Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator

The SU（1,1） coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU（1,1） coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr Sommerfeld quantization rule yields the exact expression for the energy spectrum.     

Critical scattering in heavy ion reactions

Heavy ion reactions at intermediate energies are mostly described by the VUU equations. Using methods of quantum statistics we derive a useful relation between the width of s.p. states and the density — density correlation function. Near the point of instability from this relation there follows the increase of the collision integral and enhanced equilibration.The author thanks Yu. Kalinovsky, R. Nazmitdinov and M. Di Toro for valuable discussions.     

Color degrees of freedom in a quark-glue plasma at finite baryon density

We derive the color-singlet partition function for a quark-glue plasma with finite quark (baryon number) density by a projection method. Due to colorlessness there is a gradual “freezing” (reduction) of internal degrees of freedom as compared to the Stefan-Boltzmann limit. We find here that this non-perturbative effect is reduced by a finite quark density. A relation between the requirement of colorlessness of all physical states and QCD is proposed.     

Functional Integral Approach to a Many Fermion System with Bound States

The equation of state for a dense system of interacting Fermions is derived using functional integral technique. To investigate the formation of bound states, a Hubbard-Stratonovich transformation is applied which expresses the action as functional of pair fields. The evaluation of the partition function in lowest orders with respect to the pair fields leads to a result which can be interpreted as the contribution of two-particle states, accounting for density corrections as Pauli blocking and self-energy shift. Comparison is performed with results for the equation of state of hot dense matter (plasmas, nuclear matter) obtained within a Green's function approach.     

Fractal behavior in quantum statistical physics

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.     

Coherent State Path Integral for the Harmonic Oscillator and a Spin Particle in a Constant Magnetic field

Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP¹ model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.     

Explicit relations for the radial distribution functions for one-dimensional lattice (arbitrary spacing) fluids and solutions

It is pointed out that the size of the matrix required to formulate the grand partition function for a one-dimensional lattice fluid for a fixed and finite range of the interatomic potential varies linearly with the density of lattice points used and hence is much smaller and more manageable than the expected size (which varies exponentially with the same quantity) and thus allows very fine grids to be examined. Using the matrix treatment of the grand partition function, it is shown that the radial distribution function for a one-dimensional fluid or solution can be formulated as an explicit matrix product which is simply performed by computer. The resulting distribution functions (which can be extrapolated to the continuum by varying the lattice spacing) are useful as starting solutions for the iterative solution of integral equations for three-dimensional fluids.     

Partition function for an electron in a random potential

We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.Work supported by the National Science Foundation.     

Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral b ₂(V) related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.     

Factorization of correlation functions and the replica limit of the Toda lattice equation

Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and non-Hermitian theories in the Wigner–Dyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive two-point correlation functions for class A and class AIII as well as several one-point correlation functions in class AIII. In the non-Hermitian case the average spectral density of non-Hermitian complex random matrices in the weak non-Hermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the Hatano–Nelson model) which is known from earlier work. New results are obtained for the average spectral density in the weak non-Hermiticity limit of a quenched chiral random matrix model at non-zero chemical potential. These results apply to the ergodic or ϵ domain of the quenched QCD partition function at non-zero chemical potential. Our results have been checked against numerical results obtained from a large ensemble of random matrices. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one- and two-point functions into a product of a bosonic (non-compact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic partition functions, the bosonic partition functions and the supersymmetric partition function are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

Hadronic density of states from string theory

We present an exact calculation of the finite temperature partition function for the hadronic states corresponding to a Penrose-Güven limit of the Maldacena-Nù?ez embedding of the N=1 super Yang-Mills (SYM) into string theory. It is established that the theory exhibits a Hagedorn density of states. We propose a semiclassical string approximation to the finite temperature partition function for confining gauge theories admitting a supergravity dual, by performing an expansion around classical solutions characterized by temporal windings. This semiclassical approximation reveals a hadronic energy density of states of a Hagedorn type, with the coefficient determined by the gauge theory string tension as expected for confining theories. We argue that our proposal captures primarily information about states of pure N=1 SYM theory, given that this semiclassical approximation does not entail a projection onto states of large U(1) charge.     

Simplicial vs. continuum string theory and loop equations

We derive loop equations in a scalar matrix field theory. We discuss their solutions in terms of simplicial string theory—the theory describing embeddings of two-dimensional simplicial complexes into the spacetime of the matrix field theory. This relation between the loop equations and the simplicial string theory gives further arguments that favor one of the statements of the paper hep-th/0407018. The statement is that there is an equivalence between the partition function of the simplicial string theory and the functional integral in a continuum string theory—the theory describing embeddings of smooth two-dimensional world-sheets into the spacetime of the matrix field theory in question.     

Relativistic quantum and classical kinetic equations in the tomographic-probability representation

Using the Radon integral transform of the relativistic kinetic equation for a spin-zero particle, we obtain the classical and quantum evolution equations for the tomographic probability density (tomogram) describing the states of the particle in both the classical and quantum pictures. The Green functions (propagators) of the evolution equations of a free particle are constructed. The examples of the evolution of Gaussian tomogram is considered.     

Calculation of Canonical Properties and Excited States by Path Integral Numerical Methods

Path integral Monte Carlo method in the expanded ensemble is used for calculation of the ratio of partition functions for different classes of permutations treating the problem of several interacting identical particles (fermions) in an external field. Wang‐Landau algorithm is used for adjustment of balancing factors. For systems consisting of greater than two number of particles we propose an advanced variant of our approach which implies calculation of the ratio of positive and negative contributions to the partition function. Densities and energies of the sequence of excited states starting from the ground state for a system of non interacting quantum particles are calculated in turn, one by one, by means of considering systems with artificially excluded lowest energy levels and further obtaining of the ”ground state” of each next system constructed in this way. The idea of evaluation of densities of excited states for a quantum system with interparticle interaction by evaluating density difference between systems of different number of noninteracting Fermi‐copies of the system of interest is realized in terms of cyclic expansion formalism for a simple 2D system of two spinless fermions interacting via Coulomb repulsion (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一维互作用扭折-声子气体的统计理论

本文系统地研究并改进了位移型相变中一维互作用扭折-声子气体模型的统计理论。通过对扭折和声子的配分函数各自采用适当的路径积分表式并改进声子路径积分的计算方法,得到一个较合理且较普遍的巨配分函数表达式。在基态近似下,它可简化为一个与文献[6]相似的结果;在经典近似下,由它计算出的平均扭折密度与计算机模拟实验结果比文献上的符合得要好。     

纠缠态投影算符的积分

借助湮没算符的本征值方程及其本征态的完备性,证明了纠缠态的完备性．在此基础上,利用纠缠态所满足的本征值方程,得到了非对称纠缠态投影算符的积分． 关键词： 完备性 纠缠态 投影算符 积分