Hagen Kleinert

Hagen Kleinert
Freie Universität Berlin | FUB · Department of Physics

Professor of Physics

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377
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11,915
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Publications

Publications (377)
Article
A detailed analysis of electron–positron pair creation induced by a spatially non-uniform and static electric field from vacuum is presented. A typical example is provided by the Sauter potential. For this potential, we derive the analytic expressions for vacuum decay and pair production rate accounted for the entire range of spatial variations. In...
Article
Full-text available
We analyze the functional integral for quantum Conformal Gravity and show that with the help of a Hubbard-Stratonovich transformation, the action can be broken into a local quadratic-curvature theory coupled to a scalar field. A one-loop effective action calculation reveals that strong fluctuations of the metric field are capable of spontaneously g...
Article
In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric g_μν(x) = 2M[V(x)-E]δ_μν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger e...
Article
We show how the phenomenon of spontaneous symmetry breakdown is affected by the presence of a sea of fermions in the system. When its density exceeds a critical value, the broken symmetry can be restored. We calculate the critical value and discuss the consequences for three different physical systems: First, for the standard model of particle phys...
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The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Article
Using semiclassical WKB-methods, we calculate the rate of electron-positron pair-production from the vacuum in the presence of two external fields, a strong (space- or time-dependent) classical field and a monochromatic electromagnetic wave. We discuss the possible medium effects on the rate in the presence of thermal electrons, bosons, and neutral...
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Free and weakly interacting particles perform approximately Gaussian random walks with collisions. They follow a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. By contrast, the fields of strongly interacting particles extremize more involved effective actions obeying fractional wave equations with anomalous dimensi...
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In 1936, Weisskopf showed that for vanishing electric or magnetic fields the strong-field behavior of the one loop Euler-Heisenberg effective Lagrangian of quantum electro dynamics (QED) is logarithmic. Here we generalize this result for different limits of the Lorentz invariants \(\vec{E}^2-\vec{B}^2\) and \(\vec{B}\cdot\vec{E}\). The logarithmic...
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We formulate the Collective Quantum Field Theory for three-dimensional bosonic optical lattices and evaluate its consequences in a mean-field approximation to two collective fields, proposed by Cooper, and in a lowest-order Variational Perturbation Theory (VPT). It is shown that the mean-field approximation predicts some essential features of the e...
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Due to the chiral nature of the Dirac equation, overlying of an electrical superlattice (SL) can open new Dirac points on the Fermi-surface of the energy spectrum. These lead to novel low-excitation physical phenomena. A typical example for such a system is neutral graphene with a symmetrical unidirectional SL. We show here that in smooth SLs, a se...
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While free and weakly interacting particles are well described by a a second-quantized nonlinear Schr\"odinger field, or relativistic versions of it, the fields of strongly interacting particles are governed by effective actions, whose quadratic terms are extremized by fractional wave equations. Their particle orbits perform universal L\'evy walks...
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We calculate dc-conductivities of ballistic graphene undulated by a overlying moving unidirectional electrical superlattice (SL) potential whose SL-velocity is smaller than the electron velocity. We obtain no dependence of the conductivity on the velocity along the direction of the superlattice wavevector. In the orthogonal direction however, the d...
Article
We study uncertainty relations as formulated in a crystal-like universe, whose lattice spacing is of order of Planck length. For Planck energies, the uncertainty relation for position and momenta has a lower bound equal to zero. Connections of this result with double special relativity, and with't Hooft's deterministic quantization proposal, are br...
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We point out that electromagnetism with Dirac magnetic monopoles harbors an extra local gauge invariance called monopole gauge invariance. The gauge transformations act on a gauge field of monopoles and are independent of the ordinary electromagnetic gauge invariance. The extra invariance expresses the physical irrelevance of the shape of the Dirac...
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The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor exp[-βVeff cl(x0)], where Veff cl(x0) is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average . We show how to...
