In describing quantum phenomena, theory has so far outstripped experiment that it is not possible to distinguish where physics ends and mathematics begins in this area. The RIA Novosti correspondent spoke with the participants of the international scientific school held at the Joint Institute for Nuclear Research (JINR) in Dubna about what kind of mathematics is needed by quantum physics and what problems are solved by representatives of the two most rigorous sciences.
School "Statistical Sums and Automorphic Forms" attracted about eighty young researchers and teachers from all over the world, including Hermann Nicolai, director of the Albert Einstein Institute (Germany).
Its organizers from the Laboratory of Mirror Symmetry and Automorphic Forms of the Faculty of Mathematics of the Higher School of Economics emphasize that leading scientific schools have become more active in Russia, representing the vanguard of research in many areas.
The success of our mathematicians is closely related to the achievements of theoretical physicists who are looking for new manifestations of quantum physics. This is literally the other world, the existence of which is assumed outside the Newtonian and Einstein's reality. In order to consistently describe going beyond the laws of classical physics, scientists invented string theory in the 1970s. She claims that the universe can be judged not in terms of point particles, but with the help of quantum strings.
The concepts “point”, “line”, “plane”, familiar to every student, blur in the quantum world, the boundaries disappear, and the very same string theory acquires a very complex internal structure. To understand such unusual objects requires something special. Namely, mirror symmetry, which was suggested by string physicists in the early 1990s. This is a prime example of how new mathematical structures emerge from physical intuition.
In the ordinary world, such symmetry appears, for example, when we see our reflection in a mirror. In the quantum world, this is an immeasurably more complex, abstract view that explains how two different-looking theories actually describe one system of elementary particles at different levels of interaction in multidimensional space-time.
The mathematical program for studying the effect discovered by physicists - the hypothesis of homological mirror symmetry - was proposed in 1994 by mathematician Maxim Kontsevich. Four years later, he won the Fields Prize, the Nobel Prize for the mathematical world.
In Russia, the American mathematician of Bulgarian origin Lyudmila Katsarkova, a graduate of the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University, was invited to develop the direction of mirror symmetry. His project and the creation of a laboratory at the HSE at the end of 2016 were supported by the Russian government under the mega-grant program. Being one of the co-authors of Kontsevich, Katsarkov attracted him to work.
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From intuition to proof
Most of the school's lecturers work in this dynamic field related to space-time geometry and dual field and string theories, directly or indirectly helping to piece the puzzle of the quantum world. One of the main objects of research for them is very large systems containing an infinite number of particles. To describe these systems in thermodynamic equilibrium, physicists compute quantities called partition functions.
Mirror symmetry of manifolds, Nekrasov's instanton partition functions and other concepts introduced into string theory and quantum field theory turned out to be completely new objects for mathematicians, which they began to analyze with interest. It turned out, for example, that it is convenient to describe state sums using automorphic forms - a special class of functions that has long been well studied in number theory.
The concepts “point”, “line”, “plane”, familiar to every student, blur in the quantum world, the boundaries disappear, and the very same string theory acquires a very complex internal structure. To understand such unusual objects requires something special. Namely, mirror symmetry, which was suggested by string physicists in the early 1990s. This is a prime example of how new mathematical structures emerge from physical intuition.
In the ordinary world, such symmetry appears, for example, when we see our reflection in a mirror. In the quantum world, this is an immeasurably more complex, abstract view that explains how two different-looking theories actually describe one system of elementary particles at different levels of interaction in multidimensional space-time.
The mathematical program for studying the effect discovered by physicists - the hypothesis of homological mirror symmetry - was proposed in 1994 by mathematician Maxim Kontsevich. Four years later, he won the Fields Prize, the Nobel Prize for the mathematical world.
In Russia, the American mathematician of Bulgarian origin Lyudmila Katsarkova, a graduate of the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University, was invited to develop the direction of mirror symmetry. His project and the creation of a laboratory at the HSE at the end of 2016 were supported by the Russian government under the mega-grant program. Being one of the co-authors of Kontsevich, Katsarkov attracted him to work.
From intuition to proof
Most of the school's lecturers work in this dynamic field related to space-time geometry and dual field and string theories, directly or indirectly helping to piece the puzzle of the quantum world. One of the main objects of research for them is very large systems containing an infinite number of particles. To describe these systems in thermodynamic equilibrium, physicists compute quantities called partition functions.
Mirror symmetry of manifolds, Nekrasov's instanton partition functions and other concepts introduced into string theory and quantum field theory turned out to be completely new objects for mathematicians, which they began to analyze with interest. It turned out, for example, that it is convenient to describe state sums using automorphic forms - a special class of functions that has long been well studied in number theory.
The artist's idea of mirror symmetry. Illustration by RIA Novosti. Alina Polyanina
There are many examples of the opposite effect of mathematics on theoretical physics.
“I was working on the creation of a theory for a new class of special functions called 'elliptic hypergeometric integrals'. Then it turned out that these objects are in demand by physicists as a statistical sum of a special type,”says the mathematical physicist Vyacheslav Spiridonov from the Laboratory of Theoretical Physics at JINR.
Spiridonov introduced his integrals in 2000, and eight years later two physicists from Cambridge came to the same integrals, calculating superconformal indices (or supersymmetric partition functions) in the framework of Seiberg's duality theory.
