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Elliott H. Lieb

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Elliott H. Lieb
Born(1932-07-31)July 31, 1932(age 91)
Boston,Massachusetts, U.S.
EducationMassachusetts Institute of Technology(BS)
University of Birmingham(PhD)
Known forAraki–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
AwardsHeineman Prize for Mathematical Physics(1978)
Max Planck medal
Birkhoff Prize(1988)
Boltzmann medal(1998)
Rolf Schock Prizesin 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
FieldsMathematics, Physics
InstitutionsPrinceton University
Doctoral advisorSamuel Frederick Edwards
Gerald Edward Brown
Doctoral studentsRafael Benguria
Jennifer Tour Chayes
Robert McCann
Jan Philip Solovej
Horng-Tzer Yau

Elliott Hershel Lieb(born July 31, 1932) is an Americanmathematical physicist.He is a professor of mathematics and physics atPrinceton University.Lieb's works pertain toquantumandclassical many-body problem,[1][2][3]atomic structure,[3]thestability of matter,[3]functional inequalities,[4]the theory ofmagnetism,[2]and theHubbard model.[2]

Biography[edit]

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 fromBessarabiaand worked as a secretary.[5]

Lieb received hisB.S.in physics from theMassachusetts Institute of Technologyin 1953[6]and his PhD in mathematical physics from theUniversity of Birminghamin England in 1956.[6][7]Lieb was aFulbright FellowatKyoto University,Japan (1956–1957),[6]and worked as the StaffTheoretical PhysicistforIBMfrom 1960 to 1963.[6]In 1961–1962, Lieb was on leave as professor of applied mathematics atFourah Bay College,theUniversity of Sierra Leone.[6]In 1963, he joined theYeshiva Universityas an associate professor.[5]He has been a professor at Princeton since 1975,[6]following a leave from his professorship at MIT.

Lieb is married to fellow Princeton professorChristiane Fellbaum.

For years, Lieb has rejected the standard practice of transferring copyright of his research articles toacademic publishers.Instead, he would only give publishers his consent to publish.[8]

Awards[edit]

Lieb has been awarded several prizes in mathematics and physics, including theHeineman Prize for Mathematical Physicsof theAmerican Physical Societyand theAmerican Institute of Physics(1978),[9]theMax Planck Medalof theGerman Physical Society(1992),[10]theBoltzmann medalof theInternational Union of Pure and Applied Physics(1998),[11]theSchock Prize(2001),[12]theHenri Poincaré Prizeof theInternational Association of Mathematical Physics(2003),[13]and theMedal of the Erwin Schrödinger Institute for Mathematics and Physics(2021).[14]

In 2022 Lieb was awarded theMedal for Exceptional Achievement in Researchfrom theAmerican Physical Societyfor ″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″[15] and theCarl Friedrich Gauss Prizeat theInternational 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.″[16]Also in 2022 he received theDirac Medalof the ICTP[17]jointly withJoel LebowitzandDavid Ruelle.

Lieb is a member of theU.S. National Academy of Sciences[18]and has twice served (1982–1984 and 1997–1999) as the president of theInternational Association of Mathematical Physics.[19]Lieb was awarded theAustrian Decoration for Science and Artin 2002.[20]In 2012 he became a fellow of theAmerican Mathematical Society[21]and in 2013 aForeign Member of the Royal Society.[22]

In 2023 Lieb receivedKyoto Prize in Basic Sciencesfor his achievements in many-body physics.[23]

Works[edit]

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.[1][2][3][4]More details can also be found in two books published byEMS Pressin 2022 on the occasion of his 90th birthday.[24]His research is reviewed there in more than 50 chapters.

Statistical mechanics, soluble systems[edit]

Lieb is famous for many groundbreaking results instatistical mechanicsconcerning, in particular, soluble systems. His numerous works have been collected in the Selecta″Statistical mechanics″[1]and″Condensed matter physics and exactly soluble models″,[2]as well as in a book with Daniel Mattis.[25]They treat (among other things)Ising-type models,models forferromagnetismandferroelectricity,the exact solution of thesix-vertex modelof ice in two dimensions, the one-dimensional delta Bose gas (now called theLieb-Liniger model) and theHubbard model.

Together with Daniel Mattis and Theodore Schultz, Lieb solved in 1964 the two-dimensionalIsing model(with a new derivation of the exact solution byLars Onsagervia theJordan-Wigner transformationof the transfer matrices) and in 1961 theXY model,an explicitly solvable one-dimensional spin-1/2 model. In 1968, together withFa-Yueh Wu,he gave the exact solution of the one-dimensional Hubbard model.

In 1971 Lieb andNeville Temperleyintroduced theTemperley-Lieb algebrain order to build certain transfer matrices. This algebra also has links withknot theoryand thebraid group,quantum groupsand subfactors ofvon Neumann algebras.

Together withDerek W. Robinsonin 1972, Lieb derived bounds on the propagation speed of information in non-relativistic spin systems with local interactions. They have become known asLieb-Robinson boundsand play an important role, for instance, in error bounds in thethermodynamic limitor inquantum computing.They can be used to prove the exponential decay of correlations in spin systems or to make assertions about the gap above theground statein higher-dimensional spin systems (generalized Lieb-Schultz-Mattis theorems).

In 1972 Lieb andMary Beth Ruskaiproved thestrong subadditivity of quantum entropy,a theorem that is fundamental forquantum information theory.This is closely related to what is known as thedata processing inequalityin 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.[26]

In the years 1997–99, Lieb provided a rigorous treatment of the increase of entropy in thesecond law of thermodynamicsandadiabatic accessibilitywithJakob Yngvason.[27]

Many-body quantum systems and the stability of matter[edit]

See also:stability of matter

In 1975, Lieb andWalter Thirringfound a proof of thestability of matterthat was shorter and more conceptual than that ofFreeman Dysonand Andrew Lenard in 1967. Their argument is based on a new inequality in spectral theory, which became known as theLieb-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, theLieb-Thirring inequalityhas also generated a huge interest in the spectral theory of Schrödinger operators.[28]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″[3]as well as in his book″The stability of matter in quantum mechanics″withRobert Seiringer.[29]

Based on the original Dyson-Lenard theorem of stability of matter, Lieb together withJoel Lebowitzhad already provided in 1973 the first proof of the existence of thermodynamic functions for quantum matter. With Heide Narnhofer he did the same forJellium,also called thehomogeneous electron gas,which is at the basis of most functionals inDensity Functional Theory.

In the 1970s, Lieb together withBarry Simonstudied several nonlinear approximations of the many-body Schrödinger equation, in particularHartree-Fock theoryand theThomas-Fermi modelof 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 andHaïm Brezis,he studied several variations of theThomas-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 withIsrael Michael Sigal,Barry SimonandWalter Thirring,he proved, for the first time, that the excess charge is asymptotically small compared to the nuclear charge.

Together withJakob Yngvason,Lieb gave a rigorous proof of a formula for the ground state energy of dilute Bose gases. Subsequently, together withRobert SeiringerandJakob Yngvasonhe studied the Gross-Pitaevskii equation for the ground state energy of dilute bosons in a trap, starting with many-body quantum mechanics.[30]Lieb's works with Joseph Conlon andHorng-Tzer Yauand withJan Philip Solovejon what is known as thelaw 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 theLieb-Oxford inequality[31]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 withMathieu LewinandRobert Seiringerhe gave the first rigorous justification of theLocal-density approximationfor slowly varying densities.[32]

Analysis[edit]

In the 70s Lieb entered the mathematical fields ofcalculus of variationsandpartial differential equations,where he made fundamental contributions. An important theme was the quest of best constants in several inequalities offunctional analysis,which he then used to rigorously study nonlinear quantum systems. His results in this direction are collected in the Selecta″Inequalities″.[4]Among the inequalities where he determined the sharp constants are Young's inequality and the Hardy-Littlewood-Sobolev inequality, to be further discussed below. He also developed tools now considered standard in analysis, such asrearrangement inequalitiesor theBrezis-Lieb lemmawhich provides the missing term inFatou's lemmafor sequences of functions converging almost everywhere.

With Herm Brascamp andJoaquin Luttinger,Lieb proved in 1974 a generalization of theRiesz rearrangement inequality,stating that certain multilinear integrals increase when all the functions are replaced by theirsymmetric decreasing rearrangement.WithFrederick Almgren,he clarified the continuity properties of rearrangement. Rearrangement is often used to prove the existence of solutions for some nonlinear models.

