Path integration is carried out in the field of topological defects. The topological defects being considered include a screw dislocation and a disclination in solid. The screw dislocation give rise to torsion, while the disclination generates curvature in the surrounding space. We consider a particle bound in the vicinity of the defect by a short range repulsive and long range attractive force. By path integration we obtain the energy spectrum and the corresponding eigenfunctions.

The calculation of one loop integrals at finite temperature requires the evaluation of certain series, which converge very slowly or can even be divergent. Here we review a new method, recently devised by the author, for obtaining accelerated analytical expressions for these series. The fundamental properties of the new series are studied and an application to a physical example is considered. The relevance of the method to other physical problems is also discussed.

We describe, how to construct and compute unambiguously path integrals for particles moving in a curved space, and how these path integrals can be used to calculate Feynman graphs and effective actions for various quantum field theories with external gravity in the framework of the worldline formalism. In particular, we review a recent application of this worldline approach and discuss vector and antisymmetric tensor fields coupled to gravity. This requires the construction of a path integral for the N=2 spinning particle, which is used to compute the first three Seeley-DeWitt coefficients for all p-form gauge fields in all dimensions and to derive exact duality relations.

We propose the another, in principe nonperturbative, method of the evaluation of the Wiener functional integral for φ⁴ term in the action. All infinite summations in the results are proven to be convergent. We find the "generalized" Gelfand-Yaglom differential equation implying the functional integral in the continuum limit.

A path integral method was developed according to the usual momentum-wavefunction relation, it is different from the Feynman's path integral. In order to investigate its validity and usefulness, the path integral method was applied to hydrogen atom,the energy levels were calculated out with the same fine structure as the calculation of the Dirac wave equation, the electronic spin effect was also calculated out correctly when the hydrogen atom is put in a magnetic field. The path integral method would be useful for some physical systems when for which the Dirac equation can not be solved exactly, it was pointed out that the path integral method is a rapid quantum computation method.

We propose a CFT description for a closed one-dimensional fully frustrated ladder of quantum Josephson junctions with Mobius boundary conditions; we show how such a system can develop topological order thanks to flux fractionalization. Such a property is crucial for its implementation as a protected solid state qubit.

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model - the Hilbert space approach of Koopman and von Neumann is used to study the evolution of an ensemble of such classical systems. With the help of the supersymmetry algebra, the corresponding Liouville operator can be decomposed into two contributions with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. In this way, choosing suitable phase space coordinates, the classical Liouville equation becomes a functional Schrödinger equation of a genuine quantum field theory. Quantization here is intimately related to the constraint, which selects the part of Hilbert space where the Hamilton operator is positive. This is interpreted as dynamical symmetry breaking in an extended model, introducing a mass scale which discriminates classical dynamics beneath from emergent quantum mechanical behaviour.

Recently N. Kumano-go succeeded in proving that piecewise linear time slicing approximation to Feynman path integral with integrand F(γ) actually converges to the limit as the mesh of division of time goes to 0 if the functional F(γ) of paths γ belongs to a certain class of functionals with polynomial growth at the infinity. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties. The aim of this note is to explain that the use of piecewise classical paths naturally leads us to an analytic formula for the second term of the semi-classical asymptotic expansion of the Feynman path integrals under a little stronger assumptions than that in Kumano-go's. If F(γ)=1, this second term coincides with the one given by G. D. Birkhoff.

The cosmological constant appearing in the Wheeler-De Witt equation is considered as an eigenvalue of the associated Sturm-Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. The case of massive gravitons is discussed.

We discuss nonlocal quantum mechanical effects in mesoscopic devices, for studying which, path integral has been shown to provide quite a powerful and compact approach. In particular, we focus onto geometrical phase effects. As a former example, we discuss how a geometrical phase due to spin-orbit coupling may affect Aharonov-Bohm conductance oscillations in a mesoscopic ring. As a latter example, we show that a pertinent cycling in parameter space may induce a robust Berry phase in a quantum dot tuned close to a three-level degeneracy. In both cases, we propose to detect geometrical phase effects by means of an appropriate DC transport measurement.

