Doc Madhattan: October 2014

Regge theory

In quantum physics, Regge theory is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer but is allowed to take any complex value. The nonrelativistic theory was developed by Tullio Regge in 1957.

Following Chew and Frautschi (pdf), the key papers by Tullio Regge are:

Regge T. (1959). Introduction to complex orbital momenta, Il Nuovo Cimento, 14 (5) 951-976. DOI: http://dx.doi.org/10.1007/bf02728177 (pdf)

In this paper the orbital momentumj, until now considered as an integer discrete parameter in the radial Schrödinger wave equations, is allowed to take complex values. The purpose of such an enlargement is not purely academic but opens new possibilities in discussing the connection between potentials and scattering amplitudes. In particular it is shown that under reasonable assumptions, fulfilled by most field theoretical potentials, the scattering amplitude at some fixed energy determines the potential uniquely, when it exists. Moreover for special classes of potentials $V(x)$, which are analytically continuable into a function $V(z)$, $z=x+iy$, regular and suitable bounded in $x > 0$, the scattering amplitude has the remarcable property of being continuable for arbitrary negative and large cosine of the scattering angle and therefore for arbitrary large real and positive transmitted momentum. The range of validity of the dispersion relations is therefore much enlarged.

Regge T. (1960). Bound states, shadow states and mandelstam representation, Il Nuovo Cimento, 18 (5) 947-956. DOI: http://dx.doi.org/10.1007/bf02733035

In a previous paper a technique involving complex angular momenta was used in order to prove the Mandelstam representation for potential scattering. One of the results was that the number of subtractions in the transmitted momentum depends critically on the location of the poles (shadow states) of the scattering matrix as a function of the complex orbital momentum. In this paper the study of the position of the shadow states is carried out in much greater detail. We give also related inequalities concerning bound states and resonances. The physical interpretation of the shadow states is then discussed.

The importance of the model is summarized by the following:

As a fundamental theory of strong interactions at high energies, Regge theory enjoyed a period of interest in the 1960s, but it was largely succeeded by quantum chromodynamics. As a phenomenological theory, it is still an indispensable tool for understanding near-beam line scattering and scattering at very large energies. Modern research focuses both on the connection to perturbation theory and to string theory.

During the 1980s, Regge is interested also in the mathematical art, using Anschauliche Geometrie by David Hilbert and Stefan Cohn-Vossen like inspiration for a lot of mathematical objects.

Good bye, Mr. Regge, and thanks for all...

Alan Guth, eternal inflation and the multiverse

At the beggining of October, Alan Guth was at the workshop Fine-Tuning, Anthropics and the String Landscape at Madrid, and he concluded his talk with the following slide:

The complete talk, without question time, follows:

One of the first classifications that you learn when you start to study the behavior of matter interacting with electricity is between conductors and insulators: a conductor is a material that easily allows the passage of electric charges; on the other hand, an insulator prevents it (or makes it difficult). It is possible to characterize these two kinds of materials through the physical characteristics of the atoms that compose them. Indeed, we know that an atom is characterized by having a positive nucleus with electron clouds which rotate around it: to characterize a material is precisely the behavior of the outer electrons, those of the external band. On the other hand, the energy bands of every atom are characterized by specific properties: there are the valence bands, where the electrons are used in the chemical bonds, and the conduction bands, where the electrons are free to move, the "mavericks" of the atom, used for ionic bonds. At this point I hope it is simple to characterize a conductive material such as the one whose atoms have electrons both in the valence band, both in the conduction band, while an insulating material is characterized by having full only the valence band.
Now, in band theory, the probability that an electron occupies a given band is calculated using the Fermi-Dirac distribution: this means that there is a non-zero probability that an insulator's electron in the valence band is promoted to the conduction band, but it is extremely low because of the large energy difference between the two levels. Moreover, there is an energy level said Fermi level that, while in the conductors is located within the conduction band, in the insulation is located between the two bands, the conduction and valence, allowing a valence electron to jump more easily in the conduction band.

Teachers for the peace

The Nobel Prize for Peace 2014 is awarded to Kailash Satyarthi and Malala Yousafzai, teachers and activists for children rights,

for their struggle against the suppression of children and young people and for the right of all children to education

Carlo Rubbia and the discoveries of the weak bosons

On that day 30 years ago, I was almost certainly at school. Physics still was not my passion. Of course I started very well: when the teacher asked what is the space, I thought immediately to the universe, but the question was not referring to that "space", but in another, the geometric. But it is not about those memories that I have to indulge, but on a particular photo, in which Carlo Rubbia and Simon van der Meer, with two goblets, presumably of wine in hand, are celebrating the announcement of the Nobel Prize for Physics

for their decisive contributions to the large project, which led to the discovery of the field particles W and Z, communicators of weak interaction

The story of this Nobel, however, began eight years earlier, in 1976. In that year, in fact, SPS, the Super Proton Synchrotron, begins to operate at CERN, originally designed to accelerate particles up to an energy of 300 GeV.

The same year David Cline, Carlo Rubbia and Peter McIntyre proposed transforming the SPS into a proton-antiproton collider, with proton and antiproton beams counter-rotating in the same beam pipe to collide head-on. This would yield centre-of-mass energies in the 500-700 GeV range⁽¹⁾.

On the other hand antiprotons must be somehow collected. The corresponding beam was then

(...) stochastically cooled in the antiproton accumulator at 3.5 GeV, and this is where the expertise of Simon Van der Meer and coworkers played a decisive role⁽¹⁾.

Sudoku clues

The arXiv's paper is published two years ago, but I think that every time is a good time to play sudoku!

The sudoku minimum number of clues problem is the following question: what is the smallest number of clues that a sudoku puzzle can have? For several years it had been conjectured that the answer is 17. We have performed an exhaustive computer search for 16-clue sudoku puzzles, and did not find any, thus proving that the answer is indeed 17. In this article we describe our method and the actual search. As a part of this project we developed a novel way for enumerating hitting sets. The hitting set problem is computationally hard; it is one of Karp's 21 classic NP-complete problems. A standard backtracking algorithm for finding hitting sets would not be fast enough to search for a 16-clue sudoku puzzle exhaustively, even at today's supercomputer speeds. To make an exhaustive search possible, we designed an algorithm that allowed us to efficiently enumerate hitting sets of a suitable size.

In the following video by Numberphile, James Grime discusses the paper results:

From Nash equilibria to collective behavior

The Nash equilibrium is an important tool in game theory:

[It] is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will's decision, and Will is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others in the game.

Nash equilibria may, for example, be found in the game of coordination, in the prisoner's dilemma, in the paradox of Braess⁽⁶⁾, or more generally in any strategy game. In particular, given a game, we can ask whether it has or not a Nash equilibrium: apparently deciding the existence of Nash equilibria is an intractable problem, if there is no restriction on the relationships among players. In addition for a strong Nash equilibrium, the problem is on the second level of the polynomial hierarchy, which is a scale for the classification problem based on the complexity of resolution⁽¹⁾.
In addition to this study about Nash equilibria, Gianluigi Greco (one of my high school classmates), together with Francesco Scarcello, also studied the Nash equilibria (in this case the forced equilibria) in graphical games, where graphical game is a game represented in a graphical manner, through a graph⁽²⁾.

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