Articles

Fields of Research

Introduction

A large variety of physical systems can be decribed in a unified way with the help of quantum field theory. The fields represent fluctuating physical quantities such as atomic positions, electric and magnetic fields, directions of molecules, etc., at each spacetime point. The materials can be solids, liquids, liquid crystals, or nuclei. Fields can also be used to study the changing shapes of membranes which are formed by bilipids or detergents in biophysics and chemical physics.

Field fluctuations are described most efficiently with the help of functional integrals, which form a common mathematical framework for explaining the phenomenon encountered in such systems. This description leads to a universal understanding of all continuous phase transitions. It also renders important insights into discontinuous transitions, such as the melting transition of crystals.

The research of this group is devoted to applying the functional techniques to novel systems. The purpose is to improve the available mathematical methods whenever necessary. Parallels are established with the theory of elementary particles.

1. Theory of Defect-Induced Phase Transitions

Solids and superfluids undergo phase transitions which can be explained by the large configurational entropy of line-like defects or vortex lines. In contrast to Landau's order field theory of phase transition, a powerful new disorder field theory has been developed in the textbook Gauge Fields in Condensed Matter. There it is shown that the long-range forces between the line structures can conveniently be described by a gauge theory. This makes the ensuing disorder field theories structurally very similar to the well-known Ginzburg-Landau theory of superconductivity, which is advantageous for extracting the physical consequences.

2. Quark Confinement and String Physics

Quarks are held together by color-electric flux tubes. Their formation can be studied best in a dual formulation of the Abelian Higgs model developed in my Book 1, and further in the Theory of Quark Confinement developed in Refs. 203, 205, 211.

The properties of the flux tubes are investigated with the help of a string model with curvature stiffness, which was first proposed by us (Ref. 149) and, independently, by A. M. Polyakov. The paper has instigated much research elsewhere. We have found that contrary to earlier expectations, the color-electric flux tube formed between a quark and an antiquark has a negative stiffness (Ref. 241).

A new model was found which has this property and, in addition, a smooth phase without the undesirabel phenomenon of "plumber's nightmare" (Ref. 276).

3. Quantum Mechanics

A new variational approach to path integrals developed in collaboration with R.P. Feynman (Ref. 159, see also here) has been extended systematically and has lead to exponentially fast convergent perturbation expansions. (Ref. 213, 220, 229 and Book 4). A further recent discovery is a variational approach to tunneling processes (Refs. 214, 221, 231). This help improving resummation techniques for divergent perturbation series (Refs. 228, 220).

It has led to the most precise theoretical predictions of the critical properties of statistical systems near phase transitions. See details the Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics , which has appeared in German under the title Pfadintegrale in Quantenmechanik, Statistik und Polymer Physics, B.I.-Wissenschaftsverlag, Mannheim, 1993 (Book 4). A nonholonomic mapping principle was found by which the laws quantum physics in spaces with curvature and torsion can be determined from the known laws in euclidean space by a nonholonomic mapping (Refs. 199, 202, 258; Path Integrals, Chapters 10, 11).

4. Classical and Statistical Mechanics in Spaces with Curvature and Torsion

The quantum equivalence principle necessitates the study of its classical limit. A surprising result was that the action principle for the derivation of the equations of motion is changed in an essential way with respect to previous formalisms if a space carries torsion (Ref. 219). As applications of the new action principle, we have derived correctly the Euler equations of motion of a rigid body in the body-fixed coordinate frame (Ref. 224). We have also derived the Fokker-Planck equation governing the Brownian motion of a particle in a space with curvature and torsion which determines the diffusion of classical particles through crystals with defects (Ref. 233).

5. Classical Statistics

The mechanism by which continuous phase transitions in systems with line-like excitations (such as vortex or defect lines) become first-order has been explained with the help of disorder fields (Refs. 131, 194; Books 1 and 2).

An important example is the melting transition, where we have found the first model undergoing two successive continuous melting transitions (Refs. 174, 179; Books 1 and 2).

