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We propose a novel method to determine the structure of symbols for any family of polylogarithmic Feynman integrals. Using the d log-bases and simple formulas for the leading order and next-to-leading contributions to the intersection numbers, we give a streamlined procedure to compute the entries in the coefficient matrices of canonical differential equations, including the symbol letters and the rational coefficients. We also provide a selection rule to decide whether a given matrix element must be zero. The symbol letters are deeply related to the poles of the integrands and also have interesting connections to the geometry of Newton polytopes. Our method can be applied to many cutting-edge multi-loop calculations. The simplicity of our results also hints at the possible underlying structure in perturbative quantum field theories. [ABSTRACT FROM AUTHOR]

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples. [ABSTRACT FROM AUTHOR]

We show in several important cases that the A-hypergeometric system attached to a Feynman diagram in Lee–Pomeransky form, obtained by viewing the coefficients of the integrand as indeterminates, has a normal underlying semigroup. This continues a quest initiated by Klausen and studied by Helmer and Tellander. In the process, we identify several relevant matroids related to the situation and explore their relationships. [ABSTRACT FROM AUTHOR]

We study generating series of Gromov-Witten invariants of E P¹ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are - just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals - complex analytic path integrals involving a product of propagators (equal to the Weierstrass-)-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in ET P1 T of so-called leaky degree. [ABSTRACT FROM AUTHOR]

Abstract. In the classical physics, the quantum mechanics can be formulated in terms of path integrals. A path integral is an operator calculus, as pointed out by Feynman. The gauge approach to path integrals was first presented by Henstock in 1970's, see [5]. This approach was elaborated further in [8]. In this paper, we present a path integral for scalar fields Φ(x, t), where x, t 2 [0,∞), using the gauge approach called Henstock approach in this paper. The equivalent theorem between the integral defined by using Henstock approach and the classical definition is proved. The sequentially equivalent definitions are discussed. [ABSTRACT FROM AUTHOR]

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman-Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock fixes" the defects of the Feynman integral. Possible applications where the potential in the Schr ödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper. [ABSTRACT FROM AUTHOR]

We propose a method for using functional equations to calculate Feynman integrals analytically. We describe the algorithm for solving the functional equations and show that a solution of a functional equation for the Feynman integral is a combination of several integrals with fewer kinematic variables. In some cases, using the functional equations, we can also reduce these integrals to integrals with even fewer variables. Such a stepwise application of the functional equations leads to integrals that can be calculated more simply than the original integral. We apply the proposed method to several one-loop integrals. For the three-point and four-point integrals with massless propagators and an arbitrary space dimension d, we obtain analytic expressions in terms of hypergeometric functions. [ABSTRACT FROM AUTHOR]

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

Abstract We prove that the Feynman quadrics-motive of the massive sunset graph is "generic" not mixed-Tate. Moreover, we explicitly describe its "extra" complexity in terms of a Prym variety. [ABSTRACT FROM AUTHOR]

The paper puts forward a new method of construction of a probabilistic representation of solutions to initial-boundary value problems for a number of evolution equations (in particular, for the Schrödinger equation) in a bounded subdomain of R2 with smooth boundary. Our method is based on the construction of a special extension of the initial function from the domain to the entire plane. For problems with Neumann boundary condition, this method produces a new approach to the construction of a Wiener process "reflected from the boundary," which was first introduced by A. V. Skorokhod. [ABSTRACT FROM AUTHOR]

In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by the fact that their parametric integrand is much larger and more involved. It is, moreover, only implicitly given as the result of certain differential operators applied to the scalar integrand exp ( − Φ Γ ∕ Ψ Γ ) , where Ψ Γ and Φ Γ are the Kirchhoff and Symanzik polynomials of the Feynman graph Γ . In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from Ψ Γ and Φ Γ . In the end, it can be expressed explicitly as a sum over products of new types of graph polynomials which have a combinatoric interpretation via simple cycle subgraphs of Γ . [ABSTRACT FROM AUTHOR]

This paper introduces the notion of an analytic diffusion process. Every process of this type is the limit of some sequence of random walks; however, the limit is understood not in the sense of convergence of measures but in the sense of convergence of generalized functions. Using the analytic diffusion processes it is possible to obtain a probabilistic approximation of solutions to Schr¨odinger evolution equations, whose right-hand side contains the elliptic operator with variable coefficient. [ABSTRACT FROM AUTHOR]