Article
Full-text available
A detailed analysis of electron-positron pair creation induced by a spatially nonuniform and static electric field from vacuum is presented. A typical example is provided by the Sauter potential. For this potential, we derive the analytic expressions for vacuum decay and pair production rate accounted for the entire range of spatial variations. In...
Article
We point out that there is a natural geometric procedure for constructing the quantum theory of a particle in a general metric-affine space with curvature and torsion. Quantization rules are presented and expressed in the form of a simple path integral formula which specifies compactly a new combined equivalence and correspondence principle. The as...
Article
We formulate generalized uncertainty relations in a crystal-like universe whose lattice spacing is of order of Planck length -- a "world crystal". For energies near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momentum does not pose any lower bound. We apply these results to micro black h...
Article
Full-text available
We formulate generalized uncertainty relations in a crystal-like universe whose lattice spacing is of order of Planck length --- a "world crystal". For energies near the border of the Brillouin zone, i.e. for Planckian energies, the uncertainty relation for position and momentum does not pose any lower bound. We apply these results to micro black h...
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Full-text available
We study the quantum phase transition from a superfluid to a Mott insulator in optical lattices using a Bose-Hubbard Hamiltonian. For this purpose we develop a field theoretical approach in terms of path integral formalism to calculate the second-order quantum corrections to the energy density as well as to the superfluid fraction in cubic optical...
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We argue that part of "dark matter" is not made of matter, but of the singular world-surfaces in the solutions of Einstein's vacuum field equation G_{\mu\nu}=0. Their Einstein-Hilbert action governs also their quantum fluctuation. It coincides with the action of closed bosonic "strings" in four spacetime dimensions, which appear here in a new physi...
Article
Full-text available
We study uncertainty relations as formulated in a crystal-like universe, whose lattice spacing is of order of Planck length. For Planck energies, the uncertainty relation for position and momenta has a lower bound equal to zero. Connections of this result with double special relativity, and with 't Hooft's deterministic quantization proposal, are b...
Article
Full-text available
We report on the development of a systematic variational perturbation theory for the euclidean path integral representation of the density matrix based on new smearing formulas for harmonic correlation functions. As a first application, we present the lowest-order approximation for the radial distribution function of an electron in a hydrogen atom.
Article
In this paper, we quantify the statistical coherence between financial time series by means of the Rényi entropy. With the help of Campbell’s coding theorem, we show that the Rényi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others. This accentuation is controlled with Rény...
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We consider metallic carbon nanotubes with an overlying unidirectional electrical chiral (wavevector out of the radial direction, where the axial direction is included) superlattice potential. We show that for superlattices with a wavevector close to the axial direction, the electron velocity assumes the same value as for nanotubes without superlat...
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We extend the theory of Bose-Einstein condensation from Bogoliubov's weak-coupling regime to arbirarily strong couplings.
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According to the Maupertuis principle, the movement of a classical particle in an external potential $V(x)$ can be understood as the movement in a curved space with the metric $g_{\mu\nu}(x)=2M[V(x)-E]\delta_{\mu\nu}$. We show that the principle can be extended to the quantum regime, i.e., we show that the wave function of the particle follows a Sc...
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Full-text available
A d-wave high temperature cuprate superconductor exhibits a nematic ordering transition at zero temperature. Near the quantum critical point, the coupling between gapless nodal quasiparticles and nematic order parameter fluctuation can result in unusual behaviors, such as extreme anisotropy of fermion velocities. We study the disorder effect on the...
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We calculate the current-voltage characteristic of a homogeneously strained metallic carbon nanotube adsorbed on a substrate. The strain generates a gap in the energy spectrum leading to a reduction of the current. In the elastic regime, the current-voltage characteristic shows a large negative differential conductance at bias voltages of around $...
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Superstatistics permits the calculation of the Feynman propagator of a relativistic particle in a novel way from a superstatistical average over non-relativistic single-particle paths. We illustrate this for the Klein-Gordon particle in the Feshbach-Villars representation, and for the Dirac particle in the Schroedinger-Dirac representation. As a by...
