We present a recursive procedure, which is based on the small time expansion of the propagator, in order to generate a semi-classical expansion of the \textit{quantum action} for a quantum mechanical potential in arbitrary dimensions. In the method we use the spectral information emerges from the singularities of the propagator on the complex $t$ plane, which are handled by the $i\ve$ prescription and basic complex analysis. This feature allows for generalization to higher dimensions. We illustrate the procedure by providing simple examples in non-relativistic quantum mechanics.
We analyze the partition function of two dimensional Yang-Mills theory on a family of surfaces of infinite genus. These surfaces have a recursive structure, which was used by one of us to compute the partition function that results in a generalized Migdal formula. In this paper we study the `small area' (weak coupling) expansion of the partition function, by exploiting the fact that the generalized Migdal formula is analytic in the (complexification of the) Euler characteristic. The structure of the perturbative part of the weak coupling expansion suggests that the moduli space of flat connect…
I derive a formula for the coupling-constant derivative of the coefficients of the operator product expansion (Wilson OPE coefficients) in an arbitrary curved space, as the natural extension of the quantum action principle. Expanding the coefficients themselves in powers of the coupling constants, this formula allows to compute them recursively to arbitrary order. As input, only the OPE coefficients in the free theory are needed, which are easily obtained using Wick's theorem. I illustrate the method by computing the OPE of two scalars $ϕ$ in hyperbolic space (Euclidean Anti-de Sitter space) u…
In this and subsequent paper arXiv:1011.5185 we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not yet been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniqu…
In a recent work [1] we consider the topological expansion for the non-mixed observables (including the free energy) for the formal Cauchy matrix model. The only restriction in [1] was the fact that all the branch points have to be simple. This excludes a very interesting situation not encountered in the literature before, namely the case in which two branch points merge in such a way that no cycle is pinched. In this work we focus on this situation and derive new formulas that apply to non-simple and non-singular branch-points.
In this paper, we investigate the expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes. First, we propose two types of recursive expansions of tree level EYM amplitudes with an arbitrary number of gluons, gravitons and traces by those amplitudes with fewer traces or/and gravitons. Then we give many support evidence, including proofs using the Cachazo-He-Yuan (CHY) formula and Britto-Cachazo-Feng-Witten (BCFW) recursive relation. As a byproduct, two types of generalized BCJ relations for multitrace EYM are further proposed, which will be useful in the BCFW proof. After one ap…
We further elaborate on the general construction proposed in arXiv:1812.03369, which connects, via tree-level double copy, massless string amplitudes with color-ordered QFT amplitudes that are given by Cachazo-He-Yuan formulas. The current paper serves as a detailed study of the integration-by-parts procedure for any tree-level massless string correlator outlined in the previous letter. We present two new results in the context of heterotic and (compactified) bosonic string theories. First, we find a new recursive expansion of any multitrace mixed correlator in these theories into a logarithmi…
A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \bar W = e^{ϕ/\hbar}e^{\bar X/\hbar}, and the auxiliary operators X,\bar X and the function ϕare assumed to have \hbar-expansions X = X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and ϕ= …
The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \hbar and reduces to the dispersionless KP hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients X_n of an \hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). Given th…
We derive a novel formula for the derivative of operator product expansion (OPE) coefficients with respect to a coupling constant. The formula only involves the OPE coefficients themselves, and no further input, and is in this sense self-consistent. Furthermore, unlike other formal identities of this general nature in quantum field theory (such as the formal expression for the Lagrangian perturbation of a correlation function), our formula is completely well-defined from the start, i.e. requires no further UV-renormalization. This feature is a result of a cancelation of UV-divergences between …
We present a diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of $\hbar $. Considering $\hbar $ as a running parameter, a differential equation connecting two diagonalization processes for two very close values of $\hbar $ is derived. The integration of this differential equation allows the recursive determination of the series expansion in powers of $\hbar $ for the diagonalized Hamiltonian. This approach results in effective Hamiltonians with Berry phase corrections of higher order in $\hbar $, and deepens previous works on the semiclassical …
Recursive relations from diffeomorphism in the Randall-Sundrum model
College of Physics and Communication Electronics, Jiangxi Normal University（江西师范大学物理与通信电子学院）；Laboratoire de Physique Theorique et Hautes Energies, UMR 7589, Sorbonne Université & CNRS（索邦大学与法国国家科学研究中心理论物理与高能物理联合实验室）
The quantum correlations of scalar fields are examined as a power series in derivatives. Recursive algebraic equations are derived and determine the amplitudes; all loop integrations are performed. This recursion contains the same information as the usual loop expansion. The approach is pragmatic and generalizable to most quantum field theories.
An extensive number of numerical computations of energy 1/$N$ series using a recursive Taylor series method are presented in this paper. The series are computed to a high order of approximation and their behaviour on increasing the order of approximation is examined.
We present an alternative technique for evaluating multiloop Feynman diagrams, using the integration by fractional expansion method. Here we consider generic diagrams that contain propagators with radiative corrections which topologically correspond to recursive constructions of bubble type diagrams. The main idea is to reduce these subgraphs, replacing them by their equivalent multiregion expansion. One of the main advantages of this integration technique is that it allows to reduce massive cases with the same degree of difficulty as in the massless case.
We present a method, based on loop equations, to compute recursively all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model. We illustrate the method by computing the first subleading term, i.e. the free energy of a statistical physics model on a discretized torus.
We derive a recursive formula for the alpha'-expansion of superstring tree amplitudes involving any number N of massless open string states. String corrections to Yang-Mills field theory are shown to enter through the Drinfeld associator, a generating series for multiple zeta values. Our results apply for any number of spacetime dimensions or supersymmetries and chosen helicity configurations.
Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from th…
We present a recursive method to calculate the $α'$-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of $Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order $α'^7$ is made available on the website http://repo.or.c…
We present a new, recursive approach to stochastic inflation which is self-consistent and solves multiple problems which plagued a certain number of previous studies, in particular in realistic contexts where the background spacetime is taken to be dynamical, where there is more than one field present, especially with a mass hierarchy, or where the role played by back-reaction is suspected to be important. We first review the formalism of stochastic inflation as it is usually heuristically presented, that is, deriving the Langevin equations from the field equations of motion, and summarize pre…
In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration …
