Elliott H. Lieb facts for kids

Quick facts for kids Elliott H. Lieb
Born	(1932-07-31) July 31, 1932 (age 91) Boston, Massachusetts, US
Alma mater	Massachusetts Institute of Technology University of Birmingham
Known for	Araki–Lieb–Thirring inequality Borell–Brascamp–Lieb inequality Brezis–Lieb lemma Carlen-Lieb extension Temperley–Lieb algebra Lieb conjecture Lieb's square ice constant Lieb–Liniger model stability of matter Strong Subadditivity of Quantum Entropy Lieb–Thirring inequality Brascamp–Lieb inequality Lieb–Oxford inequality AKLT model Lieb–Robinson bounds Lieb–Yngvason Entropy principle Choquard equation Wehrl entropy conjecture 1-D Hubbard model Lieb lattice Adiabatic accessibility
Awards	Heineman Prize for Mathematical Physics (1978) Max Planck medal Birkhoff Prize (1988) Boltzmann medal (1998) Rolf Schock Prizes in Mathematics (2001) Levi L. Conant Prize (2002) Henri Poincaré Prize (2003) Medal of the Erwin Schrödinger Institute (2021) APS Medal for Exceptional Achievement in Research (2022) Carl Friedrich Gauss Prize (2022) Dirac Medal (2022) Kyoto Prize in Basic Sciences (2023)
Scientific career
Fields	Mathematics, Physics
Institutions	Princeton University
Doctoral advisor	Samuel Frederick Edwards Gerald Edward Brown

Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.

Lieb is a prolific author, with over 400 publications both in physics and mathematics. In particular, his scientific works pertain to quantum and classical many-body problem, atomic structure, the stability of matter, functional inequalities, the theory of magnetism, and the Hubbard model.

Biography

Lieb was born in Boston in 1932, the family moved to New York when he was five. His father came from Lithuania and was an accountant, his mother came from Bessarabia and worked as a secretary.

He received his B.S. in physics from the Massachusetts Institute of Technology in 1953 and his PhD in mathematical physics from the University of Birmingham in England in 1956. Lieb was a Fulbright Fellow at Kyoto University, Japan (1956–1957), and worked as the Staff Theoretical Physicist for IBM from 1960 to 1963. In 1961–1962, Lieb was on leave as professor of applied mathematics at Fourah Bay College, the University of Sierra Leone. In 1963, he joined the Yeshiva University as an associate professor. He has been a professor at Princeton since 1975, following a leave from his professorship at MIT.

He is married to fellow Princeton professor Christiane Fellbaum.

For years, he has rejected the standard practice of transferring copyright of his research articles to academic publishers. Instead, he would only give publishers his consent to publish.

Awards

Lieb has been awarded several prizes in mathematics and physics, including the Heineman Prize for Mathematical Physics of the American Physical Society and the American Institute of Physics (1978), the Max Planck Medal of the German Physical Society (1992), the Boltzmann medal of the International Union of Pure and Applied Physics (1998), the Schock Prize (2001), the Henri Poincaré Prize of the International Association of Mathematical Physics (2003), and the Medal of the Erwin Schrödinger Institute for Mathematics and Physics (2021).

In 2022 he was awarded the Medal for Exceptional Achievement in Research from the American Physical Society for ″major contributions to theoretical physics through obtaining exact solutions to important physical problems, which have impacted condensed matter physics, quantum information, statistical mechanics, and atomic physics″ and the Carl Friedrich Gauss Prize at the International Congress of Mathematicians ″for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory.″ Also in 2022 he received the Dirac Medal of the ICTP jointly with Joel Lebowitz and David Ruelle.

Lieb is a member of the U.S. National Academy of Sciences and has twice served (1982–1984 and 1997–1999) as the president of the International Association of Mathematical Physics. Lieb was awarded the Austrian Decoration for Science and Art in 2002. In 2012 he became a fellow of the American Mathematical Society and in 2013 a Foreign Member of the Royal Society.

In 2023 he received Kyoto Prize in Basic Sciences for his achievements in many-body physics.

Works

Lieb has made fundamental contributions to both theoretical physics and mathematics. Only some of them are outlined here. His main research papers are gathered in four Selecta volumes. More details can also be found in two books published by EMS Press in 2022 on the occasion of his 90th birthday. His research is reviewed there in more than 50 chapters.

Statistical mechanics, soluble systems

Lieb is famous for many groundbreaking results in statistical mechanics concerning, in particular, soluble systems. His numerous works have been collected in the Selecta ″Statistical mechanics″ and ″Condensed matter physics and exactly soluble models″, as well as in a book with Daniel Mattis. They treat (among other things) Ising-type models, models for ferromagnetism and ferroelectricity, the exact solution of the six-vertex model of ice in two dimensions, the one-dimensional delta Bose gas (now called the Lieb-Liniger model) and the Hubbard model.

Together with Daniel Mattis and Theodore Schultz he solved in 1964 the two-dimensional Ising model (with a new derivation of the exact solution by Lars Onsager via the Jordan-Wigner transformation of the transfer matrices) and in 1961 the XY model, an explicitly solvable one-dimensional spin-1/2 model. In 1968, together with Fa-Yueh Wu, he gave the exact solution of the one-dimensional Hubbard model.

In 1971 he and Neville Temperley introduced the Temperley-Lieb algebra in order to build certain transfer matrices. This algebra also has links with knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.

Together with Derek W. Robinson in 1972 he derived bounds on the propagation speed of information in non-relativistic spin systems with local interactions. They have become known as Lieb-Robinson bounds and play an important role, for instance, in error bounds in the thermodynamic limit or in quantum computing. They can be used to prove the exponential decay of correlations in spin systems or to make assertions about the gap above the ground state in higher-dimensional spin systems (generalized Lieb-Schultz-Mattis theorems).