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Full-text available
We calculate the current-voltage characteristic of metallic nanotubes at high bias voltage showing that a bottleneck exists for short nanotubes in contrast to large ones. We attribute this to a redistribution of lower-lying acoustic phonons caused by phonon-phonon scattering with hot optical phonons. The current-voltage characteristic and the elect...
Article
Full-text available
We formulate generalized uncertainty relations in a crystal-like universe whose lattice spacing is of the order of Planck length -- "world crystal". In the particular case when energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momenta does not pose any lower bound on involved...
Article
We fit the volatility fluctuations of the S&P 500 index well by a Chi distribution, and the distribution of log-returns by a corresponding superposition of Gaussian distributions. The Fourier transform of this is, remarkably, of the Tsallis type. An option pricing formula is derived from the same superposition of Black–Scholes expressions. An expli...
Article
The statistical mechanics of line-like excitations of many-body systems can most easily be formulated with the help of multivalued fields which do not satisfy the Schwarz integrability conditions. These fields must be included into the hydrodynamic description of such systems in order to allow for a proper understanding of the observed phase transi...
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We derive the exact rate of pair production of oppositely charged scalar particles by a smooth potential proportional to tanh kz in three dimensions. As a check we recover from this the known results for an infinitely sharp step as well as for a uniform electric field. Comment: Removed the labels and improved the paper
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We calculate analytically the phase diagram of a two-dimensional planar crystal and its wrapped version with defects under external homogeneous stress as a function of temperature using a simple elastic square lattice model that allows for defect formation. The temperature dependence turns out to be very weak. The results are relevant for recent st...
Article
Changes of field variables may lead to multivalued fields which do not satisfy the Schwarz integrability conditions. Their quantum field theory needs special care as is illustrated here in applications to superfluid and superconducting phase transitions. Extending the notions that first qantization governs fluctuating orbits while second quantizati...
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Full-text available
We set up a recursion relation for the partition function of a fixed number of harmonically confined bosons. For an ideal Bose gas this leads to the well-known results for the temperature dependence of the specific heat and the ground-state occupancy. Due to the diluteness of the gas, we include both the isotropic contact interaction and the anisot...
Article
The theory presented is based on a simple Hamiltonian for a vortex lattice in a weak impurity background which includes linear elasticity and plasticity, the latter in the form of integer valued fields accounting for defects. By using the variational approach of Mézard and Parisi established for random manifolds, we obtain the phase diagram includi...
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Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v=sigma;{2} play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We de...
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Treating the production of electron and positron pairs by a strong electric field from the vacuum as a quantum tunneling process we derive, in semiclassical approximation, a general expression for the pair production rate in a $z$-dependent electric field $E(z)$ pointing in the $z$-direction. We also allow for a smoothly varying magnetic field para...
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Full-text available
We use a simple elastic Hamiltonian for the vortex lattice in a weak impurity background which includes defects in the form of integer-valued fields to calculate the free energy of a vortex lattice in the deep H_{c2} region. The phase diagram in this regime is obtained by applying the variational approach of M{\'e}zard and Parisi developed for rand...
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Temperature fluctuations in the normal direction of planar crystals such as graphene are quite violent and may be expected to influence strongly their melting properties. In particular, they will modify the Lindemann melting criterium. We calculate this modification in a self-consistent Born approximation. The result is applied to graphene and its...
Article
Treating the production of electron and positron pairs in vacuum by a strong electric field as a quantum tunneling process, we derive in semiclassical approximation the pair production rate for nonuniform fields E(z) pointing the z-direction. In addition, we discuss tunneling processes in which an empty atomic bound state is spontaneously filled wi...
Book
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 potentia...
Article
The energy bands of a semiconductor are lowered by an external magnetic field. When a field is switched on, the straight-line trajectories near the top of the occupied valence band are curved into Landau orbits and Bremsstrahlung is emitted until the electrons have settled in their final Fermi distribution. We calculate the radiated energy, which s...