In two papers (one in 1976 with Herm Brascamp and another one alone in 1990), Lieb determined the validity and the best constants of a whole family of inequalities that generalizes, for instance, theHölder's inequality,Young's inequality for convolutions,and theLoomis-Whitney inequality.This is now known as theBrascamp-Lieb inequality.The spirit is that the best constant is determined by the case where all functions are Gaussians. TheBrascamp-Lieb inequalityhas found applications and extensions, for instance, in harmonic analysis.

Using rearrangement inequalities and compactness methods, Lieb proved in 1983 the existence of optimizers for theHardy-Littlewood-Sobolev inequalityand of theSobolev inequality.He also determined the best constant in some cases, discovering and exploiting the conformal invariance of the problem and relating it, viastereographic projection,to a conformally equivalent, but more tractable problem on the sphere. A new rearrangement-free proof was provided later with Rupert Frank, allowing to treat the case of the Heisenberg group.[33]

In a 1977 work, Lieb also proved the uniqueness (up to symmetries) of the ground state for the Choquard-Pekar equation, also calledSchrödinger–Newton equation,[34]which can describe a self gravitating object or an electron moving in a polarizable medium (polaron). With Lawrence Thomas he provided in 1997 a variational derivation of theChoquard-Pekar equationfrom a model in quantum field theory (theFröhlich Hamiltonian). This had been solved earlier byMonroe DonskerandSrinivasa Varadhanusing a probabilistic path integral method.

In another work with Herm Brascamp in 1976, Lieb extended thePrékopa-Leindler inequalityto other types of convex combinations of two positive functions. He strengthened the inequality and theBrunn-Minkowski inequalityby introducing the notion of essential addition.

Lieb also wrote influential papers on harmonic maps, among others withFrederick Almgren,Haïm BrezisandJean-Michel Coron.In particular, Algrem and Lieb proved a bound on the number of singularities of energy minimizing harmonic maps.

Finally, his textbook″Analysis″withMichael Lossshould be mentioned.[35] It has become a standard for graduate courses in mathematical analysis. It develops all the traditional tools of analysis in a concise, intuitive and eloquent form, with a view towards applications.

Selected publications[edit]

Books
  • Lieb, Elliott H.;Seiringer, Robert.The stability of matter in quantum mechanics.Cambridge University Press,2010ISBN978-0-521-19118-0[29]
  • Lieb, Elliott H.;Loss, Michael.Analysis.Graduate Studies in Mathematics,14. American Mathematical Society, Providence, RI, 1997. xviii+278 pp.ISBN0-8218-0632-7[35]
  • 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.ISBN978-3-7643-7336-8;3-7643-7336-9[30]
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.ISBN3-540-22297-9[1]
  • 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.ISBN3-540-22298-7[2]
  • 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.ISBN978-3-540-22212-5[3]
  • 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.ISBN3-540-43021-0[4]
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.ISBN978-0-12-448750-5[25]
Other

These are two books published byEMS Presson 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[edit]

References[edit]

