Collective Classical and Quantum Fields

in Plasmas, Superconductors, Superfluid 3He, and Liquid Crystals
Hagen Kleinert
pp. 1-424, World Scientific, Singapore 2017
ISBN: 978-981-3223-93-6 (hardcover)
ISBN: 978-981-3223-94-3 (softcover)
ISBN: 978-981-3223-96-7 (ebook)
World Scientific,

This is an introductory book dealing with collective phenomena in many-body systems. A gas of bosons or fermions can show oscillations of various types of densities. These are described by different combinations of field variables. Especially delicate is the competition of these variables. In superfluid 3He, for example, the atoms can be attracted to each other by molecular forces, whereas they are repelled from each other at short distance due to a hardcore repulsion. The attraction gives rise to Cooper pairs, and the repulsion is overcome by paramagnon oscillations. The combination is what finally led to the discovery of superfluidity in 3He. In general, the competition between various channels can most efficiently be studied by means of a classical version of the Hubbard-Stratonovich transformation.

A gas of electrons is controlled by the interplay of plasma oscillations and pair formation. In a system of rod- or disc-like molecules, liquid crystals are observed with directional orientations that behave in unusual five-fold or seven-fold symmetry patterns. The existence of such a symmetry was postulated in 1975 by the author and K. Maki. An aluminium material of this type was later manufactured by Dan Shechtman which won him the 2014 Nobel prize. The last chapter presents some solvable models, one of which was the first to illustrate the existence of broken supersymmetry in nuclei.

Particles and Quantum Fields

Hagen Kleinert
pp. 1-1628, World Scientific, Singapore 2015
ISBN: 978-981-4740-89-0 (hardcover)
ISBN: 978-981-4740-90-6 (softcover)
ISBN: 978-981-4740-92-0 (ebook)
World Scientific,

This is an introductory book on elementary particles and their interactions. It starts out with many-body Schrödinger theory and second quantization and leads, via its generalization, to relativistic fields of various spins and to gravity. The text begins with the best known quantum field theory so far, the quantum electrodynamics of photon and electrons (QED). It continues by developing the theory of strong interactions between the elementary constituents of matter (quarks). This is possible due to the property called asymptotic freedom. On the way one has to tackle the problem of removing various infinities by renormalization. The divergent sums of infinitely many diagrams are performed with the renormalization group or by variational perturbation theory (VPT). The latter is an outcome of the Feynman-Kleinert variational approach to path integrals discussed in two earlier books of the author, one representing a comprehensive treatise on path integrals, the other dealing with critial phenomena. Unlike ordinary perturbation theory, VPT produces uniformly convergent series which are valid from weak to strong couplings, where they describe critical phenomena.

The present book develops the theory of effective actions which allow to treat quantum phenomena with classical formalism. For example, it derives the observed anomalous power laws of strongly interacting theories from an extremum of the action. Their fluctuations are not based on Gaussian distributions, as in the perturbative treatment of quantum field theories, or in asymptotically-free theories, but on deviations from the average which are much larger and which obey power-like distributions.

Exactly solvable models are discussed and their physical properties are compared with those derived from general methods. In the last chapter we discuss the problem of quantizing the classical theory of gravity.

Path Integrals

in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
Hagen Kleinert
5th edition, pp. 1-1624, World Scientific, Singapore 2009
ISBN: 978-981-4273-55-8 (hardcover)
ISBN: 978-981-4273-56-5 (softcover)
ISBN: 978-981-4365-26-0 (ebook)
World Scientific,

This is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations.

In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions.

The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena.

Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders.

Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern–Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.

The relevance of path integrals to financial markets is discussed, and improvements of the famous Black–Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions.

  • Spanish translation by Daniel Olguin (, with the help of Pablo Medina Zepeda (
    eae-publishing, 2017, ISBN: 978-363-9533-04-0,
  • Chinese translation by Dr. Cong-Feng Qiao (

Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity

Hagen Kleinert and Robert T. Jantzen (eds.)
pp. 1-3120, World Scientific, Singapore 2008
ISBN: 978-981-2834-26-3 (hardcover)
ISBN: 978-981-4470-03-2 (ebook)
World Scientific,

The Marcel Grossmann Meetings are three-yearly forums that meet to discuss recent advances in gravitation, general relativity and relativistic field theories, emphasizing their mathematical foundations, physical predictions and experimental tests. These meetings aim to facilitate the exchange of ideas among scientists, to deepen our understanding of space-time structures, and to review the status of ongoing experiments and observations testing Einstein's theory of gravitation either from ground or space-based experiments. Since the first meeting in 1975 in Trieste, Italy, which was established by Remo Ruffini and Abdus Salam, the range of topics presented at these meetings has gradually widened to accommodate issues of major scientific interest, and attendance has grown to attract more than 900 participants from over 80 countries. This proceedings volume of the eleventh meeting in the series, held in Berlin in 2006, highlights and records the developments and applications of Einstein's theory in diverse areas ranging from fundamental field theories to particle physics, astrophysics and cosmology, made possible by unprecedented technological developments in experimental and observational techniques from space, ground and underground observatories. It provides a broad sampling of the current work in the field, especially relativistic astrophysics, including many reviews by leading figures in the research community.

Multivalued Fields

in Condensed Matter, Electromagnetism, and Gravitation
Hagen Kleinert
pp. 1-2800, World Scientific, Singapore 2008
ISBN: 978-981-2791-70-2 (hardcover)
ISBN: 978-981-2791-71-9 (softcover)
ISBN: 978-981-3101-34-0 (ebook)
World Scientific,

This book lays the foundations of the theory of fluctuating multivalued fields with numerous applications. Most prominent among these are phenomena dominated by the statistical mechanics of line-like objects, such as the phase transitions in superfluids and superconductors as well as the melting process of crystals, and the electromagnetic potential as a multivalued field that can produce a condensate of magnetic monopoles. In addition, multivalued mappings play a crucial role in deriving the physical laws of matter coupled to gauge fields and gravity with torsion from the laws of free matter. Through careful analysis of each of these applications, the book thus provides students and researchers with supplementary reading material for graduate courses on phase transitions, quantum field theory, gravitational physics, and differential geometry.

  • Chinese translation by Dr. Ying Jiang (

Critical Properties of Phi4-Theories

Hagen Kleinert and V. Schulte-Frohlinde
pp. 1-512, World Scientific, Singapore 2001
ISBN: 978-981-0246-58-7 (hardcover)
ISBN: 978-981-0246-59-4 (softcover)
ISBN: 978-981-4490-79-5 (ebook)
World Scientific,

This book explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition.

Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a method for finding recursively all Feynman diagrams to any order in the coupling strength. Algebraic computer programs are supplied on accompanying World Wide Web pages. The diagrams correspond to integrals in momentum space. They are evaluated in 4-ε dimensions, where they possess pole terms in 1/ε. The pole terms are collected into renormalization constants.

The theory of the renormalization group is used to find the critical scaling laws. They contain critical exponents which are obtained from the renormalization constants in the form of power series. These are divergent, due to factorially growing expansion coefficients. The evaluation requires resummation procedures, which are performed in two ways: (1) using traditional methods based on Padé and Borel transformations, combined with analytic mappings; (2) using modern variational perturbation theory, where the results follow from a simple strong-coupling formula. As a crucial test of the accuracy of the methods, the critical exponent α governing the divergence of the specific heat of superfluid helium is shown to agree very well with the extremely precise experimental number found in the space shuttle orbiting the earth (whose data are displayed on the cover of the book).

The phi4-theories investigated in this book contain any number N of fields in an O(N)-symmetric interaction, or in an interaction in which O(N)-symmetry is broken by a term of a cubic symmetry. The crossover behavior between the different symmetries is investigated. In addition, alternative ways of obtaining critical exponents of phi4-theories are sketched, such as variational perturbation expansions in three rather than 4-ε dimensions, and improved ratio tests in high-temperature expansions of lattice models.


in Quantenmechanik, Statistik und Polymerphysik
Hagen Kleinert
pp. 1-850, Spektrum Akademischer Verlag, Heidelberg 1993
ISBN: 3-86025-613-0 (hardcover)

Dieses umfassende Lehrbuch aber Theorie und Anwendung von Pfadintegralen enthält erstmals Lösungen für eine Vielzahl nichttrivialer Systeme. Darunter fallen das einfachste System der Atomphysik und das für die Gravitationsphysik relevante System eines Teilchens in einem Raum mit Krümmung und Torsion. Ermöglicht werden diese Lösungen durch zwei neue Entwicklungen eine euklidische Pfadintegralformel, die im Gegensatz zur Feynmanschen Formel auch bei attraktiven 1/r- und 1/r2-Potentialen existiert, und ein einfaches Aequivalenzprinzip, die Übertragung von Pfadintegralen aus dem flachen Raum in einen nichteuklidischen Raum regelt. Damit zusammenhängend wird gezeigt, daß die Ableitung der richtigen klassischen Bewegungsgleichungen in Räumen mit Torsion eines Wirkungsprinzips bedarf.

Das Feynman-Kleinert-Variationsverfahren wird vorgestellt und systematisch verbessert. Es ermöglicht eine sehr genaue Berechnung analytisch nicht lösbarer Pfadintegrale.

Eine ausführliche Behandlung erfährt die Theorie der Tunnelprozesse mit Anwendungen auf die Lebensdauer von Supraströmen, die Stabilität metastabiler Phasen und das von Störungsreihen. Mit Hilfe eines neuen Variationsverfahrens wird der bisher auf hohe Tunnelbarrieren eingeschränkte Gültigkeitbereich auf niedrige Barrieren erweitert und vermittelt dadurch Kenntnis aber Störungskoeffizienten niedrigerer Ordnung.

Besondere Aufmerksamkeit gilt Pfadintegralen mit topologischen Einschränkungen, die für die Statistik quantenmechanischer Vielteilchensysteme und Verschlingungserscheinungen in der Polymerphysik von Bedeutung sind. Die Chern-Simons-Theorie der wird eingeführt, ihre mögliche Relevanz für die Hochtemperatur-Supraleitung erläutert und die auf ihr fußende Erklärung des Quanten-Hall-Effekts geliefert.

Gauge Fields in Condensed Matter

Vol. 1: Superflow and Vortex Lines (Disorder Fields, Phase Transitions)
Vol. 2: Stresses and Defects (Differential Geometry, Crystal Melting)
Hagen Kleinert
pp. 1-1524, World Scientific, Singapore 1989
ISBN: 978-997-1502-10-2 (hardcover)
World Scientific,

This book is the first to develop a unified gauge theory of condensed matter systems dominated by vortices or defects and their long-range interactions. Gauge fields provide the only means of describing these interactions in terms of local fields, rendering them accessible to standard field theoretic techniques. Two particularly important examples, superfluid systems and crystals, are treated in great detail. The theory is developed in close contact with physical phenomena and evolves naturally from conventional descriptions of the systems. In addition to gauge fields, the book introduces the important new concept of disorder fields for ensembles of line-like defects. The combined field theory allows for a new understanding of the important phase transitions superfluid ‘normal and solid’ liquid. Apart from the above, the book presents the general differential geometry of defects in spaces with curvature and torsion and establishes contact with the modern theory of gravity with torsion. This book is written for condensed matter physicists and field theorists. It can be used as a textbook for a second-year graduate course or as supplementary reading for courses in the areas of condensed matter and solid state physics, statistical mechanics, and field theory.



[0] H. Kleinert
The Asymmetric Energy of Single-Closed-Shell Nuclei in Present-day Microscopic Theory M.S. Thesis, 1-49 (1964)

[1] H. Kleinert
Group Dynamics of Elementary Particles, University of Colorado Thesis, Fortschritte der Physik 6, 1-74 (1968) (Ph.D. Thesis)

[2] H. Kleinert
Isoscalar Nucleon Form Factor from O(4,2) Dynamics, Phys. Rev. 163, 1807 (1968)

[3] H. Kleinert
Relativistic Current of the H-Atom in O(4,2) Dynamics, Phys. Rev. 168, 1827 (1968)

[4] H. Kleinert
Group Dynamics of the Hydrogen Atom, Lectures in Theoretical Physics, edited by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968, pp. 427-482.