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Full-text available
It is shown that superpositions of path integrals with arbitrary Hamilto-nians and different scaling parameters v ("variances") obey the Chapman-Kolmogorov relation for Markovian processes if and only if the corresponding smearing distributions for v have a specific functional form. Ensuing "smear-ing" distributions substantially simplify the coupl...
Article
We show that the minute fluctuations of S&P 500 and NASDAQ 100 indices show Boltzmann statistics over a wide range of positive as well as negative returns, thus allowing us to define a market temperature for either sign. With increasing time the sharp Boltzmann peak broadens into a Gaussian whose volatility σσ measured in 1/min is related to the te...
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Full-text available
We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of non-interacting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-T...
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We discuss compact (2+1)-dimensional Maxwell electrodynamics coupled to fermionic matter with N replica. For large enough N, the latter corresponds to an effective theory for the nearest neighbor SU(N) Heisenberg antiferromagnet, in which the fermions represent solitonic excitations known as spinons. Here we show that the spinons are deconfined for...
Article
Full-text available
Changes of field variables may lead to multivalued fields which do not satisfy the Schwarz integrability conditions. Their quantum field theory needs special care as is shown in an application to the superfluid and superconducting phase transitions.
Article
The power α of the Lévy tails of stock market fluctuations discovered in recent years are generally believed to be universal. We show that for the Chinese stock market this is not true, the powers depending strongly on anomalous daily index changes short before market closure, and weakly on the opening data.
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Full-text available
We calculate perturbatively the effect of a dipolar interaction upon the Bose-Einstein condensation temperature. This dipolar shift depends on the angle between the symmetry axes of the trap and the aligned atomic dipole moments, and is extremal for parallel or orthogonal orientations, respectively. The difference of both critical temperatures exhi...
Article
Full-text available
The theory presented is based on a simple Hamiltonian for a vortex lattice in a weak impurity background which includes linear elasticity and plasticity, the latter in the form of integer valued fields accounting for defects. Within a quadratic approximation in the impurity potential, we find a first-order Bragg-glass, vortex-glass transition line...
Article
Full-text available
The minute fluctuations of of S&P 500 and NASDAQ 100 indices display Boltzmann statistics over a wide range of positive as well as negative returns, thus allowing us to define a {\em market temperature} for either sign. With increasing time the sharp Boltzmann peak broadens into a Gaussian whose volatility $ \sigma $ measured in $1/ \sqrt{{\rm min}...
Article
Full-text available
We set up a melting model for vortex lattices in high-temperature superconductors based on the continuum elasticity theory. The model is Gaussian and includes defect fluctuations by means of a discrete-valued vortex gauge field. We derive the melting temperature of the lattice and predict the size of the Lindemann number. Our result agrees well wit...
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A dilute Bose system with Bose-Einstein condensate is considered. It is shown that the Hartree-Fock-Bogolubov approximation can be made both conserving as well as gapless. This is achieved by taking into account all physical normalization conditions, that is, the normalization condition for the condensed particles and that for the total number of p...
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We set up a harmonic lattice model for two-dimensional defect melting which, in contrast to earlier simple cubic models, resides on a triangular lattice. Integer-valued plastic defect gauge fields allow for the thermal generation of dislocations and disclinations. The model produces universal formulas for the melting tempera-ture expressed in terms...
Article
’t Hooft’s derivation of quantum from classical physics is analyzed by means of the classical path integral of Gozzi et al. It is shown how the key element of this procedure—the loss of information constraint—can be implemented by means of Faddeev–Jackiw’s treatment of constrained systems. It is argued that the emergent quantum systems are identica...
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Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occu...
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We present a path-integral formulation of ’t Hooft’s derivation of quantum physics from classical physics. The crucial ingredient of this formulation is Gozzi et al.’s supersymmetric path integral of classical mechanics. We quantize explicitly two simple classical systems: the planar mathematical pendulum and the Rössler dynamical system.
Article
Full-text available
We present a path integral formulation of 't Hooft's derivation of quantum from classical physics. Our approach is based on two concepts: Faddeev-Jackiw's treatment of constrained systems and Gozzi's path integral formulation of classical mechanics. This treatment is compared with our earlier one [quant-ph/0409021] based on Dirac-Bergmann's method.
