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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]

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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]

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]

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.

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]