The random-walk representation of classical spin systems and correlation inequalities

We use the random-walk representation to prove the first few of a new family of correlation inequalities for ferromagnetic ?⁴ lattice models. These inequalities state that the finite partial sums of the propagator-resummed perturbation expansion for the 4-point function form an alternating set of rigorous upper and lower bounds for the exact 4-point function. Generalizations to 2n-point functions are also given. A simple construction of the continuum ? _d ⁴ quantum field theory (d<4), based on these inequalities, is described in a companion paper.     

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

Different perturbation theory treatments of the Ginzburg‐Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ⁴ model with O(n) symmetry. As a result, equations for calculation of the two‐point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.     

Torus n-Poi nt Fu nctio ns for $${\mathbb{R}}$$ -graded Vertex Operator Superalgebras a nd Co nti nuous Fermio n Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay’s trisecant identity for elliptic functions. Partial support provided by NSF, NSA and the Committee on Research, University of California, Santa Cruz. Supported by a Science Foundation Ireland Frontiers of Research Grant, and by Max-Planck Institut für Mathematik, Bonn.     

Analyticity properties and many-particle structure in general quantum field theory

In the framework of L.S.Z. field theory in the case of a single massive scalar field, the two-particle irreducible parts of then-point functions (in any single channel and for arbitraryn) are defined as the solutions of a system of integral equations suggested by the perturbative framework. These solutions enjoy the analytic and algebraic properties of generaln-point functions (up to possible polar singularities of generalized C.D.D. type). Morever it is shown that the completeness of asymptotic states in the two-particle spectral region is equivalent to the analyticity of the two-particle irreduciblen-point functions in the corresponding regions of complex momentum space.     

Correlation functions in the Itzykson-Zuber model

Then-point function for the integral over unitary matrices with Itzykson-Zuber measure is reduced to the integral over the Gelfand-Tzetlin table; the integrand (for genericn) is given by linear exponential times the rational function. Forn=2 and in some cases forn>2 later in fast is the polynomial and this allows to give an explicit and simple expression for all 2-point and a set ofn-point functions. For the most generaln-point function a simple linear differential equation is constructed.     

Catoptric tadpoles

Fermionic string perturbation theory is known to suffer from an ambiguity in the form of a total derivative in the moduli space. For a class of backgrounds (including R¹⁰, orbifolds and theories with no U(1) factors in gauge group) we show that these ambiguities for the partition function of heterotic string theory at any genus are proportional to massless physical tadpoles in the theory at lower genera and hence vanish in stable vacua. We also find that in R¹⁰ the cosmological constant at a given genus is proportional to the cosmological constant at lower genera. This enables us to give an inductive argument for the vanishing of the cosmological constant in R¹⁰ to all orders in string perturbation theory. We also address the ambiguity and finiteness of n-point functions. Our results indicate that in R¹⁰ the ambiguity can be absorbed by a renormalization of the string coupling constant and the string tension. The expected sources of divergence in the n-point function in arbitrary tachyon-free backgrounds, besides the usual infrared divergences for d ≤ 4, are shown to be proportional to tadpoles of physical massless fields. For type II strings in arbitrary backgrounds, we show by explicit calculations that the ambiguity vanishes at g = 2.     

A theorem concerning the positive metric

It is proved that if then-point correlation functions of a system vanish for alln>N then they vanish for alln>2. The theorem is valid for a wide variety of formalisms and an explicit proof is given for a Bose system with the canonical commutation relations; a proof is sketched out for a relativistic field theory of the Wightman type. The essential property used in the proof is the positive definite metric.     

Higher-order approximations for the particle-particle propagator

A new type of approximations for many-body Green's functions proposed recently is applied to the particle-particle (pp) propagator for anN-particle fermion system. The new approach which is referred to as the algebraic diagrammatic construction (ADC) is based on an exact resummation of the perturbation series for the pp-propagator in terms of a simple algebraic form introducing energy-independent effective interaction matrix elements and transition amplitudes. These effective quantities are represented by perturbation expansions and can be determined consistently through a given ordern of perturbation theory by comparing the algebraic form with the diagrammatic perturbation series of the pp-propagator through ordern. By this procedure one obtaines a systematic set of approximation schemes (ADC(n)) that represent infinite partial summations for the pp-propagator being complete throughnth order of perturbation theory. The explicit ADC equations forn=1 and 2 are presented and discussed. Comparison is made with the particle-particle random phase approximation (RPA). It is demonstrated that the second-order ADC scheme constitutes an essential step beyond the RPA which is consistent only through first order.     

Statistical Properties of the Burgers Equation with Brownian Initial Velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its n-point correlations. In the same limit, we derive the n-point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.     

Currents,stress tensor and generalized unitarity in conformal invariant quantum field theory

Matrix elements of internal symmetry currents and energy momentum density tensor are constructed in Migdal Polyakov conformal invariant bootstrap field theory. Their 3-point functions satisfy Bethe Salpeter equations which determine any free coefficients that may still occur in the conformal invariant Ansatz. Ward identities are verified for alln-point functions. They imply correct equal time current commutation relations. A proof of generalized unitarity is also given. Various equivalent forms of the propagator bootstrap are discussed. Our algebraic techniques also yield an eigenvalue equation for first order correction to the exactly conformal invariant theory, assuming the latter is Gell-Mann Low large momentum asymptote of a renormalizable finite mass theory.     