[5] H. Kleinert
Decay Rates of the Decouplet resonances from O(3,1) times SU(3) Dynamics. Definite SU(3) Symmetry Breaking, Phys. Rev. Letters 18, 1027 (1967)

[6] A.O. Barut and H. Kleinert
The Solution of the Relativistic Discrete Mass Problem with Internal Degrees of Freedom and Further Developments, Proc. of the Fourth Coral Gables Conference, p. 76-105, edited by A. Perlmutter and B. Kursungoglu, W.H. Freeman, San Francisco, 1967.

[7] A.O. Barut and H. Kleinert
Resonance Decays from O(3,1) Dynamics. A Regularity in the Partial Decay Widths, Phys. Rev. Letters 18, 754 (1967)

[8] A.O. Barut and H. Kleinert
Calculation of Relativistic Transition Probabilities and Form Factors from Non-Compact Groups, Phys. Rev. 156, 1546 (1967) (PR ONLINE)

[9] A.O. Barut and H. Kleinert
Transition Probabilities of the H-Atom from Noncompact Dynamical Groups, Phys. Rev. 156, 1541 (1967)

[10] A.O. Barut and H. Kleinert
Current Operators and Majorana Equation for the Hydrogen Atom from Dynamical Groups, Phys. Rev. 157, 1180 (1967)

[11] A.O. Barut and H. Kleinert
Dynamical Group O(4,2) for Baryons and the Behaviour of Form factors, Phys. Rev. 161, 1464 (1967) (PR ONLINE)

[12] A.O. Barut and H. Kleinert
Transition Form Factors of H-Atom, Phys. Rev. 160, 1149 (1967)

[13] A.O. Barut and H. Kleinert
Tensor Operators and Mass Formula in the Minimal Extension of U(3) by Charge Conjugation, J. Math. Phys. 8, 685 (1967)

[14] A.O. Barut and H. Kleinert
Near Forward Peaks in the K^-p- and pi^- p Charge-Exchange Scattering, Phys. Rev. Letters 16, 950 (1966) (PRL ONLINE)

[15] A.O. Barut, D. Corrigan, and H. Kleinert
Derivation of Mass Spectrum and Magnetic Moments from Current Conservation in Relativistic O(3,2) and O(4,2) Theories, Phys. Rev. 167, 1527 (1968) (PR ONLINE)

[16] A.O. Barut, D. Corrigan, and H. Kleinert
Magnetic Moments, Form Factors and Mass Spectrum of Baryons, Phys. Rev. Letters 20, 167 (1968) (PRL ONLINE)

[17] A.O. Barut, H. Kleinert, and S. Malin,
The Anomalous Zitterbewegung of Composite Particles, Nuovo Cimento A 57, 835 (1968)

[18] B. Hamprecht and H. Kleinert
Nonfactorizing Saturation of Current Algebra, Phys. Rev. 180, 1410 (1969) (PR ONLINE)

[19] B. Hamprecht and H. Kleinert
Decays of Baryon resonances in an O(3,1) Model, Fortschr. Phys. 16, 635-661 (1969)

[20] H. Kleinert
Baryon Current Solving SU(3) Charge Current Algebra, Springer Tracts in Modern Physics, Vol. 49 (1969)

[21] H. Kleinert and K.H. Muetter,
Invariant Functions and Discrete Symmetries, Nuovo Cimento A 63, 657-674 (1969)

[22] D. Corrigan, B. Hamprecht and H. Kleinert
Formfactors of Delta (123) from O(4,2) Dynamics, Nucl. Phys. B 11, 1-6 (1969)

[23] F. Buccella, H. Kleinert, and C.A. Savoy,
A Mixing Operator for the SU(3) times SU(3) Chiral Algebra, Il Nuovo Cimento, A 69, 133-149 (1970)

[24] H. Kleinert
Veneziano Amplitude for any Intercept from Local Lagrangian, Lettere Nuovo Cimento 4, 285 (1970)

[25] H. Kleinert and P.H. Weisz,
Broken Scale Invariance and sigma pi pi, sigma A_1 A_1, sigma A_1 pi, and A_1 sigma pi Vertices, Lettere Nuovo Cimento 4, 1091-1096 (1970)

[26] H. Kleinert and P.H. Weisz,
Field Dimension and Gravitational Vertex, Nucl. Phys. B 27, 23-32 (1971)

[27] H. Kleinert and P.H. Weisz,
On the Dimension of Chiral and Conformal Symmetry Breaking, Nuovo Cimento A 3, 479-490 (1971)

[28] H. Kleinert, F. Steiner, and P.H. Weisz,
Do Present Meson Baryon Scattering Data Really Support (3,_3) + (_3,3) Dominance in SU(3) times SU(3) Symmetry Breaking? Physics Letters B 34, 312 (1971)

[29] J. Baacke and H. Kleinert
Backward Dispersion Relations for pi N - > pi Delta Scattering and N delta gamma Couplings, Phys. Letters B 35, 159-162 (1971)

[30] H. Kleinert, L. Staunton, and P.H. Weisz,
Hard Meson Cal culation of sigma pi pi , A sigma pi , and sigma A A Vertices, Nucl. Phys. B 38, 104-124 (1972)

[31] J. Baacke and H. Kleinert
Large Isovector Magnetic Moment for Roper Nucleon Transition, Lettere Nuovo Cimento 2, 463-467 (1971)

[32] H. Kleinert and P.H. Weisz,
On the Helicity-Flip Property of A_2 NN Coupling, Lettere Nuovo Cimento 2, 459-462 (1971)

[33] H. Kleinert, L. Staunton, and P.H. Weisz,
On the Electromagnetic Coupling of f and sigma Mesons, Nuclear Physics B 38, 87-103 (1972)

[34] J. Baacke and H. Kleinert
Electromagnetic Couplings from the Combined Use of Forward and Backward Dispersion Relations, Nuclear Physics, 42 42, 301-324 (1972)

[35] J. Baacke, T.H. Chang, and H. Kleinert
Compton Scattering and the Couplings of f, sigma , A_2, and pi to Photons and Nucleons, Il Nuovo Cimento A 12, 21-61 (1972)

[36] H. Kleinert
A Model for Deep Inelastic Structure Functions, Nucl. Phys. B 43, 301-330 (1972)

[37] A. Grillo and H. Kleinert
Is Nature Scale Invariant on the Light Cone, FU-Berlin preprint, December 1971.

[38] H. Kleinert
Low Energy Aspects of Broken Scale Invariance, Acta Phys. Austriaca Suppl IX, 533-604 (1972)

[39] H. Kleinert
Broken Scale Invariance, Fortschr. Phys. 21, 1-55 (1973)

[40] H. Kleinert
Algebraization of Regge Couplings, Phys. Letters B 39, 511-513 (1972)

[41] H. Kleinert
Scaling Properties of Infinite Component Fields, in Scale and Conformal Symmetry, Addison Wesley, N.Y. (1973)

[42] H. Kleinert
Regge Couplings From SU(2) times SU(2) Saturation Schemes, Fortschr. Phys. 21, 377-396 (1973)

[43] H. Kleinert, L. R. Ram Mohan,
Predictions on the Coupling of f and rho Trajectories to Mesons from Chiral Symmetry, Nucl. Phys. B 52, 253-279 (1973)

[44] H. Kleinert
Algebra of Regge Residues, Lettere Nuovo Cimento 6, 583 (1973)

[45] H. Kleinert
Bilocal Form Factors and Regge Couplings, Nucl. Phys. B 65, 77-111 (1973)

[46] H. Kleinert
Superalgebra of currents and Regge Couplings, Erice Lectures 1973, in Properties of the Fundamental Interactions\/, Editrice Compositori, Bologna, Italy.

[47] H. Kleinert
Quarks and Reggeons, Nucl. Phys. B 79, 526-545 (1974)

[48] H. Kleinert
The Origin of Exchange Degeneracy, Erice Lectures 1974, in Highlights in Particle Physics, Editrice Compositori, Bologna, Italy.

[49] R. Acharya and H. Kleinert
A Gauge Invariance in Gribov's Field Theory and the Intercept of the Pomeron, Lettere Nuovo Cimento 14, 395 (1975)

[50] H. Kleinert
Quark Pairs inside Hadrons, Phys. Letters B 59, 163 (1975)

[51] H. Kleinert
Quark Masses, Phys. Letters B 62, 77 (1976)

[52] H. Kleinert
Hadronization of Quark Theories and a Bilocal form of QED, Phys. Letters B 62, 429 (1976)

[52a] H. Kleinert Hadronization of Quark Theories and Bilocal QED, Lecture presented at Tbilisi Conference 1976 publ. in Proceedings.

[53] H. Kleinert
On the Hadronization of Quark Theories, Lectures presented at the Erice Summer Institute 1976, in Understanding the Fundamental Constituents of Matter, Plenum Press, New York, 1978, A. Zichichi ed., pp. 289-390.

[54] H. Kleinert
Field Theory of Collective Excitations; I. A Soluble Model, Phys. Lett. B 69, 9 (1977)

[55] H. Kleinert
Collective Quantum Fields, Lectures presented at the First Erice Summer School on Low-Temperature Physics, 1977, in Fortschr. Physik 26, 565-671 (1978)

[56] H. Kleinert
Non-Linear Field Equations in Collective Phenomena, Lecture presented at the Istanbul Conference on Mathematical Physics 1977, Nonlinear Equations in Physics and Mathematics, 355-374, Reidel 1978.

[57] H. Kleinert, Y.L. Lin-Liu and K. Maki,
Stability of Uniform Textures in ^3He-A in the Presence of Superflow, Phys. Lett. A 70, 27 (1979)

[58] H. Kleinert and H. Reinhardt
Semiclassical Approach to Large Amplitude Collective Nuclear Excitations, Nucl. Phys. A 332, 33 (1979)

[59] H. Kleinert, Y.L. Lin-Liu, and K. Maki,
Existence of Helical Textures around Superflow in ^3He-A; Paper presented at the Grenoble conference on Low-Temperature Physics, August 1978, Journal des Physique 39, C6-59 (1978)

[60] H. Kleinert
What can a Particle Physicist learn from Superliquid ^3He? Lectures presented at the International School of Subnuclear Physics, Erice, 1978, publ. in The New Aspects of Subnuclear Physics, Plenum Press 1980, A. Zichichi, ed.

[61] H. Kleinert
The Two Superflows in ^3He-A, Phys. Lett. A 71, 66 (1969)

[62] H. Kleinert and K. Maki,
High Frequences Conductivity of Charge Density Wave Condensates at Low Temperatures, Phys. Rev. B 12, 6238 (1979)

[63] H. Kleinert
Collective Field Theory of Superliquid ^3He, FU-Berlin preprint, 1978

[64] W. Janke and H. Kleinert
Summing Paths for a Particle in a Box, Lettere Nuovo Cimento 25, 297 (1979)

[65] H. Duru and H. Kleinert
Solution of the Path Integral for the H-Atom, Phys. Letters B 84, 185 (1979)

[66] R. Kaul and H. Kleinert
Surface Energy and Textural Boundary Conditions Between A and B Phases of ^3He, J. Low. Temp. Phys. 38, 539 (1980)

[67] H. Kleinert
Fate of False Vacua - Field Theory Applied to Superflow, Erice Lectures (1979), publ. in Pointlike Structures Inside and Outside Hadrons, Plenum Press 1982, A. Zichichi, ed.