6. Quantum Statistics

The above variational approach is being used to develop very accurate approximations to quantum statistical partition functions and particle densities (see Chapter 5 in Path Integrals). The operator quantum Langevin equation has been derived for the first time from the forward-backward path integral approach (Ref. 230).

Field Theory of Critical Phenomena The universal critical exponents at a second-order phase transition can be studied by renormalization group techniques. Our group was the first to determine exactly, in collabotration with a Russian group, the epsilon expansion of an O(N)-symmetric phi^4-theory up to five loops (Ref. 208). Also the asymmetric case was solved (Ref. 225). The variational approach to quantum mechanics was extended to quantum field theory resulting in a fully-fledged strong-coupling quantum field thory. This has led to a novel approach to critical phenomena with extremely accurate results for critical exponents, without use of the renormalization group (Refs. 257, 263, 275, 290). The new theory was so successful that it led to Book 6.

7. Polymer Physics

A field theory is under construction to solve the entanglement problem of polymer physics. For a first formulation see the above textbook on Path Integrals.

8. Field Theory of Liquid Crystals

Smectic, nematic, and cholesteric liquid crystals possess interesting phase transitions which are studied by field theoretic techniques. A disorder field theory opens new perspectives for understanding the transition (Ref. 102). A quasicrystalline phase with icosahedral structure was theoretically proposed by us 3 years before the experimental discovery of such a material (Ref. 75). Click here, for more details  .

9. Fluctuation Effects in Membranes

Many physical systems contain flexible membranes. Examples are red blood cells, soap layers between oil and water, and bilipid vesicles. The phase transition in such systems are studied via a suitably constructed field theory. The universal constant in the pressure law of an ideal gas of layered membranes was first determined by us with great precision via Monte Carlo calculations (Refs. 133, 143, 184), and recently analytically (225). An experimentally-observed spiky superstructure on membranes was found theoretically (266).

10. Superfluid Helium 3

Functional methods have been used to develop a theory of the long-wavelength phenomena in superfluid Helium 3 on the basis of collective fields (Ref. 55). A new texture was discovered theoretically (Ref. 59) and found later experimentally (Refs. 133, 143, 184).

11. Superconductivity

A disorder field theory was developed for superconductors in Refs. 97, 98, 1, and has allowed us to predict a tricritical point in the superconducting phase transition which was confirmed recently by Monte Carlo simulations. It has also permitted a first determination of the critical exponents of the transition (Ref. 226).

12. Mathematical Physics

Exact solution methods are developed for path integrals. The most prominent method was found in 1979 in collaboration with Duru (Refs. 65, 83). It has meanwhile led to the solution of many path integrals which previously appeared unsolvable (Book 11). Recent examples are the path integrals for a point particle on spherical surfaces and on group spaces (Ref. 202), as well as for the relativistic Coulomb system (Ref. 232). Another important result obtained in mathematical physics is the solution of the problem of defining products of distributions such as delta- and Heaviside functions. It results from the requirement of invariance of path integrals under coordinate tranformations. This is necessary for the equivalence to Schroedinger theory. This fixed all products of distributions, as shown in Ref. 303. The result is the same as can be found from dimensional resularization (see Ref. 305, also Chapter 10 of the textbook on Path Integrals).

13. Stochastic Physics

The powerful Duru-Kleinert method for solving path integrals is being extended to Markov processes. In collaboration with A. Pelster, a transformation has been found in Ref. 249, by which Fokker-Planck equations of different Markov processes can be transformed into each other. This allows us to relate unsolved to solved problems, and may the key to finding solutions for many as yet untackled equations.

14. Supersymmetry in Nuclear Physics

Nuclear spectra show broken supersymmetry as was first pointed out by us in the 1978 Erice School on The New Aspects in Nuclear Physics. For details see here.

