Functional Closure of Schwinger-Dyson Equations in Quantum Electrodynamics: 1. Generation of Connected and One-Particle Irreducible Feynman Diagrams

Using functional derivatives with respect to free propagators and interactions we derive a closed set of Schwinger-Dyson equations in quantum electrodynamics. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the n-point functions and the vacuum energy together with their correct weights.     

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.     

Theπ π πNÑ-five-point function

Choosing a suitable covariant decomposition of theπ π πNÑ-five-point function the crossing relations are derived and a complete reduction of them is given. The implications of the known factorization properties at the one-particle singularities of the invariant functions on their assumed Regge behaviour is investigated and a set of asymptotic relations between them is found. A mass degenerate parity doublet of the nucleon can remove the relations connected with the nucleon poles.     

The Galilean and translationally invariant operator algebra describing nuclei

The deviation of the commutation relations of the Galilean and translationally invariant field operators (given in part I) from the exact Fermi commutation relations is investigated. For this purpose the invariant field operators are integrated with one-particle functions describing a compactn-particle system of nucleons and its low excitations. With this assumption an approximate expression for the (subtractive) deviation of both commutation rules from one another is given. It is carefully estimated that the exact deviation, taken in the operator norm, is of ordern ^?2/3. It is shown that the degree of compactness of nuclear matter in the phase space plays an essential role in the determination of this estimate.     

Inequalities for Ising models and field theories which obey the Lee-Yang Theorem

A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, then-point Schwinger functions ofeven φ⁴ models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ⁴ field theory with a moment generating function which is entire of exponential order at most two. It is also noted that ifany (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.     

A dispersion analysis of pion photoproduction from nucleons. I

The paper gives an account of numerical results forl=0 andl=1 multipole amplitudes of pion photoproduction. They have been obtained by solving integral equations which originate from fixed-angle dispersion relations that have been deduced in an earlier paper. In this Part I outlined are some difficulties usually encountered in course of solving the problem of pion photoproduction by dispersion techniques, and proposed is an approach which introduces arbitrary parameters — cutoffs in the integral equation for transition amplitudes. Numerical results show some implications of this philosophy on the rôle played by the terms representing physical singularities and terms representing left-hand singularities controlled by one-particle (Born) contributions.     

Tensor representations of conformal algebra and conformally covariant operator product expansion

A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O(4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O(4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n-point functions from causality are described.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

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.     

Micromagnetic simulations of spinel ferrite particles

This paper presents the results of simulations of the magnetization field ac response (at 2-12 GHz) of various submicron ferrite particles (cylindrical dots). The ferrites in the present simulations have the spinel structure, expressed here by M₁−_nZn_nFe₂O₄ (where M stands for a divalent metal), and the parameters chosen were the following: (a) for n=0: M={Fe, Mn, Co, Ni, Mg, Cu }; (b) for n=0.1: M = {Fe, Mg} (mixed ferrites). These runs represent full 3D micromagnetic (one-particle) ferrite simulations. We find evidences of confined spin waves in all simulations, as well as a complex behavior nearby the main resonance peak in the case of the M = {Mg, Cu} ferrites. A comparison of the n=0 and n=0.1 cases for fixed M reveals a significant change in the spectra in M = Mg ferrites, but only a minor change in the M=Fe case. An additional larger scale simulation of a 3 by 3 particle array was performed using similar conditions of the Fe₃O₄ (magnetite; n=0, M = Fe) one-particle simulation. We find that the main resonance peak of the Fe₃O₄ one-particle simulation is disfigured in the corresponding 3 by 3 particle simulation, indicating the extent to which dipolar interactions are able to affect the main resonance peak in that magnetic compound.     

Momentum scale expansion of sharp cutoff flow equations

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organized by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations - the O(p^M) approximations - result from discarding from these parts, all terms of higher than the M^th degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond O(p⁰).     

A new method to solve the non-perturbative renormalization group equations

We propose a method to solve the non-perturbative renormalization group equations for the n-point functions. In leading order, it consists in solving the equations obtained by closing the infinite hierarchy of equations for the n-point functions. This is achieved: (i) by exploiting the decoupling of modes and the analyticity of the n  -point functions at small momenta: this allows us to neglect some momentum dependence of the vertices entering the flow equations; (ii) by relating vertices at zero momenta to derivatives of lower order vertices with respect to a constant background field. Although the approximation is not controlled by a small parameter, its accuracy can be systematically improved. When it is applied to the O(N)$O(N)$ model, its leading order is exact in the large-N limit; in this case, one recovers known results in a simple and direct way, i.e., without introducing an auxiliary field.     

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.     

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.     

Greens functions of 2-dimensional Yang-Mills theories on nonorientable surfaces

By using the path integral method, we calculate the Green functions of the field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge.We show that the non-gauge invariant correlators consist of a free part and an almost x-independent part. We also show that the gauge invariant n-point functions are those corresponding to the free part, as in the case of orientable surfaces.     

Virasoro Correlation Functions on Hyperelliptic Riemann Surfaces

N-point functions of holomorphic fields in conformal field theories can be calculated by methods from algebraic geometry. We establish explicit formulas for the 2-point function of the Virasoro field on hyperelliptic Riemann surfaces of genus g ≥  1. Virasoro N-point functions for higher N are obtained inductively, and we show that they have a nice graph representation. We discuss the 3-point function with application to the (2,5) minimal model.     

Operator approach to conformal invariant quantum field theories and related problems

Conformal quantum field theory is analysed from a global point of view. The use of the covering transformations leads to a global decomposition theory in which the basic building blocks are nonlocal quantum fields which live on each light cone separately. As an application of the new formalism arbitrary order-disorder mixed n-point functions of the Ising field theory are explicitly calculated.     

Gradient properties of the renormalisation group equations in multicomponent systems

The existence of a metric, which enables the renormalisation group β functions of a multicomponent field theory to be written as a gradient, has very important implications for the asymptotic behavior of the renormalisation group equations. It is shown that a very simple metric exists in a field theory with n-component Bose fields and arbitrary φ⁴ interaction, when the β functions are calculated perturbatively up to and including the 2-loop diagrams. This same metric is shown to work for all irreducible diagrams, but it must and can be modified to accommodate reducible 3-loop contributions. The prospects and outlook of this aspect of the renormalisation group are discussed.     

On a possible Wess-Zumino term in the heterotic string

We examine the low-energy limits of the one-particle irreducible one-loop amplitudes in the heterotic superstring for five gravitons, for three gravitons and two gauge bosons, and for one graviton and four gauge bosons. In agreement with general arguments, explicit calculation shows that there is a Wess-Zumino term in the low-energy effective action for D = 10 supergravity.