[68] H. Kleinert
Depairing Critical Current in ^3He-B. Including Gap Distortion, J. Low Temp. Phys. 39, 451 (1980)

[69] H. Kleinert
Collective Excitations in ^3He B in the Presence of Superflow Phys. Letters A 78, 155 (1980)

[70] W. Janke and H. Kleinert
Superflow in ^3He-B in the Presence of Magnetic Fields at all Temperatures, Phys. Letters A 78, 363 (1980)

[71] H. Kleinert
New Symmetries and Constants of Motion from Dynamical Groups Phys. Letters B 94, 373 (1980)

[72] R. Kaul and H. Kleinert
Surface Energy and Textural Boundary Conditions Between A and B Phases of ^3He, J. Low. Temp. Phys. 38, 539 (1980)

[73] H. Kleinert
No Pion Condensate in Nuclear Matter due to Fluctuations, Phys. Lett. B 102, 1 (1981)

[74] H. Duru. H. Kleinert and N. Uenal,
Decay Rate of Supercurrent in Thin Wire, J. Low Temp. Phys. 42, 85 (1981)

[75] H. Kleinert and K. Maki,
Lattice Textures in Cholesteric Liquid Crystals, Fortschr. Phys. 29, 1 (1981)

[76] H. Kleinert
Thermal Expansion and Bragg Reflexes in Lattice Textures of Cholesteric Liquid Crystals, Phys. Letters A 81, 141 (1981)

[77] H. Kleinert
Field Theory of Collective Excitations. II. Large Amplitude Phenomena. Lett. Nuovo Cimento 31, 521 (1981)

[78] H. Kleinert
Field Theory of Collective Excitations. III. Condensation of Four-Particle Clusters, Phys. Letters A 84, 199 (1981)

[79] H. Kleinert
Field Theory of Collective Excitations, IV. Condensation of Three and Four-Particle Clusters in Bose Systems, Phys. Letters A 84, 259 (1981)

[80] H. Kleinert
Beyond Landau's Theory of Fermi Liquids, Phys. Letters A 84, 202 (1981)

[81] H. Kleinert
Condensation of Four-Particle Clusters: A Soluble Model, Jour. Phys. G: Nucl. Phys. 8, 239 (1982)

[82] H. Kleinert
Higher Effective Actions in Bose Systems, Fortschr. Phys. 30, 187 (1982)

[83] H. Duru and H. Kleinert
Quantum Mechanics of H-Atom from Path Integrals, Fortschr. d. Phys. 30, 401 (1982)

[84] H. Kleinert
Quasiclassical Approach to Collective Nuclear Phenomena, Fortschr. d. Phys. 30, 351 (1982)

[85] H. Kleinert
No Cryptoferromagnetic State due to Fluctuations, Phys. Lett. A 83, 294(1981)

[86] H. Kleinert
Restrictions on Pion Condensate, Lett. Nuovo Cimento 34, 133 (1982)

[87] H. Kleinert
Gauge Theory of Dislocations in Solids and Melting as a Meissner-Higgs Effect Lett. Nuovo Cimento 34, 464(1982)

[88] H. Kleinert
Gauge Theory of Dislocation Melting, Phys. Lett. A 89, 294 (1982)

[89] H. Kleinert
Theory of Defect Fluctuations in Solids-Dislocations and Disclinations under Stress, Lett. Nuovo Cimento 34, 471 (1982)

[90] H. Kleinert
Towards a Unified Theory of Defects and Stresses, Lett. Nuovo Cimento 35, 41 (1982)

[91] H. Kleinert
Defect Melting as an SO(3) Lattice Gauge Theory, Phys. Lett. B 113, 395 (1982)

[92] H. Kleinert
A New Dynamical Origin for Higgs Particles, Lett. Nuovo Cimento 34, 209 (1982)

[93] H. Kleinert
Duality Transformation for Defect Melting, Phys. Lett. A 91, 295 (1982)

[94] H. Kleinert
Line-like Defects in Pion Condensate, Lett. Nuovo Cimento 34, 103 (1982)

[95] H. Kleinert
Fluctuations and Defects in the spiral state of Magnetic Superconductors, Phys. Lett. A 90, 259 (1982)

[96] H. Kleinert
Comment on Path Integral for General Time-Dependent Solvable Schr"odinger Equation, Lett. Nuovo Cimento 34, 503 (1982)

[97] H. Kleinert
Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition, Lett. Nuovo Cimento 35, 405 (1982)

[98] H. Kleinert
Gauge Theory of Vortex Lines in ^4He and the Superfluid Phase Transition, Phys. Lett. A 93, 86 (1982)

[99] H. Kleinert
Defect Lines in Pion Condensates and a Possible New Phase, Lecture presented at the 1983 Conference on High Energy Nuclear Physics at Lake Balaton, Hungary - Proceedings publ. by Budapest University Press, 1983, ed. J. Eroe.

[100] H. Kleinert
Relation between U(1) Lattice Gauge Theory and Defect Melting, Lett. Nuovo Cimento 37, 425(1983)

[101] H. Kleinert
Transition Entropy of Defect Melting, Phys. Lett. A 95, 493 (1983)

[102] H. Kleinert
Gauge Field Theory of Dislocations in Smectic A Liquid Crystals, J. Phys. 44, 353 (1983) (Paris)

[103] H. Kleinert
New Developments in Gauge Theory of Defect Melting, Lecture presented at Max-Planck-Institut fuer Festkoerperforschung und Metallforschung, Stuttgart, July 1982, publ. in Gauge Field Theories of Defects in Solids, ed. E. Kroener.

[104] H. Kleinert
Disclinations and First-Order Transition in 2D Melting, Phys. Lett. A 95, 381 (1983)

[105] H. Kleinert
Dual Model of Dislocation and Disclination Melting, Phys. Lett. A 96 302 (1983)

[106] H. Kleinert
Limitations on Coleman-Weinberg Mechanism, Phys. Lett. B 128, 69 (1983)

[107] H. Kleinert
Double Gauge Theory of Stresses and Defects, Phys. Lett. A 97, 51 (1983)

[108] H. Kleinert
Disclinations as Origin of First-Order Transitions in Melting, Lett. Nuovo Cimento 37, 295 (1983)

[109] H. Kleinert
Model of Glass, Phys. Lett. A 101, 224 (1984)

[110] H. Kleinert
Gauge Theory of Defect Melting-Status 1984 Invited lecture at the 1984 EPS Conference on Condensed Matter Physics, Den Haag, publ. in Physica 127B, 332(1984)

[111] S. Ami and H. Kleinert
Fluctuations in 2D Melting, J. Phys. Lett. 45, (Paris) 877 (1984)

[112] T. Hofsaess and H. Kleinert
Field Theory of Self-Avoiding Random Chains, Phys. Lett. A 103, 67 (1984)

[113] T. Hofsaess and H. Kleinert
Field Theory of Self-Avoiding Random Surfaces, Phys. Lett. A 102, 420 (1984)

[114] T. Matsui, H. Kleinert and S. Ami,
Two Loop Effective Action of O(N) Spin Models in 1/D Expansion, Phys. Lett. B 143, 199 (1984)

[115] T. Hofsaess and H. Kleinert
A Simple Lattice Model for Self-Avoiding Random Loops, Phys. Lett. A 105, 60 (1984)

[116] L. Jacobs and H. Kleinert
Monte Carlo Study of Defect Melting in Three Dimensions, J. Phys. A 17, L361 (1984)

[117] H. Kleinert
Higgs Particles from Pure Gauge Fields, Erice Lectures 1982, in Gauge Interactions, Plenum Press, N.Y., ed. A. Zichichi, 1984, pp. 301-326.

[118] H. Kleinert
Defect Mediated Phase Transitions in Superfluids, Solids, and Relation to Lattice Gauge Theories, Lecture presented at the 1983 Cargese Summer School on Progress in Gauge Field Theory; publ. in Progress in Gauge Field Theory, ed. by G. 't. Hooft et al., Plenum Press 1984, pp 373-401.

[119] T. Hofsaess, W. Janke and H. Kleinert
From Self-Avoiding Random Loops to Ising Systems - an Interpolation Spin Model and its Monte Carlo Study, Phys. Lett. A 105, 463, (1984)

[120] W. Janke and H. Kleinert
First Order in 2D Disclination Melting, Phys. Lett. A 105, 134 (1984)

[121] H. Kleinert
Towards a Quantum Field Theory of Defects and Stresses - Quantum Vortex Dynamics in a Film of Superfluid Helium, Int. J. Engng. Sci. 23, 927 (1985)

[122] H. Kleinert
Interaction Energy Between Defects in Higher Gradient Elasticity, Lett. Nuovo Cimento 43, 261 (1985)

[123] H. Kleinert
Path Integrals for Self-Avoiding Random Loops and Surfaces, Lecture presented at the 1985 Bielefeld Conference on ``Path Integrals from meV to MeV", ed. by M.C. Gutzwiller, A. Inomata, J.R. Klauder, L. Steit, World Scientific, Singapore, 1986, pp. 239-250.

[124] T. Hofsaess, H. Kleinert, T. Matsui,
Are Gluons Composite? A New Lattice Gauge Model with an Exact U(N) Solution, Phys. Lett. B 156, 96 (1985)

[125] T. Hofsaess and H. Kleinert
Disorder Field Theory of the Ensemble of Random Loops without Spikes, Lett. Nuovo Cimento 43, 244 (1985)

[126] W. Janke and H. Kleinert
How Good is the Villain Approximation? Nucl. Phys. B 270, 135 (1986)

[127] H. Kleinert
Gauge Theory of Time-Dependent Stresses and Defects: Quantum Defect Dynamics, J. Phys. A: Math. Gen. 19, 1855 (1986)

[128] H. Kleinert
Thermal Softening of Curvature Elasticity in Membranes, Phys. Lett. A 114, 263 (1986)

[129] H. Kleinert
Size Distribution of Spherical Vesicles, Phys. Lett. A 116, 57 (1986)

[130] H. Kleinert and W. Miller,
Renormalization of Charge in Villain Form Lattice Gauge Theory, UCSD preprint 1985, Phys. Rev. Lett. 56, 11 (1986)

[131] W. Janke and H. Kleinert
Tricritical Points in 3D XY Model with Mixed Action, Phys. Rev. Lett. 57, 279 (1986)

[132] S. Ami and H. Kleinert
Vortex Contribution to Specific Heat in 2D XY-Model, Phys. Rev. B 33, 4692 (1986)

[133] W. Janke and H. Kleinert
Fluctuation Pressure of Membrane Between Walls, Phys. Lett. A 117, 353 (1986)

[134] W. Janke and H. Kleinert
Thermodynamic Properties of 3D Defect Systems, Phys. Rev. B 33, 6346 (1986)

[135] W. Janke and H. Kleinert
First-Order Transition in Two Dimensional Laplacian Roughening Model on Square Lattice, Phys. Lett. A 114, 255 (1986)

[136] T. Matsui, T. Hofsaess, H. Kleinert
Fluctuations in the Nematic Isotropic Phase Transition, Phys. Rev. A 33, 660 (1986)

[137] H. Kleinert
Microemulsions in the Three-Phase Regime, J. Chem. Phys. 84, 964 (1986)

[138] H. Kleinert
Path Integrals and the N to Infinity Solution of U(N) Lattice Gauge Theories, Lecture presented at the 1985 Bielefeld Conference on ``Path Integrals from meV to MeV", ed. by M.C. Gutzwiller, A. Inomata, J.R. Klauder, L. Steit, World Scientific, Singapore, 1986, pp. 235-238.