Exploiting a recently constructed target space action for the exact string black hole, logarithmic corrections to the leading order entropy are studied. There are contributions from thermal fluctuations and from corrections due to α'>0 which for the microcanonical entropy appear with different signs and therefore may cancel each other, depending on the overall factor in front of the action. For the canonical entropy no such cancellation occurs. Remarks are made regarding the applicability of the approach and concerning the microstates. As a byproduct a formula for logarithmic entropy corrections in generic 2D dilaton gravity is derived.

A path integral representation is given to the Green's function for the radial Dirac equation, by constructing a countably additive path space measure on the space of continuous paths living on the real half-line. An application is suggested to a problem in quantum field theory.

A modified version of Klauder's coherent state is presented. Klauder's state is a generalized coherent state that can be constructed in terms of the energy eigenstates of a given non-degenerate system without referring to any symmetry group. It can be formed for continuous as well as discrete dynamics. The proposed modification allows us to deal with degenerate systems and to treat discrete states and continuous states in a unified manner. Some examples are given for illustration.

The Bead-Fourier path integral molecular dynamics technique, introduced earlier [S.D. Ivanov, A.P. Lyubartsev, and A. Laaksonen, Phys. Rev. E 67 (2003) 066710] for the case of distinguishable particles is reformulated in order to achieve more efficient sampling. The reformulation is carried out on the basis of the staging transformation of beads' coordinates, yielding all dynamical variables to move on similar time scales. The formalism for identical particles is presented. It is shown, that the straightforward approach leads to impossibility of the sign changes. A recipe to overcome this problem is suggested. It is demonstrated, that the developed formalism for identical particles can also be reformulated, providing efficient molecular dynamics.

Monte Carlo simulations are employed for studying a generalized three-dimensional complex |ψ|⁴ field theory with an additional fugacity term controlling the vortex-line density. It is shown that only with such an extra term, the XY type second-order phase transitions of the standard model can be tuned in certain regions of the phase diagram to become first-order. In particular, this settles a recent controversy in the standard model related to the measure of the functional integral. Also the topological excitations of the model (vortex networks) are carefully examined.

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 particular, 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. With two simple examples we discuss some of the consequences that follow from our approach.

Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac constraint quantization procedures.

We employ the su(2|1) superalgebra representation of the t-J model Hamiltonian to rigorously enforce the local constraint that guarantees a given lattice site to be either empty or singly occupied. This constraint arises because a Coulomb repulsive energy dominates over hopping energy and results in strong electron correlation which determines the basic physics of high-T_c superconductors. We apply this technique to derive a boson-spinless fermion model for the t-J Hamiltonian, which provides a microscopic scenario to take into account local spin fluctuations to the pseudogap phenomenon.

This note is an exposition of our resent papers. We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the perturbation expansion formula, the semiclassical approximation and the fundamental theorem of calculus holds in Feynman path integral.

Within the general approach which is understood as the Green function method, we develop a numerical method based on representation of the Green functions for a class of problems in the form of functional integrals with respect to Gaussian measures, and subsequent calculation of the integrals with the help of a deterministic approach. In this case the solving of the problems is reduced to evaluation of usual (Riemann) integrals of relatively low multiplicity. The method was applied to numerical solving of the Schrödinger equation and the related diffusion equation, and also to description of time evolution of some Markovian open quantum systems. The features of the method and possible area of its application are discussed.

In this work the problem of the square root operator is analyzed. To this end we considered a relativistic geometrical action of a particle in the superspace in order to quantize it and to obtain the spectrum of physical states but remaind the Hamiltonian in the natural square root form. We show, after complete quantization of the model, that the physical states that the square root Hamiltonian can operate correspond to the representations with the lowest weights λ=1/4 and λ=3/4.

A quantum field-theoretical model, which describes spatially non-homogeneous repulsive Bose gas in an external harmonic potential is considered. Two-point thermal correlation functions of the Bose gas are calculated in the framework of the functional integration approach. Successive integration over the high-energy functional variables first and then over the low-energy ones is used. The effective action functional for the low-energy variables is obtained in one loop approximation. The functional integral representations for the correlation functions are estimated by means of the stationary phase approximation. A power-law asymptotical behaviour of the correlators of the one-dimensional Bose gas is demonstrated in the limit, when the temperature is going to zero, while the volume occupied by the non-homogeneous Bose gas infinitely increases. The power-law behaviour is governed by the critical exponent dependent on the spatial arguments.