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We show that roton-like excitations are thermally induced in a two-dimensional dilute Bose gas as a consequence of the strong phase fluctuations in two dimensions. At low momentum, the roton-like excitations lead for small enough temperatures to an anomalous phonon spectrum with a temperature dependent exponent reminiscent of the Kosterlitz-Thoules...
Article
We calculate the quantum phase transition for a homogeneous Bose gas in the plane of s-wave scattering length a_s and temperature T. This is done by improving a one-loop result near the interaction-free Bose-Einstein critical temperature T_c^{(0)} with the help of recent high-loop results on the shift of the critical temperature due to a weak atomi...
Article
We calculate the quantum phase transition for a homogeneous Bose gas in the plane of s‐wave scattering length as and temperature T. This is done by improving a one‐loop result near the interaction‐free Bose‐Einstein critical temperature Tc(0) with the help of recent high‐loop results on the shift of the critical temperature due to a weak atomic rep...
Article
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We set up recursion relations for calculating all even moments of the end-to-end distance of Porod-Kratky wormlike chains in D dimensions. From these moments we derive a simple analytic expression for the end-to-end distribution in three dimensions valid for all peristence lengths. It is in excellent agreement with Monte Carlo data for stiff chains...
Article
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The wormlike chain model of stiff polymers is a nonlinear $\sigma$-model in one spacetime dimension in which the ends are fluctuating freely. This causes important differences with respect to the presently available theory which exists only for periodic and Dirichlet boundary conditions. We modify this theory appropriately and show how to perform a...
Article
We extend field theoretic variational perturbation theory by self-similar approximation theory, which greatly accelerates convergence. This is illustrated by recalculating the critical exponents of O (N) -symmetric phi(4) theory. From only three-loop perturbation expansions in 4-epsilon dimensions, we obtain analytic results for the exponents, whic...
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A cosmological model with a gravitational Lagrangian $L_g(R)\propto R+A R^n$ is set up to account for the presently observed re-acceleration of the universe. The evolution equation for the scale factor $a$ of the universe is analyzed in detail for the two parameters $n=2$ and $n=4/3$, which were preferred by previous studies of the early universe....
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In a Ginzburg–Landau theory with n fields, the anomalous dimension of the gauge-invariant nonlocal order parameter defined by the long-distance limit of Dirac's gauge-invariant two-point function is calculated. The result is exact for all n to first order in ε≡4−d, and for all d∈(2,4) to first order in 1/n, and coincides with the previously calcula...
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On the occasion of Reinhard Folk's 60th birthday, I give a brief review of the theoretical progress in understanding the critical properties of superconductors. I point out the theoretical difficulties in finding a second-order transition in the Ginzburg-Landau Model with O(N)-symmetry in 4 - ε Dimensions, and the success in predicting the existenc...
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In this paper we review recent results obtained in [quant-ph/0504200] on the path integral formulation of 't Hooft's derivation of quantum from classical physics. In partic-ular, we employ the Faddeev–Jackiw treatment of classical constrained systems to show how 't Hooft's loss of information condition may yield a genuine quantum mechanical system....
Article
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The critical temperature T_c of an interacting Bose gas trapped in a general power-law potential V(x)=\sum_i U_i|x_i|^{p_i} is calculated with the help of variational perturbation theory. It is shown that the interaction-induced shift in T_c fulfills the relation (T_c-T_c^0)/T_c^0= D_1(eta)a + D'(eta)a^{2 eta}+ O(a^2) with T_c^0 the critical temper...
Article
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We calculate the location of the quantum phase transitions of a Bose gas trapped in an optical lattice as a function of effective scattering length a(eff) and temperature T. Knowledge of recent high-loop results on the shift of the critical temperature at weak couplings is used to locate a nose in the phase diagram above the free Bose-Einstein crit...