[139] W. Janke and H. Kleinert
First Order Phase Transition in 3D XY Model with Mixed Action, Nucl. Phys. B 270, 399 (1986)

[140] H. Kleinert
Tricritical Ratio of Length Scales in 4D Abelian Higgs Model, Phys. Rev. Lett. 56, 1441 (1986)

[141] H. Kleinert
The Neighborhood of the Three-Phase Regime in Microemulsions, J. Chem. Phys. 85, 4148 (1986)

[142] W. Janke and H. Kleinert
Self-Avoiding Random Loops Versus Ising Model in Three Dimensions, Phys. Lett. A 128, 463(1988)

[143] W. Janke and H. Kleinert
Fluctuation Pressure in a Stack of Membranes, Phys. Rev. Lett. 58, 144 (1987)

[144] H. Kleinert
Path Integral for Lagrangian L = (k/2) ddot x^2 + (m^2/2) dot x^2 + (k/2)x^2, J. Math. Phys. 27, 3003 (1986)

[145] H. Kleinert
Particle Distribution from Effective Classical Potential, Phys. Lett. A 118, 267 (1986)

[146] H. Kleinert
Path Integral for Coulomb System with Magnetic Charges Phys. Lett. A 116, 201 (1986)

[147] T. Hofsaess and H. Kleinert
Gaussian Curvature in an Ising Model of Microemulsions, J. Chem. Phys. 86 (6), 3565 (1987)

[148] H. Kleinert
Effective Classical Potential for Fluctuating Field Systems of Finite Size, Phys. Lett. A 118, 195 (1986)

[149] H. Kleinert
The Membrane Properties of Condensing Strings Phys. Lett. B 174, 335 (1986)

[150] H. Kleinert
The Two Gauge fields of Elasticity and Plasticity Annals of the New York Academy of Sciences 452, 349 (1985)

[151] H. Kleinert
Effective Potentials from Effective Classical Potentials, Phys. Lett. B 181, 324 (1986)

[152] S. Ami and H. Kleinert
Functional Measures of Membrane Fluctuations, FU-Berlin preprint 1986.

[153] W. Janke and H. Kleinert
Effective Classical Potential and Particle Distribution of a Coulomb System, Phys. Lett. A 118, 371 (1986)

[154] W. Janke and H. Kleinert
The Effective Classical Potential of the Double-Well Potential, Chem. Phys. Lett. 137, 162 (1987)

[155] H. Kleinert
Recent Results on 2D Defect Melting, Lecture presented at the Boston Conference on Solid State Physics, 1986.

[156] H. Kleinert and W. Miller,
Renormalization of Charge Due to Magnetic Monopoles in the Villain-Form of U(1) Lattice Gauge Theory, Phys. Rev. D 38, 1239(1988)

[157] S. Ami and H. Kleinert
Renormalization of Curvature Elastic Constants for Elastic and Fluid Membranes, Phys. Lett. A 120, 207 (1987)

[158] H. Kleinert
Floppy Membranes, FU-Berlin preprint 1986.

[159] R.P Feynman and H. Kleinert
Effective Classical Partition Functions, Phys. Rev. A 34, 5080 (1986)

[160] H. Kleinert
Quark Mass Dependence of 1/R Term in String Potential,

[161] T. Hofsaess and H. Kleinert
A Generalization of Widom's Model of Microemulsions, J. Chem. Phys. 88(2), 1156(1988)

[162] H. Kleinert
Thermal Deconfinement Transition for Spontaneous Strings, Phys. Lett. B 189, 187 (1987)

[163] H. Kleinert
How to do the Time-Sliced Path Integral of H-Atom, Phys. Lett. A 120, 361 (1987)

[164] H. Kleinert
Spontaneous Generation of String Tension and Quark Potential, Phys. Rev. Lett. 58, 1915 (1987)

[165] H. Kleinert
Spontaneous Quantum Gravity - A Soluble Model Phys. Lett. B 196, 355 (1987)

[166] H. Kleinert
Spontaneous Strings FU-Berlin preprint 1987.

[167] H. Kleinert
Glueballs from Spontaneous Strings Phys. Rev. D 37, 1699 (1988)

[168] H. Kleinert
Universal 1/R Term in Quark Potential of Spontaneous Strings Phys. Lett. B 197, 351 (1987)

[169] H. Kleinert
Phases of Spontaneous Strings in Large Dimensions Phys. Lett. B 197, 125 (1987)

[170] H. Kleinert
Universal Entropy of Large Spheres Made of Spontaneous Strings, Mod. Phys. Lett. A 3, 531 (1987)

[171] T. Hofsaess and H. Kleinert
Towards Custom-Made Microemulsions, Chem. Phys. Lett. 145, 407 (1988)

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

[173] H. Kleinert
Smectic-nematic phase transition as wrinkling transition in a stack of membranes Phys. Lett. A. 264, 440 (2000), (cond-mat/9912023)

[174] H. Kleinert
Lattice Defect Model with Two Successive Melting Transitions Phys. Lett. A 130, 443(1988)

[175] H. Kleinert
Tangential Flow in Fluid Membranes - Absence of Renormalization Effects J. Stat. Phys. 56, 227 (1989)

[176] H. Kleinert
Fundamental Phase Space Identities for Gauge Fields in Superfluids and Solids Phys. Lett. A 130, 59(1988)

[177] H. Kleinert
Dynamical Generation of String Tension and Stiffness Phys. Lett. B 211, 151(1988)

[178] H. Kleinert
Solution of Dyson Equation For Elastic Membranes J. Math. Phys. 30, 2991 (1989)

[179] W. Janke and H. Kleinert
From First-Order to Two Continuous Melting Transitions - Monte Carlo Study of New 2D Lattice Defect Model Phys. Rev. Lett. 61(20), 2344(1988)

[180] G. German and H. Kleinert
Perturbative Two-loop Quark Potential of Spontaneous Strings in any Dimension d Phys. Rev. D 40, 1108 (1989)

[180a] G. German and H. Kleinert Quark Potential of Spontaneous Strings, Proceedings of the 1988 Eger Workshop and Conference on Frontiers in Nonperturbative Field Theory.

[181] H. Kleinert
Membrane Stiffness from v.d. Waals forces Phys. Lett. A 136, 253(1989)

[182] H. Kleinert
Effect of Nonlinear Curvature Stiffness upon Fluctuation Pressure between Membranes FU-Berlin preprint 1988.

[183] H. Kleinert
Test of New Melting Criterion - Angular Stiffness and Order of 2D Melting in Lennard-Jones and Wigner Lattices, Phys. Lett. A 136, 468(1989)

[184] W. Janke, H. Kleinert, M. Meinhart
Monte Carlo Study of Stack of Self-Avoiding Surfaces with Extrinsic Curvature Stiffness, Phys. Lett. B 217, 525(1989)

[185] G. German and H. Kleinert
Comment on ``Effective string tension in the finite-temperature smooth-string mode'' Phys. Rev. D 40, 4199 (1989)

[186] W. Janke and H. Kleinert
Finite-Size Scaling Study of the Laplacian Roughening Model, Phys. Lett. A 140, 513 (1989)

[187] H. Kleinert
Exact Interaction Energies of Vortices and Disclinations on Triangular Lattice and their Asymptotic Limits Phys. Lett. A 134, 217(1989)

[188] H. Kleinert
Exact Temperature Behaviour of Strings with Extrinsic Curvature Stiffness for d ->Infinity. Thermal Deconfinement Transition Phys. Rev. D 40, 473(1989)

[189] W. Janke and H. Kleinert
Monte Carlo Study of Two-Step Defect Melting Phys. Rev. B 41, 6848 (1990) (PRB)

[190] G. German and H. Kleinert
Two-Loop String Tension of Spontaneous Strings at Finite Temperature in any Dimension Phys. Lett. B 220, 133(1989)

[191] H. Kleinert
Relation between Fluctuation Pressure and X-ray Structure of Stack of Membranes Phys. Lett. A 138, 201(1989)

[192] H. Kleinert and F. Langhammer,
Lattice Model for the Tricritical Point of the Nematic-Smectic-A Phase Transition Phys. Rev. A 40, 5988 (1989)

[193] H. Kleinert
Model for Condensation of Lamellar Phase from Microemulsion Phys. Lett. A 137, 65(1989)

[194] W. Janke and H. Kleinert
Geometrical Origin of Tricritical Points of various U(1) Lattice Models in Proceedings of the Workshop and Conference on ``Frontiers in Non-perturbative Field Theory'' in Eger/Hungary (1988), World Scientific, 1989, p. 279 ( hep-lat/9504010)

[195] H. Kleinert
Path Collapse in Feynman Formula - Stable Path Integral Formula from Local Reparametrization Invariant Amplitude Phys. Lett. B 224, 313 (1989)

[196] W. Janke and H. Kleinert
Path Integrals of Fluctuating Surfaces with Curvature Stiffness in Proc. ``Path Integrals from meV to MeV'', Bangkok, Jan. 1989.

[197] H. Kleinert,
Path Integral over Fluctuating Non-relativistic Fermion Orbits Phys. Lett. B 225, (4) (1989)

[198] G. German and H. Kleinert
Stiffness Dependence of the Deconfinement Transition for Strings with Extrinsic Curvature Term Phys. Lett. B 225, 107(1989)

[199] H. Kleinert
Quantum Mechanics and Path Integrals in Spaces with Curvature and Torsion Mod. Phys. Lett. A 4, 2329 (1989)

[200] M. Kiometzis and H. Kleinert
Electrostatic Stiffness Properties of Charged Bilayers Phys. Lett. A 140, 520 (1989)

[201] W. Janke and H. Kleinert
Large-Order Perturbation Expansion of Three-Dimensional Coulomb Systems from Four-Dimensional Anharmonic Oscillators Phys. Rev. A 42, 2792 (1990) ( PRA ONLINE)

[202] H. Kleinert
Path Integral on Spherical Surfaces in D Dimensions and on Group Spaces with Charges and Dirac Monopoles, Phys. Lett. B 236, 315 (1990)

[203] H. Kleinert
The Extra Gauge Symmetry of String Deformations in Electromagnetism with Charges and Dirac Monopoles Int. J. Mod. Phys. A 7, 4693 (1992)

[204] H. Kleinert and T. Sauer,
Exact Semiclassical Resistance of Thin Superconducting Wire J. of Low Temp. Physics 81, 123 (1990)

[205] H. Kleinert
Double-Gauge Invariance and Local Quantum Field Theory of Charges and Dirac Magnetic Monopoles Phys. Lett. B 246, 127 (1990)

[206] H. Kleinert and F. Langhammer
Lattice Model for Layered Structures Phys. Rev. A 44, 6686 (1991)

[207] H. Kleinert and I. Mustapic
Calculation of Pöschl-Teller and Rosen-Morse Fixed-Energy Amplitudes in Closed Form J. Math. Phys. 33, 643 (1992)

[208] H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, and S.A. Larin,
Five-Loop Renormalization Group Functions of O(n)-symmetric varphi ^4-Theory and varepsilon -Expansions of Critical Exponents up to varepsilon ^5 Phys. Lett. B 272, 39 (1991) (hep-th/9503230) , (printing error) Phys. Lett. B 319, 545 (1993)

[209] G. German, H. Kleinert, and M. Lynker,
Temperature Behaviour and Thermal Deconfinement for the Nambu-Goto String Phys. Rev. D 46, 1699 (1992) ( PRD ONLINE)

[210] H. Kleinert
Improving the Variational Approach to Path Integrals Phys. Lett. B 280, 251 (1992)

[211] H. Kleinert
Abelian Double-Gauge Invariant Continuous Quantum Field Theory of Electric Charge Confinement Phys. Lett. B 293, 168 (1992)

[212] H. Kleinert
Quantum Equivalence Principle for Path Integrals in Spaces with Curvature and Torsion in Proceedings of the XXV International Symposium Ahrenshoop on ``Theory of Elementary Particles'' in Gosen/Germany 1991, ed. by H. J. Kaiser ( quant-ph/9511020 ),

[213] H. Kleinert
Systematic Corrections to Variational Calculation of Effective Classical Potential Phys. Lett. A 173, 332 (1993)

[214] H. Kleinert
Variational Approach to Tunneling. Beyond the Semiclassical Approximation of Langer and Lipatov Phys. Lett. B 300, 261 (1993)

[215] H. Kleinert, A.M.J. Schakel
One-loop critical exponents for Ginzburg-Landau theory with Chern-Simons term FU-Berlin preprint 1993

[216] H. Kleinert
Tunneling Beyond the Semiclassical Approximation Lecture Presented at the International Conference on Path Integrals in Physics, Bangkok, January 1993. Proceedings ed. by Virulh Sa-yakanit et al., World Scientific, 1994.