Closed-form expressions are proposed for the Feynman integraloverd=D+mdimensional space with, in the special caseD=1. We show thatcan be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function, as well as generalized hypergeometricand, Hornand Appellfunctions. Several interesting relations are derived between the real and imaginary parts ofand the function. [ABSTRACT FROM PUBLISHER]

We obtain an analogue of probabilistic representation of a solution of an initialboundary value problem for the equation ∂u/∂t+(σ2/2)∂2u/∂x2 +f(x)u = 0, where σ is a complex number. [ABSTRACT FROM AUTHOR]

Abstract: We present a definition for tomographic Feynman path integral as representation for quantum tomograms via Feynman path integral in the phase space. The proposed representation is the potential basis for investigation of Path Integral Monte Carlo numerical methods with quantum tomograms. Tomographic Feynman path integral is a representation of solution of initial problem for evolution equation for tomograms. The perturbation theory for quantum tomograms is constructed. [Copyright &y& Elsevier]

The article focuses on the use of Feynman integrals through the utilization of infinite-dimensional Schrödinger equations. The utilization of a Gaussian measure in a quadratic equation is mentioned. The continuation in the space of operators is explored in terms of Feynman integrals in evolution equations.

Abstract: The fundamental solution of the Schrödinger equation for a free particle is a distribution. Such distribution is approximated by a sequence of smooth functions. For each one of these functions it is defined a complex measure on the space of paths. For certain test functions, there exists the limit of integrals of a test function with respect to the complex measures. We define the Feynman measure of one such function by this limit. [Copyright &y& Elsevier]

Abstract: Several new relationships between hypergeometric functions are found by comparing results for Feynman integrals calculated using different methods. A new expression for the one-loop propagator-type integral with arbitrary masses and arbitrary powers of propagators is derived in terms of only one Appell hypergeometric function . From the comparison of this expression with a previously known one, a new relation between the Appell functions and is found. By comparing this new expression for the case of equal masses with another known result, a new formula for reducing the function with particular arguments to the hypergeometric function is derived. By comparing results for a particular one-loop vertex integral obtained using different methods, a new relationship between functions corresponding to a quadratic transformation of the arguments is established. Another reduction formula for the function is found by analyzing the imaginary part of the two-loop self-energy integral on the cut. An explicit formula relating the function and the Gaussian hypergeometric function whose argument is the ratio of polynomials of degree six is presented. [Copyright &y& Elsevier]

The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI) development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr's quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions - all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving) many-electronic systems. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of the first variation under the Fourier-Feynman transform, and we show that the Wiener integral of the first variation behaves successfully under the Lp-analytic Fourier-Feynman transform for 1≤p≤2 for cylinder functions of the form [image omitted] where f: Rn→C∈Lp (Rn), 1≤p≤2 and {h1, h2, ..., hn} is the orthonormal class of elements in H and the Lp-analytic Fourier-Feynman transform of the first variation can be perfectly expressed as the limit of Wiener integrals of the first variation. [ABSTRACT FROM AUTHOR]

In this paper, we prove a theorem which is the inversion of Chernoff’s theorem. As a consequence of this result, we obtain conditions for the validity of the Chernoff and Trotter product formulas. [ABSTRACT FROM AUTHOR]

Abstract: A Maple package for computing Gröbner bases of linear difference ideals is described. The underlying algorithm is based on Janet and Janet-like monomial divisions associated with finite difference operators. The package can be used, for example, for automatic generation of difference schemes for linear partial differential equations and for reduction of multiloop Feynman integrals. These two possible applications are illustrated by simple examples of the Laplace equation and a one-loop scalar integral of propagator type. [Copyright &y& Elsevier]

In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors. [ABSTRACT FROM AUTHOR]

Huffman, Park and Skoug introduced a generalized Fourier–Feynman transform (GFFT) and a generalized convolution product (GCP) and they obtained the relationships between the GFFT and GCP for functionals in the Banach algebra ?? introduced by Cameron and Storvick. In this paper, we investigate various relationships among the GFFT, GCP and generalized first variation for functionals in ??. [ABSTRACT FROM AUTHOR]

Problems arising in renormalization of theories with nontrivial internal symmetry are reviewed. Different approaches to renormalization are discussed. A new method which provides automatically gauge invariance of renormalized theory independently of regularization used is proposed. [ABSTRACT FROM AUTHOR]