Passive Advection and the Degenerate Elliptic Operators <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

 We prove estimates for the stationary state n-point functions at zero molecular diffusivity in the Kraichnan model [13]. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators M _n that occur in the Hopf equations for the n-point functions. Received: 25 August 2001 / Accepted: 30 September 2002 Published online: 20 January 2003 Communicated by A. Kupiainen     

Travelling Wave Solution of Korteweg-de Vries-Burger's Equation

An attempt has been made to obtain exact analytical travelling wave solution of Korteweg-de Vries-Burger's (KdVB) equation by the so-called tanh-method. This equation can be derived for dust ion acoustic shocks by using reduction perturbation method. It is found that an exact solution of the KdVB equation is obtained by tanh-method, provided the parameters involved satisfy a constraint relation. However a special exact analytical solution can be obtained where no such restriction is necessary. This solution has the structure of a shock wave. Numerical solution is also obtained for travelling wave with or without the assumption of the constraint relation. We have also found a singular solution in terms of cosech and coth functions.     

The Non-Perturbative Approach to the Infrared Problem in QCD at Finite Temperature

A non-perturbative approach is developed for investigation of the infrared problem in QCD at T ≠ 0 in the ghost-free axial gauge. The problem is solved by using a 3-dimensional theory within the exact Slavnov-Taylor identities and Schwinger-Dyson equations. The system of two non-linear integral equations for the structural functions of the gluon polarization tensor is obtained whose solution determines the infrared behavior of the temperature Green functions in the 4-dimensional QCD. The simplest solution of these equations which is the same as the first term of the perturbation expansion shows the presence of singularities in the gluon propagator at momenta p ∼ g²T, that cannot be eliminated by any choice of the gauge. The infrared instability of QCD at T ≠ 0 caused by these singularities is discussed.     

Skeleton inequalities and the asymptotic nature of perturbation theory for Φ4-theories in two and three dimensions

We use the polymer representation of ⁴-quantum field theories to prove an infinite family of correlation inequalities, called skeleton inequalities, for the 2n-point Green's functions. As an application, we show that they imply that Feynman perturbation theory is asymptotic in less than four dimensions.     

Some remarks about the localization of states in a quantum field theory

For the case of a field theory with a nuclear space of test functions (for instance, the space of strongly decreasing test functions) compact sets of states are constructed; these correspond to sets of localized states. Only such states are considered which are elements of a fixed subspace of the entire Hilbert space. This subspace belongs to them-point functions of order less than a certain fixed 2n.Supported in part by the Air Force Office of Scientific Research under Grant AF 500–64On leave of absence from the Max Planck-Institut für Physik und Astrophysik, München.     

Some analyticity properties arising from asymptotic completeness in quantum field theory

It is well-known that two-particle unitarity of the S-Matrix in quantum field theory implies (modulo regularity assumptions) second sheet analyticity for the 2-body scattering amplitude. Here this is first used to prove off-mass-shell analyticity for the 4 point function in a complex neighborhood of any real mass-shell point under the 3-particle threshold. Then this is applied to the study of the 5 and 6-point functions near the real mass-shell of 2→3 and 3→3 processes below the 4-particle threshold: the results are those suggested by perturbation theory apart from the 3-particle cut and away from some submanifolds. The advantage of this method, which could presumably be extended to the exploitation ofn-particle unitarity, is that the regularity assumptions only refer to the physical scattering amplitudes.     

重夸克偶素的高阶变分-积分微扰修正

利用基于积分方程的改进的变分微扰方法(变分-积分微扰法)求解重夸克偶素激发态.设置含有变分参数的母哈密顿量作为零级哈密顿量,选择该母哈密顿系统的精确解作为试探波函数,得到了只有几项的高阶修正波函数和高阶能量修正.该计算结果更接近精确能量值,修正波函数的有界性和收敛性也得到了证明.结果表明,这种方法不仅可提高计算结果的精度,而且可确保微扰级数解的收敛性. 关键词： 重夸克偶素 变分-积分微扰法 高阶能量修正     

Analyticity properties and many-particle structure in general quantum field theory

The extraction of one-particle singularities from then-point functions is performed in the framework of L.S.Z. field theory in the case of a single massive scalar field. It is proved that the “one-particle irreducible” functions thus obtained enjoy the analytic and algebraic primitive structure of generaln-point functions (up to a finite number of generalized C.D.D. singularities). Finally under an additional technical assumption, it is shown that the Glaser-Lehmann-Zimmermann relations stating the completeness of asymptotic states yield similar relations satisfied in any given channel by the corresponding one-particle irreducible functions.     

Index summation in real time statistical field theory

We have written a Mathematica program that calculates the integrand corresponding to any amplitude in the closed-time-path formulation of real time statistical field theory. The program is designed so that it can be used by someone with no previous experience with Mathematica. It performs the contractions over the tensor indices that appear in real time statistical field theory and gives the result in the 1-2, Keldysh or RA basis. The program treats all fields as scalars, but the result can be applied to theories with dirac and lorentz structure by making simple adjustments. As an example, we have used the program to calculate the ward identity for the QED 3-point function, the QED 4-point function for two photons and two fermions, and the QED 5-point function for three photons and two fermions. In real time statistical field theory, there are seven 3-point functions, 15 4-point functions and 31 5-point functions. We produce a table that gives the results for all of these functions. In addition, we give a simple general expression for the KMS conditions between n-point green functions and vertex functions, in both the Keldysh and RA bases. PACS 11.10.Wx; 11.15.-q