15. Financial Markets

Path integrals are a powerful tool for studying fluctuations in financial markets which are non-Gaussian. The traditional assumption of Gaussian distribution severely underestimates the probability of large jumps in asset prices and this was the main reason for the catastrophic failure in the early fall of 1998 of the hedge fund Long Term Capital Investment, which had the Nobel price winners Scholes and Merton on the advisory board (and as shareholders). The fund contained derivative with a notional value of 1,250 Billion US$. The fund collected 2% for administrative expenses and 25% of the profits, and was initially extremely profitable. It offered its shareholders returns of 42.8% in 1995, 40.8% in 1996, and 17.1% even in the disastrous year of the Asian crisis 1997. But in September 1998, after mistakenly gambling on a convergence in interest rates, it almost went bankrupt. A number of renowned international banks and Wall Street institutions had to bail it out with 3.5 Billion US$ to avoid a chain reaction of credit failures. Starting from a path integral formulation, we have developed a generalization of Ito's stochastic calculus to non-Gaussian fluctuations (Ref. 329) and derived a new option pricing formula from this (Ref. 333). A detailed theory is contained in my textbook on path integrals.

Theories

• Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics
• Duru-Kleinert Transformation for Exact Solution of Path Integrals
• Feynman-Kleinert Variational Approach (see Collaboration with R.P.Feynman)
• Disorder Field Theory of Phase Transitions
• Dual Theory of Superconductivity
• Five-Loop Calculation of Critical Exponents in 4-epsilon Dimensions
• Strong-Coupling Theory of Critical Behavior (more here)
• Kleinert Criterion for Onset of Phase Decoherence
• Variational Perturbation Theory for Field Theory with Anomalous Dimensions turning Divergent Weak-Coupling to Convergent Strong-Couling Expansions
• Vortex Gauge Field Theory of Superfluid Phase Transition
• Defect Gauge Field Theory of Crystal Melting
• Monopole Gauge Field Theory of Confinement
  • The Extra Gauge Symmetry of String Deformations in Electromagnetism with Charges and Dirac Monopoles
  • Double-Gauge Invariance and Local Quantum Field Theory of Charges and Dirac Magnetic Monopoles
  • Abelian Double-Gauge Invariant Continuous Quantum Field Theory of Electric Charge Confinement
• Collective Quantum Fields in Superconductors and Superfluid He3
• Hadronization of Quark Theories
• Polyakov-Kleinert Stiff String
• Pressure Constant for Stacks of Membranes
• Most Accurate Specific Heat Exponent alpha of Superfluid Helium
• Algebra of Regge Couplings
• Group Dynamics of Hydrogen Atom [the group O(4,2)]
• Supersymmetry in Nuclear Physics 

  Discovery of Supersymmetry in Nuclei

  There is experimental evidence for supersymmetry in nuclei (see citations below). Its existence (in broken form) was first proposed by Hagen Kleinert during a discussion session after a lecture of Sergio Ferrara on Supersymmetric Theories of Fundamental Interactions held at the 1978 Erice School. It is published in:

  The New Aspects of Subnuclear Physics , Plenum Press, N.Y., 1980 (ed. by A. Zichichi), pp. 1--37., pp. 39--58.
  The discussion sessions are reprinted on pp. 39--58.

  It documents that upon a question posed by the Yale student Yuan K. HA (now professor at Temple University), whether there could be supersymmetry in nuclei, was answered affirmatively by Kleinert who pointed out that a simple model of even and odd nuclei exhibits broken supersymmetry. Kleinert had previously presented auch a model at the 1977 Erice Summer School on Low Temperature Physics for a different purpose [lecture is published under the title Collective Quantum Fields, Fortschr. Physik 26, 565-671 (1978). (Start to read at bottom page 5)].

  Discussion manuscript, Erice 1978 

  Experimental Evidence

  Evidence for the Existence of Supersymmetry in Atomic Nuclei, Metz1 et al., Phys. Rev. Letters 83, 1542 (1999)

  For experimetal evidence of supersymmetry in nuclei see:
  Nuclear structure of ¹⁹⁶Au: More evidence for its supersymmetric description, J. Groeger et al., Physical Review C 62, 643044 (2000)

• New Stochastic Calculus for Non-Gaussian Processes
• Financial Markets---Option Pricing for Non-Gaussian Price Fluctuations
• A World Crystal 

  World Crystal

  The World Crystal was proposed as a toy model of the universe in the paper:

  H. Kleinert, Gravity as Theory of Defects in a Crystal with Only Second-Gradient Elasticity , Ann. d. Physik, 44, 117 (1987)

  It is supposed to illustrate that curvature and torsion may arise from defects and that the universe may have a short-distance structure which may completely surprise us when we shall be able to do experiments at Planck distances. More is explained in the textbook.

