arXiv subjects
Publications and source records attributed to Wu-Sheng Dai.
In the calculation of quantum-mechanical singular-potential scattering, one encounters divergence. We suggest three renormalization schemes, dimensional renormalization, analytic continuation approach, and minimal-subtraction scheme to remove the divergence.
An exact solution of the three-dimensional spherical symmetry inverse-square-root potential $V\left( r\right) =-\fracα{\sqrt{r}}$, including scattering and bound-state solutions, is presented.
Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.
A thermo field dynamics approach for calculating the entropy of a spacetime is suggested. It is exemplified through the Rindler spacetime, the Milne spacetime, the Boulware spacetime, and the Minkowski spacetime with a moving mirror that the entropy of a spacetime is equal to the entropy of a thermo quantum field with the same temperature of the spacetime we study. This in fact suggests an thermo field dynamics approach of calculating the entropy of a spacetime. In this approach, the entropy of a spacetime is an expectation value of the entropy operator on a thermo vacuum state. The thermo vacuum state is the vacuum state on the maximal manifold which is the maximal analytic extension of the spacetime we study.
In conventional scattering theory, by large-distance asymptotics, at the cost of losing the information of the distance between target and observer, one imposes a large-distance asymptotics to achieve a scattering wave function which can be represented explicitly by a scattering phase shift. In this paper, without large-distance asymptotics, we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scattering wave functions, scattering boundary conditions, cross sections, and phase shifts are given without large-distance asymptotics. The importance of an arbitrary-dimensional scattering theory is that the dimensional renormalization procedure in quantum field theory needs an arbitrary-dimensional result. Moreover, we give a discussion of one- and two-dimensional scatterings.
An approach for solving scattering problems, based on two quantum field theory methods, the heat kernel method and the scattering spectral method, is constructed. This approach converts a method of calculating heat kernels into a method of solving scattering problems. This allows us to establish a method of scattering problems from a method of heat kernels. As an application, we construct an approach for solving scattering problems based on the covariant perturbation theory of heat-kernel expansions. In order to apply the heat-kernel method to scattering problems, we first calculate the off-diagonal heat-kernel expansion in the frame of the covariant perturbation theory. Moreover, as an alternative application of the relation between heat kernels and partial-wave phase shifts presented in this paper, we give an example of how to calculate a global heat kernel from a known scattering phase shift.
By large-distance asymptotics, in conventional scattering theory, at the cost of losing the information of the distance between target and observer, one arrives at an explicit expression for scattering wave functions represented by a scattering phase shift. In the present paper, together with a preceding paper (T. Liu,W.-D. Li, and W.-S. Dai, JHEP06(2014)087), we establish a rigorous scattering theory without imposing large-distance asymptotics. We show that even without large-distance asymptotics, one can also obtain an explicit scattering wave function represented also by a scattering phase shift, in which, of course, the information of the distance is preserved. Nevertheless, the scattering amplitude obtained in the preceding paper depends not only on the scattering angle but also on the distance between target and observer. In this paper, by constructing a scattering boundary condition without large-distance asymptotics, we introduce a scattering amplitude, like that in conventional scattering theory, depending only on the scattering angle and being independent of the distance. Such a scattering amplitude, when taking large-distance asymptotics, will recover the scattering amplitude in conventional scattering theory. The present paper, with the preceding paper, provides a complete scattering theory without large-distance asymptotics.
In conventional scattering theory, to obtain an explicit result, one imposes a precondition that the distance between target and observer is infinite. With the help of this precondition, one can asymptotically replace the Hankel function and the Bessel function with the sine functions so that one can achieve an explicit result. Nevertheless, after such a treatment, the information of the distance between target and observer is inevitably lost. In this paper, we show that such a precondition is not necessary: without losing any information of distance, one can still obtain an explicit result of a scattering rigorously. In other words, we give an rigorous explicit scattering result which contains the information of distance between target and observer. We show that at a finite distance, a modification factor --- the Bessel polynomial --- appears in the scattering amplitude, and, consequently, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase (the argument of the Bessel polynomial) appears in the scattering wave function.
In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as $a^{\dagger}b=Λ\left(N\right) $ or $N=Λ^{-1} \left( a^{\dagger}b\right)$, where $N$, $a^{\dagger}$, and $b$ denote the number, creation, and annihilation operators, i.e., $N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical distribution is the Gentile distribution, a statistical distribution in which the maximum occupation number is an arbitrary integer. As examples, we discuss the statistical distributions corresponding to various operator relations. In particular, besides Bose-Einstein and Fermi-Dirac cases, we discuss the statistical distributions for various schemes of intermediate statistics, especially various $q$-deformation schemes. Our result shows that the statistical distributions corresponding to various $q$-deformation schemes are various Gentile distributions with different maximum occupation numbers which are determined by the deformation parameter $q$. This result shows that the results given in much literature on the $q$-deformation distribution are inaccurate or incomplete.
In this paper, we consider the problem of mean first-passage time (MFPT) in quantum mechanics; the MFPT is the average time of the transition from a given initial state, passing through some intermediate states, to a given final state for the first time. We apply the method developed in statistical mechanics for calculating the MFPT of random walks to calculate the MFPT of a transition process. As applications, we (1) calculate the MFPT for multiple-state systems, (2) discuss transition processes occurring in an environment background, (3) consider a roundabout transition in a hydrogen atom, and (4) apply the approach to laser theory.
