The Lieb-Robinson bound plays a significant role in current work on the topological phases of extensive quantum systems. Together with the methods developed for its proof, it now forms part of the basis of modern quantum information theory. Addressing the fundamental physics question through the newly conceived Lieb-Thirring inequalities, they opened a new chapter in functional analysis.

Over the years it has continued to inspire new mathematical results, playing a significant role in partial differential equations. He officially came on board in At Princeton, Elliott addressed important problems including stability of matter in magnetic fields, with Jan Philip Solovej and Jakob Yngvason; conditions for the emergence of ferromagnetism, some with Michael Aizenman; and an axiomatic presentation of thermodynamics based on the concept of entropy, developed with Jakob Yngvason.

Much of the research was directed toward understanding the quantum mechanics of atoms and the lowest energy state of the Bose gas. This provides an early example of a system exhibiting what is nowadays referred to as a topological state of matter, which is a subject of great current interest.

Elliott also wrote pedagogical texts, such as the mathematics book with Michael Loss called simply Analysis. With Robert Seiringer he co-authored The Stability of Matter in Quantum Mechanics , which brought together the work on the relativistic and non-relativistic many-body theory developed over the years by Elliott and others. He is a member of five national academies, including the National Academy of Sciences, a foreign member of the Royal Society in the United Kingdom, and holds four honorary doctorates.

He was twice president of the International Association of Mathematical Physics. The Austrian government awarded him the lifetime Austrian Medal of Honor for Science and Art, which can be held by no more than 18 foreign scientists at any time. While Elliott has retired from active professorial duties he continues to do research in mathematics and physics. We wish him many years of joy in this endeavor.

Princeton University. Annual Emeriti Booklet Excerpt:. Lebowitz Academic Press, U. Zia, Phys. Reports , 45 Bergesen, Z. E 52 , Alexander and G. Eyink, Phys. E 57 , R Krug, Adv. Physics 46 , Schreckenberg and D. Wolf Springer, Singapoure, Chou and D. Lohse, Phys. MacDonald, J. Gibbs and A. Pipkin, Kinetics of biopolymerization on nucleic acid templates Biopolymers 6 Spitzer, Advances in Math. Andjel, Ann. Probability 10 , Lieb and D. Landau and E. Fisher, S.

Ma and B. Nickel, Phys. Dyson, Commun. Karlin and H. Gantmacher, Matrix Theory Chelsea, Vol. J van Leeuwen and H. Hilhorst, Physica A , Nagle, Am.

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Barma and R. Bardhan, B. Chakrabarti and A. Hansen Springer, Berlin, p. Krug and M. Schimschak, J. Phys I France, 5 , Kutner, K. Kehr, W. Renz and R. Przenioslo, J. A, 28 , A, 30 , Derrida, J. Benjamini, P. Ferrari, C. Landim, Stoch. Evans, Europhys. Krug and P. Ferrari, J. A, 29 , L Krug Burlatsky, G. Oshanin M. Morea and W. Reinhardt, Phys. Gwa and H. Spohn, Phys. A 46 , Bialas, Z. Burda and D. Johnston, Nucl. B , O'Loan, M. Evans and M. Cates, Phys. Ramaswamy and M. Barma, J. Tripathy and M.

Barma, Phys. Rubinstein, Phys. Duke, Phys. Services on Demand Journal. I Introduction In recent years the study of nonequilibrium systems has come to the fore in statistical mechanics. For example, on a periodic one-dimensional homogeneous system of N sites, the partition sum can be written as the trace of a product of N transfer matrices T : where l are the eigenvalues of the transfer matrix.

Indicating the presence of a particle by a 1 and an empty site hole by 0 the dynamics comprises the following exchanges at nearest neighbour sites The open system was studied by Krug [20] and boundary induced phase transitions shown to be possible. Thus the dynamical processes at the boundaries are These boundary conditions force a steady state current of particles J through the system. III The zero-range process The zero-range process was first introduced into the mathematical literature as an example of interacting Markov processes [10].

Figure 1. Equivalence of zero range process and asymmetric exclusion process. In the basic model described above, f n is given by Note that f n is defined only up to a multiplicative constant and we could have included a factor z n in 6. IV Generalizations We now show how the totally asymmetric, homogeneous zero-range process we have considered so far may be generalised yet retain steady states of a similar form to 5,6.

It turns out that the steady state again has the form Equation 7 is modified to Equating the terms m on each side of 18 , assuming 11 and cancelling common factors yields Inserting 17 leads to the condition which is the same as the single particle steady state condition Thus, we may write The steady state of the single particle problem random walker on a disordered one dimensional lattice [60] can be solved and one obtains This network is relevant to the disordered one-dimensional exclusion process studied in [61, 62, 63]. This idea plays an important role in attempts to develop models of real world physics based on string theory, and it provides a natural explanation for the weakness of gravity compared to the other fundamental forces.

One notable fact about string theory is that the different versions of the theory all turn out to be related in highly nontrivial ways. One of the relationships that can exist between different string theories is called S-duality. This is a relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently.

Type I string theory turns out to be equivalent by S-duality to the SO 32 heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality. Another relationship between different string theories is T-duality. Here one considers strings propagating around a circular extra dimension. For example, a string has momentum as it propagates around a circle, and it can also wind around the circle one or more times.

The number of times the string winds around a circle is called the winding number. If a string has momentum p and winding number n in one description, it will have momentum n and winding number p in the dual description. For example, type IIA string theory is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.

In general, the term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. Two theories related by a duality need not be string theories. For example, Montonen—Olive duality is example of an S-duality relationship between quantum field theories. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory.

The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena. In string theory and other related theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions.

For instance, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p , these are called p -branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane. Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics.

They have mass and can have other attributes such as charge. Physicists often study fields analogous to the electromagnetic field which live on the worldvolume of a brane. In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to a certain mathematical condition on the system known as the Dirichlet boundary condition.

Branes are frequently studied from a purely mathematical point of view, and they are described as objects of certain categories , such as the derived category of coherent sheaves on a complex algebraic variety , or the Fukaya category of a symplectic manifold. Prior to , theorists believed that there were five consistent versions of superstring theory type I, type IIA, type IIB, and two versions of heterotic string theory. This understanding changed in when Edward Witten suggested that the five theories were just special limiting cases of an eleven-dimensional theory called M-theory.

His announcement led to a flurry of research activity now known as the second superstring revolution. In the s, many physicists became interested in supergravity theories, which combine general relativity with supersymmetry. Whereas general relativity makes sense in any number of dimensions, supergravity places an upper limit on the number of dimensions. Initially, many physicists hoped that by compactifying eleven-dimensional supergravity, it might be possible to construct realistic models of our four-dimensional world.

The hope was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the strong and weak nuclear forces , and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as chirality. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions. In the first superstring revolution in , many physicists turned to string theory as a unified theory of particle physics and quantum gravity.

Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary Lagrangian.

In string theory, the possibilities are much more constrained: by the s, physicists had argued that there were only five consistent supersymmetric versions of the theory. Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation. They found that a system of strongly interacting strings can, in some cases, be viewed as a system of weakly interacting strings. This phenomenon is known as S-duality.

It was studied by Ashoke Sen in the context of heterotic strings in four dimensions [39] [40] and by Chris Hull and Paul Townsend in the context of the type IIB theory. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent. At around the same time, as many physicists were studying the properties of strings, a small group of physicists was examining the possible applications of higher dimensional objects.

In , Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes.

follow Shortly after this discovery, Michael Duff , Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. In fact, Duff and his collaborators showed that this construction reproduces exactly the strings appearing in type IIA superstring theory. Speaking at a string theory conference in , Edward Witten made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions.

Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of higher dimensional branes in string theory. Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.

This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.

The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry. This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra. Douglas , and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory , a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry.