  The Italian artist Laura Pesce was inspired by the theory and created three glass paintings of the world crystal.

  1. One of these can be seen on the home page of the artist. The Italian title (Mondo Cristallizzato) can be seen on this page (picture lower left corner).
  2. A second picture of the world crystal can be seen here on the photo with the artist on this page (photo on the right).

  

• No-Go Theorem for Tachyons 

  No-Go Theorem for tachyons

  A physical theory cannot have any tachyon. A propagator has at a nonzero value of q² a pole implies the existence of particle moving with a velocity faster than light, a so-called tachyon. The possible existence of such states was conjectured for a long time since it was an allowed state in an irreducible representation of the Poincare group. They were an imporant output of infinite-component wave equations which were a popular subject of study in the sixties. See for example this paper. They also exist in the best existing quantum field theory so far, the quantum electrodynamics of electrons and photons (QED).

  The textbook can be read online here. There they appear at very large masses which lie far above the validity of this theory. They are called Landau ghosts. They also appear in attempts to quantize gravity by making it emerge from a conformally invariant action.

  At the end of Chapter 18 of the above book a no-tachyon theorem is stated as follows:

  A particle moving faster then light is removed in nature by the system undergoing a phase transition to a state of lower energy. In field theory this is usually a state with another vacuum expectation value of the fields. This is the typical way how nature deals with all unstable physical systems: For example, a timber-framed house will be unstable if its small vibrations have some eigenfrequency with a negative square value. Upon a small vibration, the amplitudes of such oscillations grow large exponentially fast in time. As a consequence, the structure will collapse into a state of lower energy. If there is again a negative square frequency, the intermediate state will collapse again. This process will go on until the stucture has settled as a stable ruin. In the ruin, all frequencies have positive square values, and no tachyon is left.
• Dark Matter and GIMPs

Collaboration with R.P. Feynman at Caltech

Richard Feynman was one of the most fascinating characters of the 20th century, both as a physicist and as a person. My friendship with him had a rather slow start, mostly due to the respect he inspired to me and the other young people around him. I met him first in 1973 when spending a sabbatical winter semester at Caltech. I was invited by Murray Gell-Mann after he had heard a lecture of mine on Current Algebra at a winter school in Schladming, Austria (reporting on the results of a paper I had written at CERN, Geneva).

Caltech was an extremely exciting place. The theory department met every Wednesday for a luncheon seminar where everyone had a chance of presenting his problems and solutions. At that time, experimentalists had discovered point-like constituents in hadrons with the help of deep inelastic scattering of electrons. The point-like structure had been explained phenomenologically by Feynman with his parton model. Gell-Mann was trying to go further by constructing a fundamental quantum field theory which would combine the point-like structure with well-known results of his Current Algebra. So he gave partons the quantum numbers of quarks and described them by fundamental fermion fields. Together with Harald Fritzsch he had just shown that currents constructed from free quark fields would explain most of the data. What was missing at that time was an explanation why free fields worked so well although nobody was able to detect quarks as particles in the laboratory! The model had, however, an important weakness, as was immediately noticed by Feynman: It did not account for the fact that high-energy collisions produced jets of particles --- the free-quark model would lead to uniform distributions. It was Gell-Mann who realized that local color gauge fields could provide them with the necessary glue to keep them forever inside the hadrons. The point-like structure was assured by what is called asymptotic freedom of color gauge theory, which ensures that quarks inside hadrons would behave almost like free particles. This property had just been discovered independently by 't Hooft, Politzer, and by Gross and Wilczek.

