
Theoretical Physics Introduction A large variety of physical systems can be decribed in a unified way with the help of quantum field theory. The fields represent fluctuating physical quantities such as atomic positions, electric and magnetic fields, directions of molecules, etc., at each spacetime point. The materials can be solids, liquids, liquid crystals, or nuclei. Fields can also be used to study the changing shapes of membranes which are formed by bilipids or detergents in biophysics and chemical physics.
Field fluctuations are described most efficiently with the help of
The research of this group is devoted to applying the functional techniques 1. Theory of DefectInduced Phase Transitions Solids and superfluids undergo phase transitions which can be explained by the large configurational entropy of linelike 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 H. Kleinert, Gauge Fields in Condensed Matter, World Scientific, Singapore, 1989 (Books 1, 2). There it is shown that the longrange 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 wellknown GinzburgLandau theory of superconductivity, which is advantageous for extracting the physical consequences. 2. Quark Confinement and String Physics Quarks are held together by colorelectric 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 (fetch citation list of this paper from SLAC server). Recently, we have found that contrary to earlier expectations, the colorelectric 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 and citations) has been extended systematically and has lead to exponentially fast convergent perturbation expansions. (Ref. 213, 220, 229 and Book 5). 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 in this link this link and in the textbook H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World Scientific, Singapore, 1990 (Book 3), second extended edition 1995 (Book 5), 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; Book 5, Chapters 10, 11). 4. Classical and Statistical Mechanicsin Spaces with Curvature and Torsion The newly found 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 bodyfixed coordinate frame (Ref. 224). We have also derived the FokkerPlanck 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 linelike excitations (such as vortex or defect lines) become firstorder has been explained with the help of disorder fields (Refs. 131, 194; Books 1, 2). An important example is the melting transition, where we have found the first model undergoing two successive continuous melting transitions (Ref. 174, 179; Books 1, 2. See also this). 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 Book 5). The operator quantum Langevin equation has been derived for the first time from the forwardbackward path integral approach (Ref. 230). Field Theory of Critical Phenomena The universal critical exponents at a secondorder 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^4theory 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 (see here) resulting in a fullyfledged strongcoupling quantum field thory. This has led to a novel approach to critical phenomena (see more details here) with extremely accurate results for critical exponents, without use of the renormalizatiuon group (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 (Book 5). 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). See this link for more information. 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 experimentallyobserved spiky superstructure on membranes was found theoretically (266). 10. Superfluid Helium 3 Functional methods have been used to develop a theory of the longwavelength 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 (see here). 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, fetch citation list from SLAC server, 83). It has meanwhile led to the solution of many path integrals which previously appeared unsolvable (Book 3). 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 DuruKleinert 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 FokkerPlanck 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 this link. 15. Financial Markets Path integrals are a powerful tool for studying fluctuations in financial markets which are nonGaussian. 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 nonGaussian fluctuations (Ref. 329) and derived a new option pricing formula from this (Ref. 333). A detailed theory is contained in the fourth edition of my textbook on path integrals (Book 8). 

© by Hagen Kleinert 2018 