In this paper, we construct an intermediate distribution linking the Gaussian and the Cauchy distribution. We provide the probability density function and the corresponding characteristic function of the intermediate distribution. Because many kinds of distributions have no moment, we introduce weighted moments. Specifically, we consider weighted moments under two types of weighted functions: the cut-off function and the exponential function. Through these two types of weighted functions, we can obtain weighted moments for almost all distributions. We consider an application of the probability density function of the intermediate distribution on the spectral line broadening in laser theory. Moreover, we utilize the intermediate distribution to the problem of the stock market return in quantitative finance.
In this paper, we provide an approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions and discuss the application of this approach in some physical problems. Concretely, we construct the equations for these three quantities; this allows us to achieve them by directly solving equations. In order to construct the equations, we introduce shifted local one-loop effective actions, shifted local vacuum energies, and local spectral counting functions. We solve the equations of one-loop effective actions, vacuum energies, and spectral counting functions for free massive scalar fields in $\mathbb{R}^{n}$, scalar fields in three-dimensional hyperbolic space $H_{3}$ (the Euclidean Anti-de Sitter space $AdS_{3}$), in $H_{3}/Z$ (the geometry of the Euclidean BTZ black hole), and in $S^{1}$, and the Higgs model in a $(1+1)$-dimensional finite interval. Moreover, in the above cases, we also calculate the spectra from the counting functions. Besides exact solutions, we give a general discussion on approximate solutions and construct the general series expansion for one-loop effective actions, vacuum energies, and spectral counting functions. In doing this, we encounter divergences. In order to remove the divergences, renormalization procedures are used. In this approach, these three physical quantities are regarded as spectral functions in the spectral problem.
In this paper, we present an exactly solvable phase transition model in which the phase transition is purely statistically derived. The phase transition in this model is a generalized Bose-Einstein condensation. The exact expression of the thermodynamic quantity which can simultaneously describe both gas phase and condensed phase is solved with the help of the homogeneous Riemann-Hilbert problem, so one can judge whether there exists a phase transition and determine the phase transition point mathematically rigorously. A generalized statistics in which the maximum occupation numbers of different quantum states can take on different values is introduced, as a generalization of Bose-Einstein and Fermi-Dirac statistics.
In this paper, we show that spin waves, the elementary excitation of the Heisenberg magnetic system, obey a kind of intermediate statistics with a finite maximum occupation number n. We construct an operator realization for the intermediate statistics obeyed by magnons, the quantized spin waves, and then construct a corresponding intermediate-statistics realization for the angular momentum algebra in terms of the creation and annihilation operators of the magnons. In other words, instead of the Holstein-Primakoff representation, a bosonic representation subject to a constraint on the occupation number, we present an intermediate-statistics representation with no constraints. In this realization, the maximum occupation number is naturally embodied in the commutation relation of creation and annihilation operators, while the Holstein-Primakoff representation is a bosonic operator relation with an additional putting-in-by-hand restriction on the occupation number. We deduce the intermediate-statistics distribution function for magnons. On the basis of these results, we calculate the dispersion relations for ferromagnetic and antiferromagnetic spin waves. The relations between the intermediate statistics that magnons obey and the other two important kinds of intermediate statistics, Haldane-Wu statistics and the fractional statistics of anyons, are discussed. We also compare the spectrum of the intermediate-statistics spin wave with the exact solution of the one-dimensional s = 1/2 Heisenberg model, which is obtained by the Bethe ansatz method. For ferromagnets, we take the contributions from the interaction between magnons (the quartic contribution), the next-to-nearest neighbor interaction, and the dipolar interaction into account for comparison with the experiment.
It is a common impression that by only setting the maximum occupation number to infinity, which is the demand of the indistinguishability of bosons, one can achieve the statistical distribution that bosons obey -- the Bose-Einstein distribution. In this letter, however, we show that only with an infinite maximum occupation number one cannot uniquely achieve the Bose-Einstein distribution, since in the derivation of the Bose-Einstein distribution, the problem of iterated limit is encountered. For achieving the Bose-Einstein distribution, one needs to take both the maximum occupation number and the total number of particles to infinities, and, then, the problem of the order of taking limits arises. Different orders of the limit operations will lead to different statistical distributions. For achieving the Bose-Einstein distribution, besides setting the maximum occupation number, we also need to state the order of the limit operations.
The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda) from a nonuniformly convergent expansion of K(t). We calculate N(lambda) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N(lambda) are introduced.
For a given spectrum {lambda_{n}} of the Laplace operator on a Riemannian manifold, in this paper, we present a relation between the counting function N(lambda), the number of eigenvalues (with multiplicity) smaller than λ, and the heat kernel K(t), defined by K(t)=\sum_{n}e^{-lambda_{n}t}. Moreover, we also give an asymptotic formula for N(λ) and discuss when lambda \to \infty in what cases N(lambda)=K(1/lambda).
In this paper we show that for a non-ideal Bose gas there exists an upper limit on the transition temperature above which Bose-Einstein condensation cannot occur regardless of the pressure applied. Such upper limits for some realistic Bose gases are estimated. This result implies that there may also exist an upper limit on the transition temperature of superconductors.