Unfortunately, I did not participate in this fundamental endeavor since I was working on a field-theoretic derivation of the Algebra of Regge Residues (see also here), which had been postulated seven years earlier on phenomenological grounds by Cabibbo, Horwitz, and Ne'emann [Phys. Lett. 166, 1786 (1968)]. I found a derivation by generalizing Current Algebra to an algebra of bilocal quark charges of free quark fields. In fact, Yuval Ne'eman [who had discovered in 1961, simultaneously with Gell-Mann, the fact that mesons and nucleons occur in multiplets representing the symmetry group SU(3) and who became later Israel's Minister of Science and development (1982--1984) and of Energy (1990--1992)] was my office mate at Caltech in 1980 and was very happy about my derivation. We became close friends, and whenever he appeared on a ministerial visit in Berlin, he always found some spare time to meet me and discuss physics (protected by several body guards sitting at the tables around us).

Gell-Mann, however, convinced me that according to my derivation, the algebra was not exact but only approximately true due to logarithmic corrections. This made it uninteresting to him. He always said "Don't waste your time with theories which do not have a chance of being true.". He taught me that theories which are only approximate from the beginning are not worth pursuing.

This was quite different with Feynman who loved simple models which can explain things approximately. Feynman appeared regularly at the seminars, and it was an experience to witness the pointed discussions evolving between him and Gell-Mann. There was always a tension between them, a certain rivalry, which led to interesting exchanges. Some of them were plainly silly: for instance, a speaker said on the blackboard: "I am now using Feynman's parton model" and was interrupted by Gell-Mann with the provocative question "What's that?" (whose answer he knew, of course). Feynman responded with a boyish smile: "It's published"! Upon which Gell-Mann said "You mean that phenomenological model which I superseded with my quark field theory?".

The students thoroughly enjoyed seeing the greatest physicists of that time fighting like kids.

In this environment I had the opportunity of participating in a series of seminars Feynman gave to graduate students and postdocs on path integrals with applications to quantum electrodynamics. He told us that when he had first come to Caltech as a young professor he had used path integrals quite extensively in his course on quantum mechanics. Later, however, he had given up on this since he was tired of confessing to the students that he was unable solve the path integral of the most fundamental atomic system, the hydrogen atom, whose solution is so easy in Schrödinger wave mechanics. Feynman knew that I had developed the group theory of the hydrogen atom in my 1967 Ph. D. Thesis, so he challenged me to try my luck with the path integral. We tried a number of things on the blackboard which, however, did not lead to anything.

When I returned home to Berlin in spring 1974 the problem stuck in my head while I was working out my 1976 Erice Lecture "On the Hadronization of Quark Theories" [in which I calculated the mass differences between current and constituent quarks and various current algebra relations]. The hydrogen atom sufaced again in 1978 when a Turkish postdoc, H. Duru, came to me as a Humboldt fellow. He was familiar with my thesis, so I told him Feynman's challenge, and we began searching for the solution. It turned out that Feynman's definition of path integrals as a product of a large but finite number of ordinary integrals was in principle unable to describe the hydrogen atom. The situation is similar to ordinary integrals: if a function is too singular, these can not be approximated by finite sums. We published the solution in 1979, and it became the basis for solving all path integrals whose Schrödinger equation can be solved analytically (see my textbook on this subject).

This success encouraged me to loose my shyness when meeting Feynman again on another sabbatical which I spent mostly in Santa Barbara in 1984. I frequently drove to Pasadena, and met with him to discuss physics, and he always had time for me. We talked for a few hours and usually went to a diner to have a soup together. He was sometimes very funny. Once he said conspiratively to me: "Hagen, I know you have a loose mouth. I shall show you a secret, but only if you promise to tell it to everybody." I did, and he pulled out a photo which showed him in a huge bathtub with three(!) beautiful long-haired California beauties in his arms.