[217] J. Jaenicke and H. Kleinert
Loop Corrections to the Effective Classical Potential Phys. Lett. A 176, 409 (1993)

[218] H. Kleinert and A. Zhuk
Finite Size and Temperature Properties of Matter and Radiation Fluctuations in Closed Friedmann Universe Teor. Math. Phys. 108, 1236 (1996).

[219] P. Fiziev and H. Kleinert
New Action Principle for Classical Particle Trajectories In Spaces with Torsion Europhys. Lett. 35, 241 (1996) ( hep-th/9503074 )

[220] H. Kleinert and H. Meyer
Variational Calculation of Effective Classical Potential at T neq 0 to Higher Orders Phys. Lett. A 184, 319 (1994) (hep-th/95040788)

[221] R. Karrlein and H. Kleinert
Precise Variational Tunneling Rates for Anharmonic Oscillator with g <0 Phys. Lett. A 187, 133 (1994) ( hep-th/9504048 )

[222] H. Kleinert
Group Theory and Orbital Fluctuations of the Hydrogen Atom Found. of Physics 23(5) (1993)

[223] H. Kleinert
Higher-Order Variational Approach to Non-Borel Systems. The Energies of the Double-Well Potential. Phys. Lett. A 190, 131 (1994)

[224] P. Fiziev and H. Kleinert
Euler Equations for Rigid Body -- A Case for Autoparallel Trajectories in Spaces with Torsion FU-Berlin preprint 1994 ( hep-th/9503075 )

[225] H. Kleinert and V. Schulte-Frohlinde
Exact Five-Loop Renormalization Group Functions of phi^4 -Theory with O(N)-Symmetric and Cubic Interactions. Critical Exponents up to epsilon ^5 Phys. Lett. B 342, 284 (1995) ( cond-mat/9503038) ( PLB ONLINE)

[226] M. Kiometzis, H. Kleinert, A. Schakel
Critical Exponents of the Superconducting Phase Transition Phys. Rev. Lett. 73, 1975 (1994) ( cond-mat/9503019 ) ( PRL ONLINE)

[227] H. Kleinert
Theory of Fluctuating Nonholonomic Fields and Applications: Statistical Mechanics of Vortices and Defects and New Physical Laws in Spaces with Curvature and Torsion in: Proceedings of a NATO Advanced Study Institute on Formation and Interactions of Topological Defects at the University of Cambridge, England, in Formation and Interactions of Topological Defects (NATO Asi Series B, Physics, Vol 349) Anne-Christine Davis and R. Brandenberger (eds.) Plenum Press, New York, 1995, p. 201 ( cond-mat/9503030 )

[228] W. Janke and H. Kleinert
Convergent Strong-Coupling Expansions from Divergent Weak-Coupling Perturbation Theory Phys. Rev. Lett. 75, 2787 (1995) ( quant-ph/9502019 ) ( PRL ONLINE)

[229] W. Janke and H. Kleinert
Scaling property of variational perturbation expansion for general anharmonic oscillator with x^p-potential Phys. Lett. A 199, 287 (1995) ( quant-ph/9502018) ( PLA ONLINE)

[230] H. Kleinert and S.V. Shabanov,
Quantum Langevin equation from forward-backward path integral Phys. Lett. A 200, 224 (1995) ( quant-ph/9503004 ) ( PLA ONLINE)

[231] H. Kleinert and I. Mustapic,
Decay Rates of Metastable States in Cubic Potential by Variational Perturbation Theory Int. J. Mod. Phys. A 11,4383 (1996) ( quant-ph/9502027 )

[232] M. Kiometzis, H. Kleinert, A. Schakel
Dual description of the superconducting phase transition Fortschr. Phys. 43, 697 (1995) (cond-mat/9508142 )

[233] H. Kleinert
Path integral for a relativistic spinless Coulomb system Phys. Lett. A 212, 15 (1996) ( hep-th/9504024) ( PLA ONLINE)

[234] H. Kleinert and S.V. Shabanov,
Theory of Brownian motion of a massive particle in spaces with curvature and torsion J. Phys. A: Math. Gen. 31, 7005 (1998) ( cond-mat/9504121 ) ( JPA ONLINE)

[235] H. Kleinert and W. Janke,
Convergence behavior of variational perturbation expansion--- A method for locating Bender-Wu singularites Phys. Lett. A 206, 283 (1995) (quant-ph/9509005) ( PLA ONLINE)

[236] H. Kleinert
Variational Interpolation Algorithm between Weak- and Strong-Coupling Expansions---Application to Polaron Phys. Lett. A 207, 133 (1995) ( quant-ph/9507005 ) ( PLA ONLINE)

[237] H. Kleinert
Variational Resummation of Divergent Series with known Large-Order Behavior Phys. Lett. B 360, 65 (1995) ( quant-ph/9507009 ) ( PLB ONLINE)

[238] H. Kleinert and S. Thoms
Large-Order Behavior of Two-coupling Constant $\phi^4$-Theory with Cubic Anisotropy Phys. Rev. D 52, 5926 (1995) (hep-th/9508172) ( PRD ONLINE)

[239] H. Kleinert
Nonabelian Bosonization as a Nonholonomic Transformations from Flat to Curved Field Space Ann. Phys. 253, 121 (1997) (hep-th/9606065) (AP ONLINE))

[240] H. Kleinert, G. Lambiase, and V.V. Nesterenko
Nambu-Goto String without Tachyons Between a Heavy and a Light Quark - Real Interquark Potential at all Distances Phys. Lett. B. 384,213 (1996) (hep-th/9601019) ( PLB ONLINE)

[241] H. Kleinert and A. Chervyakov
Evidence for Negative Stiffness of QCD Strings Phys. Lett. B 381, 286 (1996) (hep-th/9601030) ( PLB ONLINE)

[242] P. Fiziev and H. Kleinert
Comment on Path Integral Derivation of Schrödinger Equation in Spaces with Curvature and Torsion J. Phys. A.: Math. Gen. 29, 7619 (1996) (hep-th/9604172 ) ( JPA ONLINE)

[243] H. Kleinert and A. Pelster,
Autoparallels From a New Action Principle Gen. Rel. Grav. 31, 1439 (1999) (gr-qc/9605028)

[244] P. Fiziev and H. Kleinert,
Anholonomic Transformations of Mechanical Action Principle Proceedings of the Workshop on the Variational and Local Methods in the Study of Hamiltonian Systems, editors A.Ambrosetti and G.F.Dell'Antonio, World Scientific, 1995, pp.166-177 (gr-qc/9605046)

[245] H. Kleinert, S. Thoms, and W. Janke
Resummation of anisotropic quartic oscillator. Crossover from anisotropic to isotropic large-order behavior Phys. Rev. A 55, 915 (1997) (quant-ph/9605033) ( PRA ONLINE)

[246] H. Kleinert
Classical and Fluctuating Paths in Spaces with Curvature and Torsion FU-Berlin preprint 1996 (quant-ph/9606001)

[247] H. Kleinert and A.M.J. Schakel
New Universality Class in Superconductive Phase Transition FU-Berlin preprint 1996 (supr-con/9606001)

[248] A. Pelster and H. Kleinert
Transformation Properties of Classical and Quantum Laws under Some Nonholonomic Spacetime Transformations Proceedings of the 5th International Conference on Path Integrals from meV to MeV, Dubna, Russia, May 27 -- 31, 1996}; Edited by V.S. Yarunin, M.A. Smondyrev, Publishing Department Joint Institute for Nuclear Research Dubna, 187--191 (1996) FU-Berlin preprint 1996 (quant-ph/9608037)

[249] A. Pelster and H. Kleinert
Relations Between Markov Processes Via Local Time and Coordinate Transformations Phys. Rev. Lett. 78, 565-569 (1997) (cond-mat/9608120) ( PRL ONLINE)

[250] H. Kleinert, S. Thoms, and V. Schulte-Frohlinde
Stability of 3D Cubic Fixed Point in Two-Coupling-Constant $\phi^4$-Theory Phys. Rev. B 56, 14428 (1997) (quant-ph/9611050) ( PRB ONLINE)

[251] H. Kleinert and A. Zhuk
Casimir effect at nonzero temperature in closed Friedmann universe Theor. Math. Phys. 109 N2, 307 (1996)

[252] H. Kleinert
Quantum Equivalence Principle Lectures presented at the 1996 Cargese Summer School FUNCTIONAL INTEGRATION: BASICS AND APPLICATIONS Berlin preprint 1996 (quant-ph/9612040)

[253] H. Kleinert and S. Shabanov
Proper Dirac Quantization of Free Particle on $D$-Dimensional Sphere Phys. Lett. A 232, 327 (1996) (quant-ph/9702006) ( PLA ONLINE)

[254] H. Kleinert and A. Schakel
Comment on "Continuum dual theory of the transition in 3D lattice superconductor" Berlin preprint 1997 (cond-mat/9702159)

[255] H. Kleinert
Systematic Improvement of Hartree-Fock-Bogoljubov Approximation with Exponentially Fast Convergence from Variational Perturbation Theory Annals of Physics 266, 135 (1998) (APS E-Print aps1997may22_001 ) ( AP ONLINE)

[256] H. Kleinert and S. Shabanov
Supersymmetry in Stochastic Processes with Higher-Order Time Derivatives Phys. Lett. A 235, 105 (1997) (quant-ph/9705042) ( PLA ONLINE)

[257] H. Kleinert
Strong-Coupling Behavior of Phi^4-Theories and Critical Exponents Phys. Rev. D 57, 2264 (1998) (cond-mat/9801167) Addendum: Phys. Rev. D 58, 107702 (1998) (cond-mat/9803268)

[258] H. Kleinert
Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion Gen. Rel. Grav. 32 , 769 (2000) (gr-qc/0203029) ( GRG ONLINE)

[259] H. Kleinert and S.V. Shabanov
Spaces with Torsion from Embedding, and the Special Role of Autoparallel Trajectories Phys. Lett B 428, 315 (1998) ( gr-qc/9709067 ) (PLB ONLINE)

[260] H. Kleinert and A. Chervyakov
Functional determinants via Wronski construction of Green functions J. Math. Phys. 40, 6044 (1999) (physics/9712048)

[261] H. Kleinert
Nonholonomic Mapping Principle for Classical Mechanics in Spaces with Curvature and Torsion. New Covariant Conservation Law Lectures presented at Workshop Gauge Theories of Gravitation, Jadwisin, Poland, 4-10 September 1997 Short version of Paper 258. Act. Phys. Pol. B 29, 1033 (1998) (gr-qc/9801003)

[262] H. Kleinert and A. Pelster
Novel Geometric Gauge Invariance of Autoparallels Lectures presented at Workshop Gauge Theories of Gravitation, Jadwisin, Poland, 4-10 September 1997 Act. Phys. Pol. B 29 , 1015 (1998) (gr-qc/9801030)

[263] H. Kleinert
Strong-Coupling $\phi^4$-Theory in $4- \epsilon$ Dimensions, and Critical Exponent Phys. Lett. B 434, 74 (1998) (cond-mat/9801167) (PLB ONLINE)

[264] H. Kleinert and A. Chervyakov
Simple Explicit Formulas for Gaussian Path Integrals with Time-Dependent Frequencies Phys. Lett. A 245, 345 (1998) (quant-ph/9803016) ( PLA ONLINE)