“Superconformal indices are a very convenient concept for describing electromagnetic dualities, generalizing the phenomenon that first manifested itself in Maxwell's equations (the presence of mutually complementary physical properties in one phenomenon. - Ed.). With the help of the constructed mathematical theory, we predicted new dualities that physicists missed. Physicists express ideas, get preliminary results, and mathematicians build an absolute, systematic analysis: they give definitions, formulate theorems, prove, without allowing any breaks in the description of the phenomenon. How many more are there? What did the physicists miss? Mathematicians answer these questions. Physicists are interested in all the variety of objects classified by mathematicians,”says Spiridonov.
In search of quantum gravity and supersymmetry
“I want to understand the nature of quantum gravity and the physics of black holes, if string theory is correct to describe nature. This is my motivation. To do this, you need to calculate physical quantities and compare them with experiment. But the fact is that these are very complex calculations, there are many mathematical problems,”says Pierre Vanhove from the Institute for Theoretical Physics (Saclay, France), an associate member of the HSE laboratory.
A physicist who wants to understand what happened before the Big Bang, to study the configuration of a black hole, is forced to deal with space, which is compressed into a point, as a result of which its geometry is greatly changed. The theory of relativity cannot explain these objects, as well as other non-classical phenomena - dark matter, dark energy. Scientists judge their existence by indirect signs, but it has not yet been possible to fix the manifestations of new physics in an experiment, including signs of quantum gravity - a theory that would combine general relativity and quantum mechanics. The Soviet physicist Matvey Bronstein stood at its origins in the mid-1930s.
By the way, scientists recorded classical (from the point of view of Einstein's theory) gravitational waves in an experiment only in 2015. To do this, they had to significantly upgrade the LIGO detector. To get a feel for the quantum nature of gravity, you need even greater instrument accuracy, unattainable at the current level of technology development.
“Right now, LIGO measurements do not give access to this new physics, it takes time to get there. Probably time consuming. We need to invent new methods, mathematical tools. Previously, only accelerators were available to us to search for new physics, the most powerful of which is the LHC; now another way is open - the study of gravitational waves,”explains Vankhov.
To explain the oddities of the observed world, for example, scientists have introduced the supersymmetry hypothesis. According to her, the elementary particles that we observe in experiments must have twins in a "different" area of our world. One of the expected manifestations of these twins is that the lightest of them forms dark matter, that is, it lives around us, but is inaccessible for observation.
“To see supersymmetry, you need to better understand the structure of particles, and this requires even more accelerator energies. For example, if in collisions of protons we see the birth of supersymmetric partners of ordinary particles, then what we are doing really exists. At the moment, at CERN, the accelerator collides particles at maximum energy, but supersymmetry has not yet been discovered. The limit of its manifestation - Planck energy - is beyond our reach,”says Ilmar Gahramanov, head of the Department of Mathematical Physics at the State University of Fine Arts named after Mimar Sinan (Istanbul, Turkey), a graduate of the MISiS.
However, supersymmetry must exist, Gahramanov believes, since its very idea, its mathematics, is "very beautiful."
“Formulas are simplified, some problems disappear, many phenomena can be explained by this theory. We want to believe that it exists, since the ideas of supersymmetry allow us to obtain interesting results for other theories that are experimentally testable. That is, the methods, technology, the mathematics that arise in it are transferred to other areas,”says the scientist.
Pure mathematics
One such area, which is developing thanks to the problems formulated in string theory, is the theory of moonshine.
"Moonshine" in English means sleepwalking and madness, "says John Duncan of Emory University (USA).
For clarity, during his speech, he shows the audience a photo of the blood-red moon over the Acropolis, taken during the January 31 super moon. Duncan was educated in New Zealand and then came to the United States to pursue his PhD. Having met there Igor Frenkel, a former Soviet mathematician, decided to tackle the Munshine theory (translated into Russian as "nonsense theory"), which was building bridges between the "monster" - the largest finite exceptional group of symmetries - and other mathematical objects: automorphic forms, algebraic curves and vertex algebras.
“From string theory came very deep mathematical ideas that changed geometry, the theory of Lie algebras, the theory of automorphic forms. The philosophical concept began to change: what is space, what is diversity. New types of geometries, new invariants appeared. Theoretical physics enriches mathematics with new ideas. We start working on them, and then we return them back to physicists. In fact, mathematics is being rebuilt now, as it already happened in the 20-30s of the XX century after the development of quantum mechanics, when it became clear that there are other structures in mathematics that have not been seen before, says Valery Gritsenko, professor at the University of Lille (France) and HSE.
Gritsenko is engaged in pure mathematics, but his results are in demand by physicists. One of his greatest achievements, obtained jointly with the mathematician Vyacheslav Nikulin, is the classification of infinite-dimensional automorphic hyperbolic Kats - Moody algebras, which has found application in string theory. It is to the description of a special hyperbolic Kats-Moody algebra of type E10, which claims to be the unifier of all physical symmetries of nature, that Herman Nicolai dedicated his lecture.
Despite the absence of experimental manifestations of string theory, supersymmetry, quantum gravity, scientists not only do not discard these concepts, but, on the contrary, continue to actively develop them. So "Not a geometer, let him not enter!" - the motto of Plato's Academy, formulated two and a half millennia ago, is most relevant in our time for theoretical physics.
Tatiana Pichugina