His office had a wall full of tightly filled notebooks which he occasionally pulled one to show me some calculations he had done earlier on the topic we were discussing. He showed me, in particular, long calculations which had not produced any interesting results so far. I still possess a copy of such notes where he calculated the properties of an analog of a polaron, where the role of the particle coupled to phonons is played by a spin. He always wanted me to find a nice physical system where his model could be applied, but I did not succeed. He put a message to his secretary Helen Tuck to send me these notes on his blackboard, which remained there until he died in February 1989. It is visible in Fig. 1a: "Feynman's last blackboard" in the February issue of Physics Today in 1989, page 88. The relevant section of the blackboard is pictured in Fig. 1. Right bottom shows note to his secretary Helen Tuck asking her to send his calculations on self-coupled spin system to Kleinert.

Fig. 1a: Feynman's last blackboard in Physics Today, February 1989 issue, p. 88.

Fig. 1b: Feynman's last blackboard (zoom to the red frame shown in a)

Feynman's notes were always filled with lots of numerical calculations, with long columns of numbers obtained with the help of a pocket calculator. It reminded me of Riemann's notes, which are filled with such columns. Feynman told me that he liked to get numerical results. This was necessary back in the forties when working in Los Alamos on the Manhattan Project. At that time the machines were quite primitive by today's standards. Students often believe that great theoreticians design only abstract formulas and leave the calculations to others. But this is not so. Feynman always insisted in going to the end to get numbers, And when these did not fit the data, this was an important source of discoveries.

One of the set of notes which he showed me to in 1982 was easier to turn into physical results. These contained the variational calculations which are published in his 1972 Benjamin book on "Statistical Mechanics". Feynman suggested to me to work out a simple generalization, leading to what we called the effective classical potential. Fortunately, a simple cheap computer (the Sinclair ZX81, see Fig. 2) had become available a year earlier, and was just becoming very cheap in the supermarkets (15$ at Woolworth).

Fig. 2 The Sinclair ZX81 Computer bought at Woolworth for 15 US$.

It connected to a television set and worked at a frequency of 3.25 MHz with a memory of 1 Kilobyte. With this little machine, I easily did the necessary calculations and wrote up the manuscript in Spring 1983. When I told him I had written up the paper Feynman came to visit me at the ITP in Santa Barbara to discuss the results. When he arrived everybody wanted to talk to him and he was pressed to give a seminar. He finally agreed, although he was not in good shape at that time. He began his talk with the words: "Sorry, that I am unprepared but I came here only to finish a paper with my friend Hagen". His talk dealt with the proliferation of vortex lines explaining the critical properties of superfluid helium. He had proposed this mechanism in a lecture in 1955, probably inspired by a similar proposal by Shockley in 1952 that the proliferation of defect lines should drive the melting transition. Both have probably known the pioneering 1949 paper by Onsager who had explained the phase transition of the two-dimensional Ising model by the proliferation of the line-like domain walls between up and down spins, and conjectured a similar mechanism for the vortex lines in superfluid helium. To his surprise, Feynman encountered strong but unjustified criticism from the young people in the audience who wanted to show that they were smart.

They had just learned to calculate critical properties by applying the renormalization group to complex scalar field theory, and did not believe that vortex lines were relevant. At that time, the disorder field description of superfluid helium was not yet common knowledge. These describe ensembles of vortex lines by fields whose Feynman diagrams are direct pictures of these lines. I had inroduced these fields in 1982 and published a full theory in a textbook in 1989 (Gauge Fields in Condensed Matter, Vols. I and II). After his talk and the fights, Feynman was extremely exhausted and disappointed by the aggressiveness of the audience. He lay down in my office and sighed "I should not have talked! Why am I doing this to myself?". When he returned to Caltech he felt quite ill and I was very worried. Of course, I did not dare to press him to go through our paper and permit me to send it off to a journal.

In February 1984 I returned to Berlin to my regular teaching duties, and forgot all about our manuscript, when in a night of May 1986 the telephone rang and Helen Tuck, Feynman's secretary, told me that Feynman found the paper OK as it was and wanted me to submit it to Physical Review A. It was interesting to observe the reaction of the referees to a paper with Feynman as an author: One wrote: "The manuscript shows the clarity and conciseness typical for Feynman's writing". The paper appeared in December 1986. Feynman never expected the method would be applicable to quantum field theory. However, I have found a way of doing this, and developed what I have named Field-Theoretic Variational Perturbation Theory .