  1. ^abcdStatistical mechanics: selecta of Elliott H. Lieb.Springer. November 29, 2004.ISBN3-540-22297-9.
  2. ^abcdefCondensed matter physics and exactly soluble models: selecta of Elliott H. Lieb.Springer. November 29, 2004.ISBN3-540-22298-7.
  3. ^abcdefThe stability of matter: from atoms to stars: selecta of Elliott H. Lieb(4th ed.). Springer. November 29, 2004.ISBN3-540-22212-X.
  4. ^abcdLoss, Michael; Ruskai, Mary Beth, eds. (2002).Inequalities: Selecta of Elliott H. Lieb.doi:10.1007/978-3-642-55925-9.ISBN978-3-642-62758-3.
  5. ^abPhysics, American Institute of (December 15, 2022)."Elliott H. Lieb".aip.org.RetrievedNovember 8,2023.
  6. ^abcdef"Lieb, Elliott H."American Institute of Physics.RetrievedJanuary 5,2020.
  7. ^"Elliott Lieb".Mathematics Genealogy Project.RetrievedJanuary 5,2020.
  8. ^Sternheimer, Daniel (April 5, 2022)."Some Multifaceted Aspects of Mathematical Physics, Our Common Denominator with Elliott Lieb".Axioms.11(10): 522.arXiv:2204.02160.doi:10.3390/axioms11100522.
  9. ^"1978 Dannie Heineman Prize for Mathematical Physics".American Physical Society.RetrievedJanuary 5,2020.
  10. ^"Preisträgerinnen und Preisträger, Max Planck Medaille".Deutsche Physikalische Gesellschaft(in German).RetrievedJanuary 5,2020.
  11. ^"The Boltzmann Award".The Web Archive.February 20, 2015. Archived fromthe originalon February 20, 2015.
  12. ^"Schock Prize 2001".Kungl. Vetenskaps-Akademien.RetrievedJanuary 5,2020.[permanent dead link]
  13. ^"The Henri Poincaré Prize".International Association of Mathematical Physics.RetrievedJanuary 5,2020.
  14. ^"ESI Medal".ESI.RetrievedJuly 2,2022.
  15. ^"2022 APS Medal for Exceptional Achievement in Research Recipient".RetrievedJune 15,2022.
  16. ^"Gauss Prize".RetrievedJuly 5,2022.
  17. ^"Dirac Medal".ICTP.RetrievedAugust 8,2022.
  18. ^"Elliott Lieb".U.S. National Academy of Sciences.RetrievedJanuary 5,2020.
  19. ^"About the IAMP – Past presidents".International Association of Mathematical Physics.RetrievedJanuary 5,2020.
  20. ^"Reply to a parliamentary question"(PDF)(in German). p. 1517.RetrievedNovember 19,2012.
  21. ^List of Fellows of the American Mathematical Society,retrieved January 27, 2013.
  22. ^"New Fellows 2013".Royal Society.RetrievedJuly 30,2013.
  23. ^"Elliott H. Lieb".Inamori Foundation.RetrievedJune 16,2023.
  24. ^abFrank, Rupert; Laptev, Ari; Lewin, Mathieu; Seiringer, Robert, eds. (2022).The Physics and Mathematics of Elliott Lieb, The 90th Anniversary Volume (vol 1 and 2).Vol. 2 (EMS Press ed.).arXiv:2202.01867.doi:10.4171/90.ISBN978-3-98547-019-8.
  25. ^abDyson, Freeman J.(1967). "Review ofMathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particlesby Elliott H. Lieb and Daniel C. Mattis ".Physics Today.20(9): 81–82.doi:10.1063/1.3034501.
  26. ^Lieb, Elliott H (December 1973)."Convex trace functions and the Wigner-Yanase-Dyson conjecture".Advances in Mathematics.11(3): 267–288.doi:10.1016/0001-8708(73)90011-X.
  27. ^Lieb, Elliott H.; Yngvason, Jakob (March 1999). "The physics and mathematics of the second law of thermodynamics".Physics Reports.310(1): 1–96.arXiv:cond-mat/9708200.Bibcode:1999PhR...310....1L.doi:10.1016/S0370-1573(98)00082-9.S2CID119620408.
  28. ^Frank, Rupert L.; Laptev, Ari; Weidl, Timo (2022).Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities.Cambridge University Press.
  29. ^abLieb, Elliott H.; Seiringer, Robert (November 5, 2009).The Stability of Matter in Quantum Mechanics.doi:10.1017/CBO9780511819681.ISBN9780521191180.
  30. ^abHoffmann-Ostenhof, T. (2007)."Book Review:The mathematics of the Bose gas and its condensation".Bulletin of the American Mathematical Society.44(3): 493–497.doi:10.1090/S0273-0979-07-01147-0.
  31. ^Lieb, Elliott H.; Oxford, Stephen (March 1981). "Improved lower bound on the indirect Coulomb energy".International Journal of Quantum Chemistry.19(3): 427–439.doi:10.1002/qua.560190306.
  32. ^Lewin, Mathieu; Lieb, Elliott H.; Seiringer, Robert (January 1, 2020). "The local density approximation in density functional theory".Pure and Applied Analysis.2(1): 35–73.arXiv:1903.04046.doi:10.2140/paa.2020.2.35.S2CID119176239.
  33. ^Frank, Rupert L.; Lieb, Elliott H. (July 1, 2012)."Sharp constants in several inequalities on the Heisenberg group".Annals of Mathematics.176(1): 349–381.doi:10.4007/annals.2012.176.1.6.
  34. ^Lieb, Elliott H. (October 1977). "Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation".Studies in Applied Mathematics.57(2): 93–105.doi:10.1002/sapm197757293.
  35. ^abLieb, Elliott H.; Loss, Michael (2001).Analysis: Second Edition.American Mathematical Society.ISBN978-0-8218-2783-3.

External links[edit]

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