Kleinert has written ~420 papers on mathematical physics and the physics of elementary particles, nuclei, solid state systems, liquid crystals, biomembranes, microemulsions, polymers, and the theory of financial markets.[2] He has written several books on theoretical physics,[3] the most notable of which, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, has been published in five editions since 1990 and has received enthusiastic reviews.[4]
As a young professor in 1972, Kleinert visited Caltech and was impressed by noted US physicist Richard Feynman. Later, Kleinert was to collaborate with Feynman[5] in some of the latter's last work.[6]
This collaboration led to a mathematical method for converting divergent weak-coupling power series into convergent strong-coupling ones. This so-called variational perturbation theory yields at present the most accurate theory of critical exponents[7]
observable close to second-order phase transitions, as confirmed for superfluid helium in satellite experiments.[8] He also discovered an alternative to Feynman's time-sliced path integral construction which can be used to solve the path integral formulations of the hydrogen atom and the centrifugal barrier, i.e. to calculate their energy levels and eigenstates, as special cases of a general strategy for treating systems with singular potentials using path integrals.[9][10]
Within the quantum field theories of quarks
he found the origin[11] of the algebra of Regge residues conjectured by N. Cabibbo, L. Horwitz, and
Y. Ne'eman (see p. 232 in reference[12]).
Together with A. Chervyakov, Kleinert developed an extension of the theory of distributions from linear spaces to semigroups by defining their products uniquely (in the mathematical theory, only linear combinations are defined). The extension is motivated by the physical requirement that the corresponding path integrals must be invariant under coordinate transformations,[21] which is necessary for the equivalence of the path integral formulation to Schrödinger theory.
- Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 743–1456, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online: Vol. I and Vol. II)
- Critical Properties of φ4-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online) (together with V. Schulte-Frohlinde)
- Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edition, World Scientific (Singapore, 2009) (also available online)
- Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online)
- Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity, World Scientific (Singapore, 2008) (together with R.T. Jantzen)
- Particles and Quantum Fields, World Scientific (Singapore, 2016) (also available online)