[265] E. Babaev and H. Kleinert
Crossover from Weak- to Strong-Coupling Superconductivity and to Normal State with Pseudogap (Berlin preprint 1998) (cond-mat/9804206)

[266] H. Kleinert
Spiky Phases of Smooth Membranes. Implications for Smooth Strings Eur. Phys. J. B 9, 651 (1999) (cond-mat/9805307) ( EPJB ONLINE)

[267] Hagen Kleinert, Werner Kuerzinger, and Axel Pelster
Smearing Formula for Higher-Order Effective Classical Potentials J. Phys. A: Math. Gen. 31, 8307 (1998) (quant-ph/9806016) (JPA ONLINE) Short version presented as a lecture at the 1998 Conference on Path Integrals in Florence, Italy

[268] M. C. Diamantini, H. Kleinert, C. A. Trugenberger
Smoothening Transition of Rough Surfactant Surfaces FU-Berlin preprint 1998 (cond-mat/9806077)

[269] H. Kleinert and E. Babaev
Two Phase Transitions in Chiral Gross-Neveu Model in 2+ eps Dimensions at Low $N$ Phys. Lett. B 438, 311 (1998) (hep-th/9809112) (PLB ONLINE)

[270] M. E. S. Borelli, H. Kleinert, and Adriaan M. J. Schakel
Derivative Expansion of One-Loop Effective Energy of Stiff Membranes with Tension Phys. Lett. A 253, 239 (1999) (cond-mat/9806184) ( PLA ONLINE)

[271] H. Kleinert
Universality Principle for Orbital Angular Momentum and Spin in Gravity with Torsion Gen. Rel. Grav. 32, 1271 (2000) (gr-qc/9807021) ( GRG ONLINE)

[272] H. Kleinert and D.H. Lin
Relativistic Trace Formula for Bound States in Terms of Classical Periodic Orbits FU-Berlin preprint 1998 (quant-ph/9807068)

[273] H. Kleinert
Solution of Coulomb Path Integral in Momentum Space Phys. Lett. A 252, 277 (1999) (quant-ph/9807073) ( PLA ONLINE)

[274] H. Kleinert
Spontaneous Generation of Torsion Coupling of Electroweak Massive Gauge Bosons Phys. Lett. B 440, 283 (1998) (gr-qc/9808022) (PLB ONLINE)

[275] H. Kleinert
Variational Resummation of $ \epsilon $-Expansions of Critical Exponents of Nonlinear O($N$)-Symmetric $ \sigma $-Model in $2+ \epsilon $ Dimensions Phys. Lett A 264 , 357 (2000) (hep-th/9808145) ( PLA ONLINE)

[276] M. C. Diamantini, H. Kleinert, C. A. Trugenberger
Strings with Negative Stiffness and Hyperfine Structure FU-Berlin preprint 1998 (hep-th/9810171)

[277] H. Kleinert
Fluctuation Pressure of Membrane between Walls Phys. Lett. A 257, 269 (1999) (cond-mat/9811308) ( PLA ONLINE)

[278] H. Kleinert
Vortex Line Nucleation of First-Order Phase Transitions in Early Universe Phys. Lett B 460, 36 (1999) (hep-th/9811185) (PLB ONLINE)

[279] H. Kleinert
Critical Exponents from Seven-Loop Strong-Coupling $\phi^4$-Theory in Three Dimensions Phys. Rev. D 60, 085001 (1999) (hep-th/9812197) (PRD ONLINE)

[280] Michael Bachmann, H. Kleinert, and Axel Pelster
Variational Perturbation Theory for Density Matrices Phys. Rev. A 60, 3429 (1999) (quant-ph/9812063) (PRA ONLINE) Short version presented as a lecture at the 1998 Conference on Path Integrals in Florence, Italy

[281] Hagen Kleinert, Axel Pelster, and Michael Bachmann
Generating Functionals for Harmonic Expectation Values of Paths with Fixed End Points. Feynman Diagrams for Nonpolynomial Interactions Physical Review E 60, 2510 (1999) (quant-ph/9902051) (PRE ONLINE)

[282] M. C. Diamantini, H. Kleinert, C. A. Trugenberger
Floppy Membranes Phys. Lett. A 269, 1 (2000) (cond-mat/9903021) ( PLA ONLINE)

[283] M. C. Diamantini, H. Kleinert, C. A. Trugenberger
Universality Class of Confining Strings Phys. Lett. B 457, 87 (1999) (hep-th/9903208) (PLB ONLINE)

[284] M. E. S. Borelli, H. Kleinert, A. M. J. Schakel
Quantum Statistical Mechanics of Nonrelativistic Membranes: Crumpling Transition at Finite Temperature Phys. Lett. A 267, 201 (2000) (cond-mat/9905241) ( PLA ONLINE)

[285] M. Bachmann, H. Kleinert, A. Pelster
Strong-Coupling Calculation of Fluctuation Pressure of Membrane Between Walls Phys. Lett. A 261, 127 (1999) (cond-mat/9905397) (PLA ONLINE)

[286] Hagen Kleinert
Theory and Satellite Experiment for Critical Exponent alpha of lambda-Transitiion in Superfluid Helium Phys. Lett. A 277, 205 (2000) (cond-mat/9906107) ( PLA ONLINE)

[287] Florian Jasch, Hagen Kleinert
Fast-Convergent Resummation Algorithm and Critical Exponents of $\phi^4$-Theory in Three Dimensions J. Math. Phys. 42, 52 (2001) (cond-mat/9906246)

[288] H. Kleinert, A. Chervyakov, B. Hamprecht
Perturbation Theory for Particle in a Box Phys. Lett. A 260, 182 (1999) (cond-mat/9906241) ( PLA ONLINE)

[289] H. Kleinert, A. Chervyakov
Reparametrization Invariance of Path Integrals Phys. Lett. B 464, 257 (1999) (hep-th/9906156) (PLB ONLINE)

[290] H. Kleinert
Critical Exponents without beta-Function Phys. Lett. B 463, 69 (1999) (cond-mat/9906359) (PLB ONLINE)

[291] H. Kleinert, B. Van den Bossche
No Spontaneous Breakdown of Chiral Symmetry in Nambu-Jona-Lasinio Model Phys. Lett. B 474 , 336 (2000) (hep-ph/9907274) (PLB ONLINE)

[292] M. Bachmann, H. Kleinert, A. Pelster
Recursive Graphical Construction of Feynman Diagrams in Quantum Electrodynamics Phys. Rev. D 61, 085017 (2000) (hep-th/9907044) ( PRD ONLINE)

[293] E. Babaev, H. Kleinert
Nonperturbative $XY$-Model Approach to Strong-Coupling Superconductivity in Two and Three Dimensions Phys. Rev. B 59, 12083 (1999) (cond-mat/9907138) ( PRB ONLINE)

[294] H. Kleinert, A. Pelster, B. Kastening, M. Bachmann
Recursive Graphical Construction of Feynman Diagrams and Their Multiplicities in phi^4- and in phi^2 A-Theory Phys. Rev. E 62, 1537 (2000) (hep-th/9907168) ( PRE ONLINE) Program and its Output for Diagrams

[295] H. Kleinert, V. Schulte-Frohlinde
Critical Exponents from Five-Loop Strong-Coupling phi^4-Theory in 4- epsilon Dimensions J. Phys. A 34 1037 (2001) (cond-mat/9907214) ( JPA ONLINE)

[296] H. Kleinert, B. Van den Bossche
No Massless Pions in Nambu-Jona-Lasinio Model due to Chiral Fluctuations Berlin Preprint 1999 (hep-ph/9908284)

[297] H. Kleinert
Perturbative Calculation of Multi-Loop Feynman Diagrams. New Type of Expansions for Critical Exponents Phys. Lett. B 465, 235 (1999) (hep-th/9908078) (PLB ONLINE)

[298] H. Kleinert
Criterion for Dominance of Directional over Size Fluctuations in Destroying Order Phys. Rev. Lett. 84, 286 (2000) (cond-mat/9908239) (PRL ONLINE)

[299] B. Kastening, H. Kleinert
Efficient Algorithm for Perturbative Calculation of Multiloop Feynman Integrals Phys. Lett. A 269, 50 (2000) (quant-ph/9909017) ( PLA ONLINE)

[300] H. Kleinert and A. Chervyakov
Reparametrization Invariance of Perturbatively Defined Path Integrals. II. Integrating Products of Distributions Phys. Lett. B 477, 373 (2000) (quant-ph/9912056) (PLB ONLINE)

[301] H. Kleinert, B. Van den Bossche
Global Derivation of the Fluctuation Determinant from Group Property of Time Evolution. FU-Berlin preprint 2000 (quant-ph/0002008)

[302] F. Ferrari, H. Kleinert, I. Lazzizzera
Second Topological Moment < m^2 > of Two Closed Entangled Polymers Phys. Lett. A 276, 1 (2000) (cond-mat/0002049) (PLA ONLINE)

[303] H. Kleinert, A. Chervyakov
Rules for Integrals over Products of Distributions from Coordinate Independence of Path Integrals Eur. Phys. J. C 19, 743-747 (2001) (quant-ph/0002067) ( EPJC ONLINE)

[304] M. E. S. Borelli, H. Kleinert
Phases of a stack of membranes in a large number of dimensions of configuration space Phys. Rev. B 63, 205414 (2001) (cond-mat/0003362) ( PRB ONLINE)

[305] H. Kleinert, A. Chervyakov
Coordinate Independence of Quantum-Mechanical Path Integrals Phys. Lett. A 269, 63 (2000) (second print because of errors caused by publisher in Phys. Lett. A 273, 1 (2000) (quant-ph/0003095) (PLA ONLINE)

[306] F. Ferrari, H. Kleinert, I. Lazzizzera
Calculation of Second Topological Moment < m^2 > of Two Entangled Polymers Eur. Phys. J B 18, 645 (2000) (cond-mat/0003355) (EPJB ONLINE)

[307] M. Bachmann, H. Kleinert, and A. Pelster
Variational Approach to Hydrogen Atom in Uniform Magnetic Field of Arbitrary Strength Phys. Rev. A 62, 52509 (2000) (quant-ph/0005074) (PRA ONLINE)

[308] F. Ferrari, H. Kleinert, I. Lazzizzera
Field Theory of N Entangled Polymers Int. J. Mod. Phys. B 14, 3881 (2000) (cond-mat/0005300)

[309] M. Bachmann, H. Kleinert, A. Pelster
Quantum Statistics of Hydrogen in Strong Magnetic Fields Phys. Lett. A 279, 23 (2001) (quant-ph/0005100) (PLA ONLINE)

[310] A. Pelster and H. Kleinert
Functional Differential Equations for the Free Energy and the Effective Energy in the Broken-Symmetry Phase of phi^4-Theory and Their Recursive Graphical Solution Physica A 323, 370-400 (2003) (hep-th/0006153) (Physica A ONLINE)

[311] H. Kleinert and H.-J. Schmidt
Cosmology with Curvature-Saturated Gravitational Lagrangian R/\sqrt{1 + l^4 R^2} Gen. Rel. Grav. 34, 1295 (2002) (gr-qc/0006074) (GRG ONLINE)

[312] M.E.S. Borelli, H. Kleinert, A.M.J. Schakel
Vertical Melting of a Stack of Membranes Eur. Phys. J. E 4, 217 (2001) (cond-mat/0004432) (EPJB ONLINE)

[313] H. Kleinert and A. Chervyakov
Covariant Effective Action for Quantum Particle with Coordinate-Dependent Mass Phys. Lett. A 299, 319 (2002) (quant-ph/0206022)

[314] H. Kleinert
Fokker-Planck and Langevin Equations from the Forward--Backward Path Integral Ann. Phys. 291, 14 (2001) (quant-ph/0008109) (AP ONLINE)