Article
Full-text available
We present a path-integral formulation of 't Hooft's derivation of quantum from classical physics. The crucial ingredient of this formulation is Gozzi et al.'s supersymmetric path integral of classical mechanics. We quantize explicitly two simple classical systems: the planar mathematical pendulum and the Roessler dynamical system.
Article
We set up and solve a recursion relation for all even moments of a two-dimensional stiff polymer (Porod–Kratky wormlike chain) and determine from these moments a simple analytic expression for the end-to-end distribution applicable for all persistence lengths.
Article
We present a method for evaluating divergent series with factorially growing coefficients of equal sign. The method is based on an analytic continuation of variational perturbation theory from the regime of alternating signs. We demonstrate its power first by applying it to the exactly known partition function of the anharmonic oscillator in zero s...
Article
From the path integral description of price fluctuations with non-Gaussian distributions we derive a stochastic calculus which replaces Itô's calculus for harmonic fluctuations. We set up a natural martingale for option pricing from the wealth balance of options, stocks, and bonds, and evaluate the resulting formula for truncated Lévy distributions...
Article
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We set up a method for a recursive calculation of the effective potential which is applied to a cubic potential with imaginary coupling. The result is resummed using variational perturbation theory (VPT), yielding an exponentially fast convergence. Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html Latest up...
Article
Fundamentals Path Integrals - Elementary Properties and Simple Solutions External Sources, Correlations, and Perturbation Theory Semiclassical Time Evolution Amplitude Variational Perturbation Theory Path Integrals with Topological Constrainsts Many Particle Orbits - Statistics and Second Quantization Path Integrals in Spherical Coordinates Fixed-E...
Article
We calculate the location of the quantum phase transition of a gas of bosons trapped in an optical lattice as a function of the ratio U/J and temperature T. A loop expansion for the effective potential is carried out, yielding a nose in the phase diagram above the critical temperature T_c^0 of the interaction-free system, thus predicting the existe...
Article
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Assuming that at small distances space-time is equivalent to an elastic medium which is isotropic in space and time directions, we demonstrate that the quantum nematic liquid arising from this crystal by spontaneous proliferation of dislocations corresponds with a medium which is merely carrying curvature rigidity. This medium is at large distances...
Article
Full-text available
We set up and solve a recursion relation for all even moments of a two-dimensional stiff polymer (Porod-Kratky wormlike chain) and determine from these moments a simple analytic expression for the end-to-end distribution at all persistence lengths.
Article
We attribute the gravitational interaction between sources of curvature to the world being a crystal which has undergone a quantum phase transition to a nematic phase by a condensation of dislocations. The model explains why spacetime has no observable torsion and predicts the existence of curvature sources in the form of world sheets, albeit with...
Article
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We present a method for extracting tunnelling amplitudes from perturbation expansions which are always divergent and not Borel-summable. We show that they can be evaluated by an analytic continuation of variational perturbation theory. The power of the method is illustrated by calculating the imaginary parts of the partition function of the anharmo...
Article
Extending recent work on QED and the symmetric phase of the euclidean multicomponent -theory, we construct the vacuum diagrams of the free energy and the e#ective energy in the ordered phase of # -theory. By regarding them as functionals of the free correlation function and the interaction vertices, we graphically solve nonlinear functional di#eren...
Article
We set up simple harmonic lattice models for elastic fluctuations in bcc and fcc lattices and the excitation of dislocations and disclinations. From these we derive, in the lowest approximation, universal formulas which predict melting temperatures in good agreement with the experiments. This new theory is more precise than Lindemann's rule by fact...
Article
INTRODUCTION In quantum field theory, the calculation of physical quantities usually relies on evaluating Feynman integrals which are pictured by diagrams. Each diagram is associated with a certain weight depending on its topology. There exist various convenient computer programs, for instance FeynArts [1--3] or QGRAF [4, 5], for constructing these...
Article
Extending recent work on QED and the symmetric phase of the euclidean multicomponent scalar φ4-theory, we construct the vacuum diagrams of the free energy and the effective energy in the ordered phase of φ4-theory. By regarding them as functionals of the free correlation function and the interaction vertices, we graphically solve nonlinear function...