It quickly led to the discovery of other important links between noncommutative geometry and various physical theories. In general relativity, a black hole is defined as a region of spacetime in which the gravitational field is so strong that no particle or radiation can escape. In the currently accepted models of stellar evolution, black holes are thought to arise when massive stars undergo gravitational collapse , and many galaxies are thought to contain supermassive black holes at their centers. Black holes are also important for theoretical reasons, as they present profound challenges for theorists attempting to understand the quantum aspects of gravity.

String theory has proved to be an important tool for investigating the theoretical properties of black holes because it provides a framework in which theorists can study their thermodynamics. In the branch of physics called statistical mechanics , entropy is a measure of the randomness or disorder of a physical system. This concept was studied in the s by the Austrian physicist Ludwig Boltzmann , who showed that the thermodynamic properties of a gas could be derived from the combined properties of its many constituent molecules.

Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give a precise definition of entropy as the natural logarithm of the number of different states of the molecules also called microstates that give rise to the same macroscopic features. In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases, the entropy scales with the volume. In the s, the physicist Jacob Bekenstein suggested that the entropy of a black hole is instead proportional to the surface area of its event horizon , the boundary beyond which matter and radiation is lost to its gravitational attraction.

The Bekenstein—Hawking formula expresses the entropy S as. Like any physical system, a black hole has an entropy defined in terms of the number of different microstates that lead to the same macroscopic features. The Bekenstein—Hawking entropy formula gives the expected value of the entropy of a black hole, but by the s, physicists still lacked a derivation of this formula by counting microstates in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the viability of any theory of quantum gravity such as string theory.

In a paper from , Andrew Strominger and Cumrun Vafa showed how to derive the Beckenstein—Hawking formula for certain black holes in string theory. In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of different ways of placing D-branes in spacetime so that their combined mass and charge is equal to a given mass and charge for the resulting black hole.

The black holes that Strominger and Vafa considered in their original work were quite different from real astrophysical black holes. One difference was that Strominger and Vafa considered only extremal black holes in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge.

Although it was originally developed in this very particular and physically unrealistic context in string theory, the entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity. Indeed, in , Strominger argued that the original result could be generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry. This is a theoretical result which implies that string theory is in some cases equivalent to a quantum field theory.

It is closely related to hyperbolic space , which can be viewed as a disk as illustrated on the left. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior. One can imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The resulting geometric object is three-dimensional anti-de Sitter space. Time runs along the vertical direction in this picture. As with the hyperbolic plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface.

This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space. An important feature of anti-de Sitter space is its boundary which looks like a cylinder in the case of three-dimensional anti-de Sitter space.

One property of this boundary is that, within a small region on the surface around any given point, it looks just like Minkowski space , the model of spacetime used in nongravitational physics. The claim is that this quantum field theory is equivalent to a gravitational theory, such as string theory, in the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory.

For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding. One reason for this is that the correspondence provides a formulation of string theory in terms of quantum field theory, which is well understood by comparison. Another reason is that it provides a general framework in which physicists can study and attempt to resolve the paradoxes of black holes.

In , Stephen Hawking published a calculation which suggested that black holes are not completely black but emit a dim radiation due to quantum effects near the event horizon. This property is usually referred to as unitarity of time evolution. The apparent contradiction between Hawking's calculation and the unitarity postulate of quantum mechanics came to be known as the black hole information paradox. This state of matter arises for brief instants when heavy ions such as gold or lead nuclei are collided at high energies.

The physics of the quark—gluon plasma is governed by a theory called quantum chromodynamics , but this theory is mathematically intractable in problems involving the quark—gluon plasma. The calculation showed that the ratio of two quantities associated with the quark—gluon plasma, the shear viscosity and volume density of entropy, should be approximately equal to a certain universal constant. In , the predicted value of this ratio for the quark—gluon plasma was confirmed at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory.

Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an insulator. A superfluid is a system of electrically neutral atoms that flows without any friction. Such systems are often produced in the laboratory using liquid helium , but recently experimentalists have developed new ways of producing artificial superfluids by pouring trillions of cold atoms into a lattice of criss-crossing lasers.