[315] F. S. Nogueira and H. Kleinert
Field Theoretical Approaches to the Superconducting Phase Transition in Order, Disorder, and Criticality - Advanced Problems of Phase Transition Theory, Yurij Holovatch ed., (World Scientific, Singapore, 2004), pp. 253-283. (cond-mat/0303485)

[316] Z. Haba and H. Kleinert
Master Equation for Electromagnetic Dissipation and Decoherence of Density Matrices Eur. Phys. J. B 21, 553 (2001) (cond-mat/0011486) (EPJB ONLINE)

[317] M. Bachmann, H. Kleinert, and A. Pelster
Fluctuation Pressure of a Stack of Membranes Phys. Rev. E 63, 51709 (2001) (cond-mat/0011281) ( PRE ONLINE)

[318] H. Kleinert, B. Van den Bossche
Three-Loop Critical Amplitude Functions and Ratios from Variational Perturbation Theory Phys. Rev. E 63, 056113 (2001) (cond-mat/0011329 ) ( PRE ONLINE)

[319] Z. Haba and H. Kleinert
Quantum-Liouville and Langevin Equations for Gravitational Radiation Damping Int. J. Mod. Phys. A 17 3729 (2002) (quant-ph/0101006)

[320] H. Kleinert
Five-Loop Critical Temperature Shift in Weakly Interacting Homogeneous Bose-Einstein Condensate Mod. Phys. Lett. B 17, 1011 (2003) (cond-mat/0210162)

[321] H. Kleinert and B. Van den Bossche
Two-Loop Effective Potential of O(N)-Symmetric Scalar QED in 4-epsilon Dimensions Nucl. Phys. B 632, 51 (2002) (cond-mat/0104102 )

[322] H. Kleinert and F.S. Nogueira
Charged fixed point in the Ginzburg-Landau superconductor and the role of the Ginzburg parameter $\kappa$ Nucl. Phys. B 651, 361 (2003) (cond-mat/0104573 )

[323] Z. Haba and H. Kleinert
Langevin Equation for Particle in Thermal Photon Bath FU-Berlin preprint 2001 (quant-ph/0106096)

[324] Z. Haba and H. Kleinert
Schrödinger Wave Functions from Classical Trajectories Phys. Lett. A 294, 139 (2002) (quant-ph/0106095)

[325] H. Kleinert, A. Pelster, and B. Van den Bossche
Recursive Graphical Construction of Feynman Diagrams and Their Weights in Ginzburg-Landau Theory Physica A 312 , 141 (2002) (hep-th/0107017)

[326] A. Pelster, H. Kleinert, and M. Schanz
High-Order Variational Calculation for the Frequency of Time-Periodic Solutions Phys. Rev. E 67, 016604 (2003) (math-ph/0208032) ( PRE ONLINE)

[327] A. Pelster, H. Kleinert, and M. Bachmann
Functional Closure of Schwinger-Dyson Equations in Quantum Electrodynamics, Part 1: Generation of Connected and One-Particle Irreducible Feynman Diagram Annals of Physics 297, 363 (2002) (hep-th/0109014) ( AP ONLINE)

[328] B. Kastening, H. Kleinert, and B. Van den Bossche
Three-Loop Ground-State Energy of O(N)-Symmetric Ginzburg-Landau Theory above T_c in 4-epsilon Dimensions with Minimal Subtraction Phys. Rev. B 65, 174512 (2002) (cond-mat/0109372) ( PRB ONLINE)

[329] H. Kleinert
Stochastic Calculus for Assets with Non-Gaussian Price Fluctuations Physica A 311, 536 (2002) (cond-mat/0203157)

[330] H. Kleinert and A. Chervyakov
Integrals over Products of Distributions from Perturbation Expansions of Path Integrals in Curved Spaces Int. J. Mod. Phys. A 17, 2019 (2002) (quant-ph/0208067)

[331] V. Folomeev, V. Gurovich, H. Kleinert and H.-J. Schmidt
Flashing Dark Matter -- Gamma-Ray Bursts from Relativistic Detonations of Electro-Dilaton Stars Grav.Cosmol. 8, 299-304 (2002) (gr-qc/0206043)

[332] H. Kleinert and F. Nogueira
Two different scaling regimes in Ginzburg-Landau model with Chern-Simons term J. Phys. Stud. 5, 327 (2001) (cond-mat/0109539)

[333] H. Kleinert
Option Pricing from Path Integral for Non-Gaussian Fluctuations. Natural Martingale and Application to Truncated Lévy Distributions Physica A 312, 217 (2002) (cond-mat/0202311)

[334] H. Kleinert, A. Pelster, and M.V. Putz
Variational Perturbation Theory for Markov Processes Phys. Rev. E 65, 066128 (2002) (cond-mat/0202378)

[335] H. Kleinert and A. Chervyakov
Integrals over Products of Distributions and Coordinate Independence of Zero-Temperature Path Integrals Phys. Lett. A 308, 85 (2003) (quant-ph/0204067)

[336] H. Kleinert, F.S. Nogueira, and A. Sudbo
Deconfinement transition in three-dimensional compact U(1) gauge theories coupled to matter fields Phys. Rev. Lett. 88, 232001 (2002) (hep-th/0201168) ( PRL ONLINE)

[337] H. Kleinert and F.S. Nogueira
Critical behavior of Ginzburg-Landau model coupled to massless Dirac fermions Phys. Rev. B 66, 012504 (2002) (cond-mat/0112030) ( PRB ONLINE)

[338] H. Kleinert and A. M. J. Schakel
Gauge-invariant critical exponents for the Ginzburg-Landau model Phys. Rev. Lett. 90, 097001 (2003) (cond-mat/0209449) ( PRL ONLINE)

[339] H. Kleinert and A. Chervyakov
Perturbatively Defined Effective Classical Potential in Curved Space Int. J. Mod. Phys. A 18 , 5521 (2003) (quant-ph/0301081)

[340] H. Kleinert and Y. Jiang
Defect Melting Models for Cubic Lattices and Universal Laws for Melting Temperatures Phys. Lett. A 313, 152 (2003) (cond-mat/0301453)

[341] B. Hamprecht and H. Kleinert
Dependence of Variational Perturbation Expansions on Strong-Coupling Behavior. Inapplicability of delta-Expansion to Field Theory Phys. Rev. D 68 065001 (2003) (hep-th/0302116) ( PRD ONLINE)

[342] B. Hamprecht and H. Kleinert
Tunneling Amplitudes from Perturbation Expansions Phys. Lett. B 564 111 (2003) (hep-th/0302124)

[343] H. Kleinert, F. S. Nogueira and A. Sudbo
Kosterlitz-Thouless-like deconfinement mechanism in the 2+1 dimensional Abelian Higgs model Nucl. Phys. B 666, 361 (2003) (hep-th/0209132) ( Nucl. Phys. B ONLINE)

[344] B. Hamprecht and H. Kleinert
Variational Perturbation Theory for Summing Divergent Non-Borel-Summable Tunneling Amplitudes J. Phys. A: Math. Gen. 37, 8561-8574 (2004) (hep-th/0303163)

[345] B. Hamprecht and H. Kleinert
End-To-End Distribution Function of Stiff Polymers for all Persistence Lengths Phys. Rev. E 71, 031803 (2005) (Berlin preprint 2004) (cond-mat/0305226)

[346] H. Kleinert and J. Zaanen
Nematic World Crystal Model of Gravity Explaining Absence of Torsion in Spacetime Phys. Lett. A 324 , 361 (2004) (gr-qc/0307033) ( Phys. Lett. A ONLINE)

[347] H. Kleinert, S. Schmidt, and A. Pelster
Reentrant Phenomenon in Phase Diagram of Mott Insulator Transition in Optical Boson Lattices at Nonzero T Phys. Rev. Lett. 93, 160402 (2004) (cond-mat/0307412 )

[348] B. Hamprecht, W. Janke, and H. Kleinert
End-to-End Distribution Function of Two-Dimensional Stiff Polymers for all Persistence Lengths Phys. Lett. A 330, 254 (2004) (cond-mat/0307530)

[349] H. Kleinert and V.I. Yukalov
Highly Accurate Critical Exponents from Self-Similar Variational Perturbation Theory Berlin preprint 2004, Phys. Rev. E 71, 026131 (2005) (cond-mat/0402163)

[350] S. F. Brandt, H. Kleinert, A. Pelster
Recursive Calculation of Effective Potential and Variational Resummation J. Math. Phys. 46, 032101 (2005) (quant-ph/0406206)

[351] H. Kleinert
Option pricing for non-Gaussian price fluctuations Lecture presented at the conference Frontier Science 2003 in Pavia Physica A 338, 151 (2004)

[352] O. Zobay, G. Metikas, H. Kleinert
Nonperturbative Effects on $T_c$ of Interacting Bose Gases in Power Traps Phys. Rev. A 71, 043614 (2005) (cond-mat/0411133)

[353] H. Kleinert
Travailler avec Feynman Pour La Science 19, 89-95 (2004) (see also here)

[354] H. Kleinert
Order of Superconductive Phase Transition Berlin Preprint 2005 publ. in Festschrift in honor of R. Folk's 60th birthday (2004).

[355] M. Blasone, P. Jizba, and H. Kleinert
Path Integral Approach to 't Hooft's Derivation of Quantum from Classical Physics. Berlin Preprint (2004), Phys. Rev. A 71 052507 (2005) (quant-ph/0409021)

[356] H. Kleinert and A.J.M. Schakel
Anomalous Dimension of Dirac's Gauge-Invariant Nonlocal Order Parameter in Ginzburg-Landau Field Theory Phys. Lett. B 611, 182 (2005)

[357] H. Kleinert, S. Schmidt, and A. Pelster
Quantum Phase Diagram For Homogeneous Bose-Einstein Condensate Ann. Phys. 14 214 (2005) (cond-mat/0308561)

[358] H. Kleinert and A. Chervyakov
Perturbation Theory for Path Integrals of Stiff Polymers J. Phys. A: Math. Gen. 39 8231 (2006) (cond-mat/0503100)

[359] H. Kleinert
Emerging Gravity from Defects in World Crystal Lecture Presented at the 2004 Conference on Emerging Gravity in Piombino Berlin preprint (2005) (Brazilian Journal of Physics 35, (2005); pdf file)

[360] H. Kleinert
Vortex Origin of Tricritical Point in Ginzburg-Landau Theory Europh. Letters 74, 889 (2006) (cond-mat/0509430)

[361] F.S. Nogueira and H. Kleinert
Quantum Electrodynamics in 2+1 Dimensions, Confinement, and the Stability of U(1) Spin Liquids Phys. Rev. Lett. 95, 176406 (2005) (cond-mat/0501022) ( PRL ONLINE)

[362] M. Blasone, P. Jizba, and H. Kleinert
Quantum Behavior of Deterministic Systems with Information Loss. Path Integral Approach Berlin Preprint (2005), Annals Phys. 320 468 (2005) (quant-ph/0504200)

[363] V.I. Yukalov and H. Kleinert
Gapless Hartree-Fock-Bogoliubov Approximation for Bose Gas Berlin Preprint (2005). Phys. Rev. A 73, 063612 (2006) (cond-mat/0606484)

[364] F.S. Nogueira and H. Kleinert
Thermally Induced Rotons in Two-Dimensional Dilute Bose Gases Phys. Rev. B 73, 104515 (2006) (cond-mat/0503523)

[365] H. Kleinert and X.J. Chen
Boltzmann Distribution and Market Temperature Physica. A 383, 583 (2007) (physics/0609209)

[366] H. Kleinert
Field Transformations and Multivalued Fields Berlin preprint 2006

[367] H. Kleinert
Stiff Quantum Polymers Phys. Rev. B 76, 052202 (2007) (arXiv:cond-mat/0701030v3)

[368] F. Nogueira and H. Kleinert
Compact quantum electrodynamics in 2+1 dimensions and spinon deconfinement: a renormalization group analysis Phys. Rev. B 77, 045107 (2008) (arXiv:0705.3541)

[369] J.W. Zhang, Y. Zhang, and H. Kleinert
Power tails of Index Distributions in Chinese Stock Market Physica A 377, 166 (2007)

[370] K. Glaum, A. Pelster, H. Kleinert, and T. Pfau
Critical Temperature of Weakly Interacting Dipolar Condensates; Physical Review Letters 98, 080407/1-4 (2007); (cond-mat/0606569);

[371] H. Kleinert and S.-S. Xue
Photoproduction in Semiconductors by Onset of Magnetic Field Eur. Phys. Letters 81, 57001 (2008).