In 1972 he and Mary Beth Ruskai proved the strong subadditivity of quantum entropy, a theorem that is fundamental for quantum information theory. This is closely related to what is known as the data processing inequality in quantum information theory. The Lieb-Ruskai proof of strong subadditivity is based on an earlier paper where Lieb solved several important conjectures about operator inequalities, including the Wigner-Yanase-Dyson conjecture.

In the years 1997–99, Lieb provided a rigorous treatment of the increase of entropy in the second law of thermodynamics and adiabatic accessibility with Jakob Yngvason.

Many-body quantum systems and the stability of matter

In 1975, Lieb and Walter Thirring found a proof of the stability of matter that was shorter and more conceptual than that of Freeman Dyson and Andrew Lenard in 1967. Their argument is based on a new inequality in spectral theory, which became known as the Lieb-Thirring inequality. The latter has become a standard tool in the study of large fermionic systems, e.g. for (pseudo-)relativistic fermions in interaction with classical or quantized electromagnetic fields. On the mathematical side, the Lieb-Thirring inequality has also generated a huge interest in the spectral theory of Schrödinger operators. This fruitful research program has led to many important results that can be read in his Selecta ″The stability of matter : from atoms to stars″ as well as in his book ″The stability of matter in quantum mechanics″ with Robert Seiringer.

Based on the original Dyson-Lenard theorem of stability of matter, Lieb together with Joel Lebowitz had already provided in 1973 the first proof of the existence of thermodynamic functions for quantum matter. With Heide Narnhofer he did the same for Jellium, also called the homogeneous electron gas, which is at the basis of most functionals in Density Functional Theory.

In the 1970s, Lieb together with Barry Simon studied several nonlinear approximations of the many-body Schrödinger equation, in particular Hartree-Fock theory and the Thomas-Fermi model of atoms. They provided the first rigorous proof that the latter furnishes the leading order of the energy for large non-relativistic atoms. With Rafael Benguria and Haïm Brezis, he studied several variations of the Thomas-Fermi model.

The ionization problem in mathematical physics asks for a rigorous upper bound on the number of electrons that an atom with a given nuclear charge can bind. Experimental and numerical evidence seems to suggest that there can be at most one, or possibly two extra electrons. To prove this rigorously is an open problem. A similar question can be asked concerning molecules. Lieb proved a famous upper bound on the number of electrons a nucleus can bind. Moreover, together with Israel Michael Sigal, Barry Simon and Walter Thirring, he proved, for the first time, that the excess charge is asymptotically small compared to the nuclear charge.

Together with Jakob Yngvason, he gave a rigorous proof of a formula for the ground state energy of dilute Bose gases. Subsequently, together with Robert Seiringer and Jakob Yngvason he studied the Gross-Pitaevskii equation for the ground state energy of dilute bosons in a trap, starting with many-body quantum mechanics. Lieb's works with Joseph Conlon and Horng-Tzer Yau and with Jan Philip Solovej on what is known as the law for bosons provide the first rigorous justification of Bogoliubov's pairing theory.

In quantum chemistry Lieb is famous for having provided in 1983 the first rigorous formulation of Density Functional Theory using tools of convex analysis. The universal Lieb functional gives the lowest energy of a Coulomb system with a given density profile, for mixed states. In 1980, he proved with Stephen Oxford the Lieb-Oxford inequality which provides an estimate on the lowest possible classical Coulomb energy at fixed density and was later used for calibration of some functionals such as PBE and SCAN. More recently, together with Mathieu Lewin and Robert Seiringer he gave the first rigorous justification of the Local-density approximation for slowly varying densities.

Selected publications

Books

• Lieb, Elliott H.; Seiringer, Robert. The stability of matter in quantum mechanics. Cambridge University Press, 2010 ISBN: 978-0-521-19118-0
• Lieb, Elliott H.; Loss, Michael. Analysis. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997. xviii+278 pp. ISBN: 0-8218-0632-7
• Lieb, Elliott H.; Seiringer, Robert; Solovej, Jan Philip; Yngvason, Jakob. The mathematics of the Bose gas and its condensation. Oberwolfach Seminars, 34. Birkhäuser Verlag, Basel, 2005. viii+203 pp. ISBN: 978-3-7643-7336-8; 3-7643-7336-9

Articles

• Statistical mechanics. Selecta of Elliott H. Lieb. Edited, with a preface and commentaries, by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+505 pp. ISBN: 3-540-22297-9
• Condensed matter physics and exactly soluble models. Selecta of Elliott H. Lieb. Edited by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+675 pp. ISBN: 3-540-22298-7
• The Stability of Matter: From Atoms to Stars. Selecta of Elliott H. Lieb (4th edition). Edited by W. Thirring, with a preface by F. Dyson. Springer-Verlag, Berlin, 2005. xv+932 pp. ISBN: 978-3-540-22212-5
• Inequalities. Selecta of Elliott H. Lieb. Edited, with a preface and commentaries, by M. Loss and M. B. Ruskai. Springer-Verlag, Berlin, 2002. xi+711 pp. ISBN: 3-540-43021-0

As editor

• Lieb, Elliott H. and Mattis, Daniel C., editors. Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles, Academic Press, New York, 1966. ISBN: 978-0-12-448750-5

Other

• The Physics and Mathematics of Elliott Lieb. Edited by R. L. Frank, A. Laptev, M. Lewin and R. Seiringer. EMS Press, July 2022, 1372 pp. ISBN: 978-3-98547-019-8

These are two books published by EMS Press on the occasion of Lieb's 90th birthday, which contain around 50 chapters about his impact on a very broad range of topics and the resulting subsequent developments. Many contributions are of an expository character and thus accessible to non-experts.

See also