These atoms initially behave as a superfluid, but as experimentalists increase the intensity of the lasers, they become less mobile and then suddenly transition to an insulating state. During the transition, the atoms behave in an unusual way. For example, the atoms slow to a halt at a rate that depends on the temperature and on Planck's constant , the fundamental parameter of quantum mechanics, which does not enter into the description of the other phases. This behavior has recently been understood by considering a dual description where properties of the fluid are described in terms of a higher dimensional black hole.

In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing models of real world physics that combine general relativity and particle physics. Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. String phenomenology is the part of string theory that attempts to construct realistic or semi-realistic models based on string theory.

Partly because of theoretical and mathematical difficulties and partly because of the extremely high energies needed to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to any of these models being a correct fundamental description of nature. This has led some in the community to criticize these approaches to unification and question the value of continued research on these problems.

The currently accepted theory describing elementary particles and their interactions is known as the standard model of particle physics. This theory provides a unified description of three of the fundamental forces of nature: electromagnetism and the strong and weak nuclear forces. Despite its remarkable success in explaining a wide range of physical phenomena, the standard model cannot be a complete description of reality.

This is because the standard model fails to incorporate the force of gravity and because of problems such as the hierarchy problem and the inability to explain the structure of fermion masses or dark matter. String theory has been used to construct a variety of models of particle physics going beyond the standard model. Typically, such models are based on the idea of compactification.

Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles. Such compactifications offer many ways of extracting realistic physics from string theory.

Other similar methods can be used to construct realistic or semi-realistic models of our four-dimensional world based on M-theory. The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. Despite its success in explaining many observed features of the universe including galactic redshifts , the relative abundance of light elements such as hydrogen and helium , and the existence of a cosmic microwave background , there are several questions that remain unanswered.

For example, the standard Big Bang model does not explain why the universe appears to be same in all directions, why it appears flat on very large distance scales, or why certain hypothesized particles such as magnetic monopoles are not observed in experiments. Currently, the leading candidate for a theory going beyond the Big Bang is the theory of cosmic inflation. Developed by Alan Guth and others in the s, inflation postulates a period of extremely rapid accelerated expansion of the universe prior to the expansion described by the standard Big Bang theory.

The theory of cosmic inflation preserves the successes of the Big Bang while providing a natural explanation for some of the mysterious features of the universe. In the theory of inflation, the rapid initial expansion of the universe is caused by a hypothetical particle called the inflaton.

The exact properties of this particle are not fixed by the theory but should ultimately be derived from a more fundamental theory such as string theory. While these approaches might eventually find support in observational data such as measurements of the cosmic microwave background, the application of string theory to cosmology is still in its early stages. In addition to influencing research in theoretical physics , string theory has stimulated a number of major developments in pure mathematics.

Like many developing ideas in theoretical physics, string theory does not at present have a mathematically rigorous formulation in which all of its concepts can be defined precisely. As a result, physicists who study string theory are often guided by physical intuition to conjecture relationships between the seemingly different mathematical structures that are used to formalize different parts of the theory.

These conjectures are later proved by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics. After Calabi—Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi—Yau manifold.

In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry. Regardless of whether Calabi—Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between different string theories has significant mathematical consequences. The Calabi—Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry , a branch of mathematics concerned with counting the numbers of solutions to geometric questions.

Enumerative geometry studies a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic illustrated on the right is an algebraic variety defined using a certain polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface. Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi—Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five.

This problem was solved by the nineteenth-century German mathematician Hermann Schubert , who found that there are exactly 2, such lines. In , geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is , By the year , most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish.

Originally, these results of Candelas were justified on physical grounds. However, mathematicians generally prefer rigorous proofs that do not require an appeal to physical intuition. Inspired by physicists' work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry. Group theory is the branch of mathematics that studies the concept of symmetry.

For example, one can consider a geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle without changing its shape. Each of these operations is called a symmetry , and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a group. In this particular example, the group is known as the dihedral group of order 6 because it has six elements.

A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a finite group.