Some symmetries can be broken in the quantization process (anomalies) and this breaking is signalled by a non-invariance of the quantum path integral measure. In this talk we show that it is possible to formulate also classical field theories via path integral techniques. The associated classical functional measure is larger than the quantum one, because it includes some auxiliary fields. For a fermion coupled with a gauge field we prove that the way these auxiliary fields transform compensates exactly the Jacobian which arises from the transformation of the fields appearing in the quantum measure. This cancels the quantum anomaly and restores the symmetry at the classical level.

Path integral effective potential method is applied to symmetric and asymmetric double-well potential systems. We calculate the temperature dependence of 2nd, 3rd and 4th order cumulants, which are useful to study vibrational effects in various spectroscopic techniques. We evaluate the Debye-Waller factors in EXAFS, EELFS and XPD and discuss the characteristic features caused by asymmetric double-well potential. We also relate this method to the quantum tunneling effect in the simple chemical reaction rate constant.

One of the outstanding problems in the numerical discretization of the Feynman-Kac formula calls for the design of arbitrary-order short-time approximations that are constructed in a stable way, yet only require knowledge of the potential function. In essence, the problem asks for the development of a functional analogue to the Gauss quadrature technique for one-dimensional functions. In PRE 69 (2004) 056701, it has been argued that the problem of designing an approximation of order ny is equivalent to the problem of constructing discrete-time Gaussian processes that are supported on finite-dimensional probability spaces and match certain generalized moments of the Brownian motion. Since Gaussian processes are uniquely determined by their covariance matrix, it is tempting to reformulate the moment-matching problem in terms of the covariance matrix alone. Here, we show how this can be accomplished.

The Titius-Bode law for planetary distances is reviewed. A model describing the basic features of this law in the "quantum-like" language of a wave equation is proposed. Some considerations about the 't Hooft idea on the quantum behaviour of deterministic systems with dissipation are discussed.

A general class of functional integrals of the exponents of polynomially growing functionals on the Hilbert space be studied. A representation formula by integrals for Gaussian measures is given for this class of functional integrals. These results are applied to provide a rigorous Feynman path integral representations for the solutions of the time dependent Schrodinger equations with a polynomially growing potentials (it is possible with alternating signs). Special self-adjoint extensions for Schrodinger differential operators with a polynomially growing potentials are obtained.

The Wiener path integral splits the net diffusion flux into infinite unidirectional fluxes, whose difference is the classical diffusion flux. The infinite unidirectional flux is an artifact of the diffusion approximation to Langevin's equation, an approximation that fails on time scales shorter than the relaxation time 1/γ. The probability of one-dimensional Brownian trajectories that cross a point in one direction per unit time Δt equals that of Langevin trajectories if γΔt=2. This result is relevant to Brownian and Langevin dynamics simulation of particles in a finite volume inside a large bath. We describe the sources of new trajectories at the boundaries of the simulation that maintain fixed average concentrations and avoid the formation of spurious boundary layers.

Both the trapping geometry and the interatomic interaction strength of a dilute ultracold fermionic gas can be well controlled experimentally. When the interactions are tuned to strong attraction, Cooper pairing of neutral atoms takes place and a BCS super fluid is created. Alternatively, the presence of Feshbach resonances in the interatomic scattering allow populating a molecular (bound) state. These molecules are more tightly bound than the Cooper pairs and can form a Bose-Einstein condensate (BEC). In this contribution, we describe both the BCS and BEC regimes, and the crossover, from a functional integral point of view. The path-integral description allows to derive the properties of the super-fluid (such as vortices and Josephson tunneling) and follow them as the system is tuned from BCS the BEC.

A general method for computing kinetic isotope effects is described. The method uses the quantum-instanton approximation and is based on the thermodynamic integration with respect to the mass of the isotopes and on the path-integral Monte-Carlo evaluation of relevant thermodynamic quantities. The central ingredients of the method are the Monte-Carlo estimators for the logarithmic derivatives of the partition function and the delta-delta correlation function. Several alternative estimators for these quantities are described here and their merits are compared on the benchmark hydrogen-exchange reaction, H+H₂->H₂+H on the Truhlar-Kuppermann potential energy surface. Finally, a qualitative discussion of issues arising in many-dimensional systems is provided.