Homepage of Hagen Kleinert

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
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 H. Kleinert,
Gauge Fields in Condensed Matter, World Scientific, Singapore, 1989
(Books 1, 2).
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
(fetch citation list of this paper from SLAC server).
Recently, 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 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 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, 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 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 (see here)
resulting in a fully-fledged strong-coupling 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 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 (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 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 this link.

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 the fourth edition of
my textbook on path integrals (Book 8).