[372] H. Kleinert, R. Ruffini, and S.-S. Xue
Electron-Positron Pair Production in Space- or Time-Dependent Electric Fields Phys. Rev. D 78, 025011 (2008)

[373] H. Kleinert,
From Landau's Order Parameter to Modern Disorder Field Theory in L.D. Landau and his Impact on Contemporary Theoretical Physics, Horizons in World Physics 264 (2008), A. Sakaji and I. Licata (preprint).

[374] H. Kleinert and P. Kienle,
Neutrino Mass Differences from Interfering Recoil Ions Lecture presented at the 3rd Stueckelberg Workshop on Relativistic Field Theories ICRANET Stueckelberg July 8-18, 2008 - ICRANet Center, Pescara (Italy), (ed.) EJTP 6, 107 (2009)

[375] A. Chervyakov and H. Kleinert,
Exact Pair Production Rate for a Smooth Potential Step (preprint) Phys. Rev. D 80, 065010 (2009)

[376] P. Jizba, H. Kleinert, and P. Haener
Perturbation Expansion for Option Pricing with Stochastic Volatility Physica A 388 (2009) 3503 (preprint)

[377] H. Kleinert,
Equivalence Principle and Field Quantization in Curved Spacetime (arxiv:0910.4034) EJTP 6, 1 (2009)

[378] J. Dietel and H. Kleinert,
Modeling two-dimensional crystals and nanotubes with defects under stress (arXiv:0812.0226) Phys. Rev. B 79, 245415 (2009)

[379] J. Dietel and H. Kleinert,
Lindemann parameters for solid membranes focused on carbon nanotubes (arXiv:0806.1656) Phys. Rev. B 79, 075412 (2009)

[380] J. Dietel and H. Kleinert,
Phase diagram of vortices in high-Tc superconductors with a melting line in the deep Hc2 region (arXiv:0807.2757) Phys. Rev. B 79, 014512 (2009)

[381] J. Dietel and H. Kleinert,
Phase diagram of vortices in high-Tc superconductors from lattice defect model with pinning (arXiv:cond-mat/0612042) Phys. Rev. B 75, 144513 (2007) Erratum: Phys. Rev. B 78, 059901 (2008)

[382] J. Dietel and H. Kleinert,
Defect-induced melting of vortices in high- Tc superconductors: A model based on continuum elasticity theory (arXiv:cond-mat/0511710) Phys. Rev. B 74, 024515 (2006)

[383] J. Dietel and H. Kleinert,
Triangular lattice model of two-dimensional defect melting (arXiv:cond-mat/0508780) Phys. Rev. B 73, 024113 (2006)

[384] P. Jizba, H. Kleinert, and F. Scardigli
Uncertainty Relation on World Crystal and its Applications to Micro Black Holes (arXiv:0912.2253) Phys. Rev. D 81, 084030 (2010)

[385] H. Kleinert
New Gauge Symmetry in Gravity and the Evanescent Role of Torsion (arxiv/1005.1460) EJTP 24, 287 (2010)

[386] H. Kleinert
The Invisibility of Torsion in Gravity Essay with Honorable Mention at the Gravity Research Foundation (2010)

[387] H. Kleinert
Converting Divergent Weak-Coupling into Exponentially Fast Convergent Strong-Coupling Expansions (arXiv:1006.2910) preprint (2010). Lectures at the Centre International de Rencontres Mathematiques in Luminy 2010, publ. in EJTP 8, 15 (2011)

[388] Petr Jizba and Hagen Kleinert Superstatistics approach to path integral for a relativistic particle
(arxiv/1007.1007.3922) Phys. Rev. D 82, 085016 (2010) (2010)

[389] J. Dietel and H. Kleinert,
Optical phonon lineshapes and transport in metallic carbon nanotubes under high bias voltage (arxiv/1006.5872) Phys. Rev. B 82, 195437 (2010)

[390] A. Karamatskou and H. Kleinert
Quantum Maupertuis Principle (quant-ph/0910.4034)

[391] H. Kleinert
Hubbard-Stratonovich Transformation: Successes, Failure, and Cure (cond-mat/1104.5161), EJTP 8, 57 (2011) (arXiv:1104.5161).

[392] H. Kleinert
Strong-Coupling Bose-Einstein Condensation (cond-mat/1105.5115)

[393] H. Kleinert
Extending Bogoliubov's Boson Theory to Strong Couplings preprint 2011

[394] H. Kleinert Challenge to find Quasicrystals with Seven-Fold Symmetry
EJTP 8, 169 (2012) preprint 2011

[395] H. Kleinert Is dark matter made entirely of the gravitational field?
Phys. Scripta T151 14081 (2012) (gr-qc/1107.2610)

[396] H. Kleinert and A. Chervyakov
On Electron-Positron Pair Production by a Spatially Nonuniform Electric Field (arXiv:1112.4120)

[397] H. Kleinert, P. Jizba, and M. Shefaat
Renyi's information transfer between financial time series Physica A 391, 2971 (2012)

[398] H. Kleinert
Quantum Field Theory of Particle Orbits with Large Fluctuations (preprint)

[399] H. Kleinert
Fractional quantum field theory, path integral, and stochastic differential equation for strongly interacting many-particle systems EPL 100 10001 (2012)

[400] H. Kleinert
Superpositions of probability distributions Phys. Rev. E 78, 031122 (2008)

[401] P. Jizba, H. Kleinert, and M. Shefaat
Rényis information transfer between financial time series Physica A 391, 297 (2012)

[402] M. Fiolhais and H. Kleinert
Higgs boson mass from tricritical point of the weak interactions Physics Letters A 377, 2195 (2013)

[403] H. Kleinert
Effective Action and Field Equation for BEC from Weak to Strong Couplings J. Phys. B 46, 175401 (2013)

[404] H. Kleinert
Conformal Gravity with Fluctuation-Induced Einstein Behavior at Long Distance (preprint)

[405] H. Kleinert
Melting of Wigner-like Lattice of Parallel Polarized Dipoles of Parallel Polarized Dipoles EPL 102, 56002 (2013)

[406] H. Kleinert and V. Zatloukal
Fractional Fokker-Planck Equations, Stochastic Differential Equations, and Path Integrals for Lévy Random Walks (preprint)

[407] H. Kleinert and Pisin Chen
Bohm Trajectories as Approximations to Properly Fluctuating Quantum Trajectories EJTP 13, 1 (2016), (preprint)

[408] C. Gruber and H. Kleinert
Observed Cosmological Reexpansion from Minimal QFT with Bose and Fermi Fields Astroparticle Physics 62, 72 (2015)

[409] H. Kleinert
Quantum Field Theory of Black-Swan Events Found. Phys. 44, 546 (2014).

[410] H. Kleinert, Z. Narzikulov, and Abdulla Rakhimov
Quantum phase transitions in optical lattices beyond the Bogoliubov approximation Phys. Rev. A 85, 063602 (2012)

[411] H. Kleinert, Z. Narzikulov, and Abdulla Rakhimov
Phase transitions in three-dimensional bosonic systems in optical lattices J. Stat. Mechanics P01003 (2014)

[412] A. Karamatskou and H. Kleinert,
Geometrization of the Schrödinger equation: Application of the Maupertuis Principle to quantum mechanics Int. J. Geom Methods 11, 145006 (2014)

[413] H. Kleinert,
The purely Geometric Part of "Dark Matter"---A Fresh Playground for "String Thery" EJTP 8 27 (2011) (

[414] H. Kleinert,
The GIMP Nature of Dark Matter Berlin preprint (2016), EJTP 13 (2016) 1-12

[415] S.-S. Xue and H. Kleinert,
Composite Fermions and their Pair States in a Strongly-Coupled Fermi Liquid Berlin preprint (2017) (longer version).

[416] S.-S. Xue and H. Kleinert,
Photoproduction in semiconductors by onset of magnetic field EPL 81 (2008)

[417] S.-S. Xue and H. Kleinert,
Critical fermion density for restoring spontaneously broken symmetry Mod. Phys. Lett. A 30, 1550122 (2015) (

[418] Jean-Philippe Aguilar, Cyril Coste, Hagen Kleinert, Jan Korbel
Regularization and analytic option pricing under α-stable distribution of arbitrary asymmetry arXiv:1611.04320

Fields of Research


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  .

Discovery of Quasicrystals; Icosahedral Phase

The existence of quasicrystals in an icosahedral phase was first conjectured and investigated in 1981 as a candiate for the blue phase in liquid crystals by

H. Kleinert and K. Maki,
Lattice Textures in Cholesteric Liquid Crystals
Fortschr. Phys. 29, 1 (1981). See p. 242 and 243.

A phase with these properties was discovered experimentally three years later by Shechtman at al. in 1984: D. Shechtman at al., Phys. Rev. Lett. 53, 1951 (1984) in a rapid solidification of the alloy Al86Mn14.

Shechtman's experimental discovery earned him the Nobel Prize 2011 for a "Paradigm Shift in Crystallography" . Chemists had a hard time believing in the existence of a five-fold symmetric crystal structure discussed in our 1981 paper with Maki.

See also the citations of Kleinert and Maki in the review article on Liquid Crystals by Tamar Seideman. A picture of the square of the icosahedral order parameter in Eq. (4,22) of above Kleinert and Maki paper can easily be plotted with the help of this short Mathematica script:
nfold=5; k[n_]:=2 Pi *n/nfold
phi=Sum[E^(I (Cos[k[n]]*x+Sin[k[n]]*y)),{n,0,nfold-1}]

Icosahedral order parameter

By changing the number nfold, the program draws a picture of the density of other two-dimensional quasicrystal symmetries, for instance a picture of the nfold=7 heptagonal order parameter:

heptagonal order parameter

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.


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.

Feynman's last blackboard
Fig. 1a: Feynman's last blackboard in Physics Today, February 1989 issue, p. 88.
Feynman's last blackboard with zoom to corner
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).

Feynman's last blackboard
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 theory allows for the arbitrarily precise calculations of strong-coupling properties of quantum field theories. In particular, it enables one to find critical exponents near second-order phase transitions in a simple way without using renormalization group theory. It makes essential use of the Wegner exponent governing the approach to scaling (which Pade and Pade-Borel methods are unable to do) and determines it with high accuracy. The theory is based on an essential extension of the original Feynman-Kleinert Variational Approach in two steps.
  1. The Peierls inequality was abandoned in order to deal with expansions of any order. This turns divergent weak-coupling expansions into convergent strong-coupling expansions.  The convergence is mostly exponentially fast.

    The details are described in Chapter 5 of my textbook Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets.

  2. A simple trick was found to accommodate the anomalous dimensions of quantum field theory. The result is Field-Theoretic Variational Perturbation Theory which has led to the most reliable results of strong-coupling properties of field theories so far. The most-accurately measured strong-coupling property is the critical exponent that governs the singularity of the specific heat of superfluid helium, which was performed with great effort in a microgravity environment in a satellite by J. Lipa and collaborators . This experiment found precisely the value which I predicted in a seven-loop calculation. For details, see my textbook on this subject Critical Properties of phi4 Theories.

    The relevant papers are:

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

picture gallery


Prof. Dr. Hagen Kleinert
Liebensteinstr. 6
14195 Berlin

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