This led to the most reliable calculations of critical properties of systems near phase transitions so far. In particular, the most-accurately known singularity of the specific heat of superfluid helium which has been measured with great effort in a satellite experiment by J. Lipa and collaborators was precisely predicted, and I have written a textbook on this subject (Critical Properties of phi⁴ Theories).

Inspired by the collaboration with Feynman I published in 1990 the first edition of my textbook on Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics which has become a real best seller. It also contains applications to financial markets. Without him, this book would have never appeared and many problems would have remained unsolved.

Another system we discussed many times was a stack of biomembranes. If they are contained between two walls they exert a pressure obeying a law very similar to the ideal gas law (Helfrich's law: p d^3=c T^2/k, where p=pressure, d=distance between the walls, T=temperature, k=stiffness of membranes). The challenge was to find the proportionality constant c which was only known from Monte-Carlo simulations. Unfortunately we could not find a solution at that time. The problem was not forgotten, however, and many years later I succeeded in solving it with the help of field-theoretic variational perturbation theory. Of all the physicists I have known, Feynman impressed me most with the simplicity and elegance with which he articulates and solves complicated problems. His course I attended was extremely clear. He never used camouflaging mathematical language to express physical facts. If someone did, he always stopped him with the question "what's that?", and insisted on a down-to-earth explanation. He never made the student feel stupid (unless he really was) and always explained his ideas to exhaustion. He was a perfect teacher. It is somewhat amazing that he has produced only a relatively small number of excellent postdocs of his own, in comparison with J.A. Wheeler or J. Schwinger. I guess the reason must be that too few students had the courage to approach him. Feynman has certainly shaped the thinking of all Caltech students who went through his classes even if they graduated with other thesis advisors. And, of course, the many students and colleagues around the world who have studied his fascinating textbooks on physics.

Documents on H. Kleinert's Collaboration with R.P. Feynman

• This is Kleinert's and Feynman's joint paper:
  Effective Classical Partition Functions
  Phys. Rev. A 34, 6, 5080 (1986)
• Excerpt on this collaboration from the biography by John and Mary Gribbin:
  "RICHARD FEYNMAN, A Life in Science", Viking, NY, 1997
• The article "Travailler avec Feynman" in the French journal POUR LA SCIENCE: download
• Printing errors in the book on path integrals by Feynman and Hibbs (collected by Tim Hatamian)

picture gallery

with P.-G. De Gennes in a Discussion in 1998 in Berlin

with Walter Kohn at a party in Santa Barbara in 1982 (together with Annemarie Kleinert, Douglas Scalapino, and his wife)

with Cecile De Witt-Morette in a Discussion in 1998 in Berlin

Signing the Documents for the IRAP Programm in 2002 with the president of the University Nice-Sophia Antipolis (seated between me and Remo Ruffini).

with Jim Hartle and Thomas Elze at Conference Party at Piombino, Italy, in Summer 2004 (together with Annemarie Kleinert (l) and Laura Pesce (r), alias Ms. Elze)

with Jack Steinberger at Cern Cafeteria in May 2005

with Murray Gell-Mann (together with Annemarie Kleinert)

with Gerard 't Hooft (together with Josef Devreese at Devreese's birthday festival in 2002)

with Annemarie Kleinert (together with Hagen Kleinert at Sebastian Brandt's Diploma party in 2003)

with Annemarie Kleinert (together with Hagen Kleinert at party in 2006)

with Annemarie Kleinert (together with Hagen Kleinert receiving the Max-Born-Prize 2008 in London)

with Gerard 't Hooft and Frank Wilczek (at 't Hooft's 60th Birthday Conference together with Betteke 't Hooft and Wilczek's wife, Betsy Devine, 2004)

with Gerard 't Hooft (at 't Hooft's 60th Birthday Conference together with Betteke 't Hooft and Annemarie Kleinert, 2004)

at 't Hooft's 60th Birthday Conference together with Julius Wess and Annemarie Kleinert, 2004

