Functional integral via functional equation

Discretization ofp-adic Grassmann-valued -model leads to a hierarchical model with the Hamtilonian given by a nontrivial functional integral over the Grassmann variables. Using renormalization group arguments, we reduce the calculation of this integral to a functional equation. The problem of the convergence of the perturbation expansion of this integral, realized as a small-divisors problem, is investigated.     

Milne's problem for anisotropic scattering medium with specular reflecting boundary

The Milne problem is investigated subject to reflecting boundary conditions. The original version of the problem with vacuum boundary condition is generalized assigning, to the surface x=0, a specular reflection coefficient . Linearly anisotropic case is studied. The integral version of the transport equation solved using trial functions based on Case's eigenvalues and exponential integral function. Solution of the Milne problem is formulated in terms of characteristic parameters such as extrapolated end point, emergent angular distribution and total neutron density. Numerical results for the analytically evaluated parameters are then present. Some of our numerical results are compared with the available published results.     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

The spin- 1 2 singleton dipole

We establish a canonical formulation for a second-order wave equation for a spin-1/2field in a 3+2 de Sitter spacetime. We make variations of the Lagrangian, keeping the surface terms that appear in the process. By demanding that the surface terms be finite, we find that the second-order wave equation must be a singleton dipole equation. The resulting field theory exhibits a very interesting dynamics on the boundary. We study the Hamiltonian of the system and we discover that, after imposing the Lorentz condition, it reduces to an integral over a two-dimensional surface at spatial infinity.     

Fractional-order diffusion-wave equation

The fractional-order diffusion-wave equation is an evolution equation of order (0, 2] which continues to the diffusion equation when 1 and to the wave equation when 2. We prove some properties of its solution and give some examples. We define a new fractional calculus (negative-direction fractional calculus) and study some of its properties. We study the existence, uniqueness, and properties of the solution of the negative-direction fractional diffusion-wave problem.     

Some first passage time problems for shot noise processes

It has recently been shown that the first passage time problem for a certain class of one-dimensional processes that includes shot noise can be formulated in terms of a set of integral equations. These are found by exact enumeration of all possible trajectories. We show that the equations can be found by more direct means for processes described by the evolution equation , wheren(t) is time-localized shot noise.     

Distribution of Singular Values of Random Band Matrices; Marchenko–Pastur Law and More

We consider the limiting spectral distribution of matrices of the form \(\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}\), where X is an \(n\times n\) band matrix of bandwidth \(b_{n}\) and R is a non random band matrix of bandwidth \(b_{n}\). We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For \(R=0\), the integral equation yields the Stieltjes transform of the Marchenko–Pastur law.     

The mechanism of complex Langevin simulations

We discuss conditions under which expectation values computed from a complex Langevin processZ will converge to integral averages over a given complex-valued weight function. The difficulties in proving a general result are pointed out. For complex-valued polynomial actions, it is shown that for a process converging to a strongly stationary process one gets the correct answer for averages of polynomials ifc (k) E(e^ikZ()) satisfies certain conditions. If these conditions are not satisfied, then the stochastic process is not necessarily described by a complex Fokker-Planck equation. The result is illustrated with the exactly solvable complex frequency harmonic oscillator.     

Inverse problem in the complex λ-plane in the case of a coulomb interaction

In view of the large body of experimental information available, there is intense interest in the problem of reconstructing the nuclear potential from data on the scattering of charged particles. The inverse problem of scattering theory is solved in the complex -plane in the case of a nuclear-Coulomb potential by a method similar to that used by Burget et al. A generalized Yukawa potential is used as nuclear potential. It is shown that the nuclear potential can be reconstructed unambiguously from the spectral function given in the -plane. The solution of the problem reduces to one of solving an integral equation of the Gel'fand-Levitan type in which the kernel is expressed in terms of the Jost solutions of the radial Schrödinger equation with a Coulomb potential and in terms of the spectral function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 116–120, September, 1970.The author thanks Professor V. V. Malyarov for useful discussion.     

Random Motions at Finite Speed in Higher Dimensions

We present a general method of studying the transport process , t≥0, in the Euclidean space ℝ ^m , m≥2, based on the analysis of the integral transforms of its distributions. We show that the joint characteristic functions of are connected with each other by a convolution-type recurrent relation. This enables us to prove that the characteristic function (Fourier transform) of in any dimension m≥2 satisfies a convolution-type Volterra integral equation of second kind. We give its solution and obtain the characteristic function of in terms of the multiple convolutions of the kernel of the equation with itself. An explicit form of the Laplace transform of the characteristic function in any dimension is given. The complete solution of the problem of finding the initial conditions for the governing partial differential equations, is given. We also show that, under the standard Kac condition on the speed of the motion and on the intensity of the switching Poisson process, the transition density of the isotropic transport process converges to the transition density of the m-dimensional homogeneous Brownian motion with zero drift and diffusion coefficient depending on the dimension m. We give the conditional characteristic functions of the isotropic transport process in terms of the inverse Laplace transform of the powers of the Gauss hypergeometric function. Some important models of the isotropic transport processes in lower dimensions are considered and some known results are derived as the particular cases of our general model by means of the method developed.     

