TIFR, and IAS have organized a course on Quantum phases of matter. The lectures can be viewed at http://qpt.physics.harvard.edu/qpm/.
• Breakthrough Prizes have been awarded https://breakthroughprize.org/News/65.

Here, I include some important papers (or books) which one (if not already) should read in this Summer. (Of course, I can miss many great which I have read or haven't).

Kawai-Lewellen-Tye Relation, see here, is a perfect tool to make connections between amplitudes. The most general of them is the relation between a tree gravity amplitude (from Einstein-Hilbert action) and a gauge theory amplitude (from Yang-Mills). We can also relate the closed strings amplitude and open strings amplitude using KLT. (Many connections are there for closed strings algebra and open strings. From a conversation with Ed Witten; Ed - The simplest is that they can be computed in similar ways by integral representations that come from the open or closed string).

From the paper (recommended);

and same paper (sec II, a short review);

More papers on this subject; here, here, and here.

I wrote a very brief discussing note on superstring formalism that was developed in the early years. In the note, I discuss it with required algebra. However, an adopted model for superstrings is good than this one. You can read it from here:

Old Superstring Formalism - Pdf

Though some papers promise to show the Lorentz invariance in non-critical string theory, I haven't found any strong evidence of it, yet. Except some papers claiming it to be consistent in some scenarios.  Evidently, the addition of longitudinal oscillators in lower dimensions is not harmful, however, we wait for a good scheme for non-critical string theory. (I also have no idea of eliminating anomalies and ghosts in superstring in non-critical dimensions.) We discuss the origin of $D=26$ (critical dimensions) in bosonic string theory.

We can use the Feynman diagrams to replicate the process of scattering with strings. For particle interactions, we can do Feynman diagrams (for , see this - just topologies). For strings, we can do the same; we call them "string diagrams." A closed string forming two closed strings is depicted by changing the point-particle by strings and word line by worldsheet.

The crossing line indicates (this one line is for collective dimensions. However, there should be definitive for each one) that there is not one for all Lorentz frame, unlike in point particle theory, but two. It can be interpreted that the point-particle Feynman diagram is just a limiting case of the string diagrams. Furthermore, one string diagram (with vertex function) can be deformed to a few particle Feynman diagrams. That is one of the reasons why there are not many string diagrams. Lorentz frames are also the reason for the absence of ultraviolet divergence because of independent defined Lorentz sites at interaction.

Among the advantages of conformal mapping, one being that there would not be the h (associated with  ) integrals in the matrix calculations. But what about the conservation of quantum numbers after topological mapping? For that, we introduce vertex operators. In the conformal image, cross-markers indicate the strings (the top one shows the far past string, and the bottom two indicates the far future newly born closed strings). The marked area is for the vertex operator. We can introduce it with the symbol , where m is for an m-type particle. This operation is effortless in a 1+1  system, which indeed we are following. The  is the operator for local absorption and emission of string states. We can introduce another operator , which is for the re-parametrization of the mapping. While W operators account for Lorentz transformation, we must also take accounts of translation. That is how we reach a well-known translation operator

In string theory, when we write the Feynman diagrams, we denote amplitude for open and closed string, as Veneziano and Virasoro-Shapiro amplitudes (a complex beta function) respectively (we will only discuss the former). Veneziano amplitude is an Euler beta function that obeys the crossing symmetry and looks*

where s and t are Mandelstem variables defined;

and $ \alpha(s) $ is Regge trajectory.

The amplitude is a result of the work on the duality between s and t channels. According to this duality, the sum of all the s channels and t channels should be equal. It was written for a model obeying the Regge trajectory which at the time was indicating not the string theory, but a QCD theory.  The Euler form of the amplitude can be written through expansions as**

because any beta function of the form

can be written as 

Note * has only one pole rather than two and ** is written in t poles. What we can do is writing ** in s poles, which then

And that is the duality. We can study various aspects of it by keeping t fixed or s fixed. This is done in the very first paper on this by Veneziano, here. Also, in integral representation, as like a beta function, this amplitude can be written as 

                  

                  

In large s and fixed t

                                               

In quantum mechanics, there are many problems; the biggest of them (according to a group) is finding the first and initial state. We are talking about the initial state of the observable universe; it sounds cliche. But people believe that if we can find the initial state, we can determine the difference between the macroscopic and microscopic world. I have not any thoughts on that. However, as like many, my respect for quantum mechanics is just fundamental (and theoretical). When you do quantum mechanics, without any evidence of research, you will likely come to believe that 'what is true and what is wrong' is differently to be seen in the subject.

Just like the statistical mechanics, we have $\rho$ the density matrix in quantum mechanics. This looks like

When we come to 'Copenhagen's Interpretation of quantum mechanics, which is heavily misguided in different ideologies, we can observe incompleteness and loss of determinism. The former and latter can be written to co-exist as 'no macroscopic evidence.' It is not entirely true. However, it is accepted that the interpretations can be completed by the completeness of matching the probabilistic theory to macroscopic lurches of evidence. Indeed, we would enjoy a version like this. Rather than this, the Copenhagen interpretation is widely celebrated as a successful interpretation.

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The reason these are half-baked problems that no one now cares about it. These problems are widely accepted as 'reality and fundamental.' Even when young people try to think about it, they fall into the prey of 'cancellation' and 'no go' which they say are occults of the subject. But, my stance is quantum mechanics is an achievement that we should sing from time to time.

Notes; A friend of mine shared with me an unlisted link of a talk by Weinberg at 'Standard Model at 50', in which he talks the same. And in fact, this post is inspired by his talk. 

https://www.youtube.com/watch?v=mBninatwq6k