with Remo Ruffini (at Marcel Grossmann Meeting 11 in Berlin together with Anita Ehlers, 2004)

with Sasha Polyakov (at Marcel Grossmann Meeting 11 in Berlin 2004)

with Francis Everitt (at Marcel Grossmann Meeting 11 in Berlin together with Annemarie Kleinert and Wolfard Janke, 2004)

with a picture of world crystal by Italian artist Laura Pesce 2005

with Laura Pesce and the picture of Instanton, 2005

with Gerard 't Hooft (at Kleinert's 60th birthday party)

with Gerard 't Hooft (at Kleinert's 60th birthday party)

with Gerard 't Hooft (at Kleinert's 60th birthday party)

with Gerard 't Hooft (at a 1978 party in Kleinert's home)

with Gerard 't Hooft (at Kleinert's 60th birthday party)

with Gerard 't Hooft (at Lisbon 1981 conference, photo taken by a street photographer for newly-married couples with an ancient plate camera)

Annemarie Kleinert with Murray Gell-Mann und Harald Fritzsch at Harald's Emeritus Party in Hofbräuhaus in 2008 (Foto taken by Ted Hänsch) )

with Academic Senate of West University of Timisoara at Dr. h.c. -ceremony in Timisoara 2009

with Rector of West University of Timisoara at Dr. h.c. -ceremony in Timisoara 2009

with Annemarie Kleinert at a Table exhibiting my books in the library of West University of Timisoara at the Dr. h.c. -ceremony in Timisoara 2009

Political discussion with the King of Romania his Majesty Mihai I at Dr. h.c. -ceremony in Timisoara 2009

Representing the Specola Vaticana at the ICRANet board of directors meeting in 2009. The rightmost person is Riccardo Giacconi, one of the fathers of X-ray astronomy.

with Murray Gell-Mann and his lady friend Michelle Gaugy (left), Agnès Rampal (Maire-Adjoint déléguée à l´Enseignement Supérieur et aux Rapatriés), Remo Ruffini, Thibault Damour, and Annemarie Kleinert in Villa Ratti, Nizza during Gell-Mann's visit in June 2012 signing Villa Ratti's wall of fame

with Murray Gell-Mann greeting Agnès Rampal (Maire-Adjoint déléguée à l´Enseignement Supérieur et aux Rapatriés), Annemarie Kleinert in garden of Villa Ratti, Nizza during Gell-Mann's visit in June 2012

with Gerard 't Hooft, Annemarie Kleinert, and Betteke 't Hooft in Taipeh at the conference "Horizons"

Kleinert's research group 1996

Kleinert's research group 1998

Kleinert's research group 2000

Kleinert's research group 2003

Kleinert's research group 2006

Kleinert's research group 2009

Kleinert's research group 2010

Kleinert's research group 2011

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Germany

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Responsible contact for any questions regarding our privacy policy:
Michael Kleinert, Liebensteinstr. 6, 14195 Berlin, GERMANY

When you access this website through your web browser, the data of your specific request is submitted to our systems. The data contains the requested web page, the type of your web browser, your computer's operating system, the hostname of your internet service provider, and your current internet protocol address (IP address). The data itself does not allow a direct reconstruction of who you are. However, collecting the data is technically required to deliver the content of the page you requested to you. This is how the internet works. Since the protection of your privacy is one of our highest priorities, your data is not stored in a permanent way on our systems. All your data is deleted after delivering the requested content to you. Since the data is not stored on a hard disk, and just kept in our system's memory for the short time of your request, it becomes unlikely that, e.g., in the worst case of a successful cyber attack, your data may be compromised.

All the data exchanged between your web browser and our systems is protected by modern encryption technologies (TLS).

We collect data that is necessary to process your request and deliver the content to your web browser. The data is deleted immediately after the connection between your web browser and our systems is closed.

You have the rights to be informed about all your data that is stored on our systems. Furthermore, you have the rights that we change or delete your data upon your request.

This privacy policy may be changed in the future to always fulfill the laws. In any case, for each request of our website the currently valid privacy policy may be found on this page.

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