Numerically stable solution of coupled channel equations: The local reflection matrix

The second order linear Schrödinger equation is transformed to a first order nonlinear differential equation for a quantityp=(iq ^–1 –)/(iq ^–1 +). In a coupled channel problem all quantities occurring in this equation includingq (the WKB wave number) are matrices andp may be calledlocal reflection matrix. This quantity is closely related to the logarithmic derivative of the Schrödinger function but has no singularities in the classically allowed region. In the asymptotic region where the potential is constant the local reflection matrix approaches the physical reflection matrix. In a pure reflection problem (with an infinite potential on one side) this is the fullS-matrix, in a transmission problem (with an activation barrier of finite height) unitarity of theS-matrix can be used to determine most quantities of physical interest fromp. While standard logarithmic derivative methods can become instable for transmission problems the solution with the local reflection matrix is completely stable both for reflection and transmission problem.     

A note on the integral formulation of Einstein’s equations induced on a braneworld

We revisit the integral formulation (or Greens function approach) of Einsteins equations in the context of braneworlds. The integral formulation has been proposed independently by several authors in the past, based on the assumption that it possible to give a reinterpretation of the local metric field in curved spacetimes as an integral expression involving sources and boundary conditions. This allows one to separate source-generated and source-free contributions to the metric field. As a consequence, an exact meaning to Machs Principle can be achieved in the sense that only source-generated (matter fields) contributions to the metric are allowed for; universes which do not obey this condition would be non-Machian. In this paper, we revisit this idea concentrating on a Randall–Sundrum-type model with a non-trivial cosmology on the brane. We argue that the role of the surface term (the source-free contribution) in the braneworld scenario may be quite subtler than in the 4D formulation. This may pose, for instance, an interesting issue to the cosmological constant problem.     

Quaternionic Electron Theory: Dirac's Equation

We perform a one-dimensional complexifiedquaternionic version of the Dirac equation based oni-complex geometry. The problem of the missing complexparameters in quaternionic quantum mechanics withi-complex geometry is overcome by a nicetrick which allows us to avoid the Diracalgebra constraints in formulating our relativisticequation. A brief comparison with other quaternionicformulations is also presented.     

Field theory of the two-dimensional Ising model: Equivalence to the free particle one-dimensional Dirac equation

The Schultz-Mattis-Lieb fermion formulation of the two-dimensional Ising model is simplified by means of long-wavelength approximations which become exact in the critical region. The resulting continuum theory has a Hamiltonian density which is shown to be identical, to within a perfect derivative, to that of free, spinless particles satisfying the one-dimensional Dirac equation. Filling the negative-energy single-particle states of momentumq and mass gives an integral over the single-particle energies -( ²+k²)^1/2. Because varies linearly with the temperature, differentiating twice gives Onsager's logarithmic singularity in the specific heat.Work supported in part by the Office of Naval Research.     

Wave equation for a magnetic monopole

We show that there is room, in the Dirac equation, for a massless monopole. The basic idea is that the Dirac equation admits a second electromagnetic minimal coupling associated to the chiral gauge , which is only valid for a massless particle, but satisfies all the symmetry laws of a monopole. In the problem of the diffusion on a central electric field, we find the Poincaré integral and the Dirac relationeg/=n/2. The latter is deduced as a consequence of the fact (which is shown in this paper) thateg/c is the projection of the total angular momentum on the symmetry axis of the system formed by the monopole and the electric charge. Another important property is that a monopole and an antimonopole have opposite helicities (as for the neutrino), but do not have opposite charges: this precludes a vacuum magnetic polarization which would be analogous to the electric one, but allows us to imagine an aether made up of monopole-antimonopole pairs. The theory is then generalized on the basis of a nonlinear equation which is the most general invariant equation under the chiral gauge law. This equation admits solutions corresponding to massive monopoles, among which there are bradyons (i.e., ordinary massive particles) and tachyons. This equation is shown to be closely related to previous works initiated by Hermann Weyl, on Dirac's theory in the framework of general relativity. In conclusion, it is suggested that massless monopoles are perhaps excited states of the neutrino and that they may be produced in some weak interactions. Consequences on the solar activity are considered.     

Bilinear quantum Monte Carlo: Expectations and energy differences

We propose a bilinear sampling algorithm in the Green's function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schrödinger equations are transformed into two equations whose solution has the form _a(x) t(x, y)_b(y), where _a and _b are the wavefunctions for the two related systems andt(x, y) is a kernel chosen to couplex andy. The Monte Carlo process, with random walkers on the enlarged configuration spacex y, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom.     

Ground state properties of the two-band Anderson-type model in one dimension

We study the ground states of the one-dimensional two-band Anderson type model in both the symmetric and the asymmetric cases. In the symmetric case the analytical expression of the charge-complex distribution function is formally derived, which is then applied to calculate the binding energy of the Kondo state. In the general asymmetric cases the behaviors of localized- and conduction-electron numbers are investigated as functions ofU and other parameters by numerically solving the integral equation. Particularly, for the asymmetric limitU2V ² and _{F a} ( _F the Fermi level, _a the localized level), when a nonintegral localized-electron valence is stabilized implying a valence fluctuation, _F lies in the gap, whereas when it is an integral valence, _F lies in the upper band. The former state is semiconducting and the latter is metallic.     

Propagating fronts and the center manifold theorem

We prove the existence of propagating front solutions for the Swift-Hohenberg equation (SH). Using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations. They describe stationary solutions and front solutions of the partial differential equation (SH). We identify the well-known amplitude equation as the lowest order approximation to the equation of motion on the center manifold.     

A riemann integral approach to Feynman's path integral

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrödinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) practical calculations — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrödinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in practical calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.