Asymptotic completeness and multiparticle structure in field theories

The ideas developed in Part I (ref. [1]) are applied to the recently constructed massive Gross-Neveu model. We define in this case an irreducible kernel satisfying a regularized Bethe-Salpeter equation which is convenient to derive asymptotic completeness in the 2-particle region. As in Part I, the method allows direct graphical definition of general irreducible kernels and is well suited to the analysis of asymptotic completeness and related results in more general energy regions.A large part of the paper is devoted to a new self-contained construction (via phase space expansion) of the Gross-Neveu model. The presentation is somewhat simpler than previous ones, is more complete on some points and is best suited to our purposes.     

Bethe-Salpeter kernel and short distance expansion in the massive Gross-Neveu model

The Bethe-Salpeter kernel is defined (non-perturbatively) for the weakly coupled massive Gross-Neveu model. Its large momentum properties are established. They are used to justify subtracted Bethe-Salpeter equations initially proposed (for ₄ ⁴ ) by K. Symanzik, and in turn to give non-perturbative proofs of the Wilson short distance expansion at first order and of 2-particle asymptotic completeness and related results.     

2-Particle asymptotic completeness and bound states in weakly coupled quantum field theories

Rigorous results on poles of the 2- and 4-point functions, which yield 2-particle asymptotic completeness and give information on the presence or absence of 2-particle bound states and resonances, are presented for weakly coupled even and non-even-field theories with mass gap in space-time dimensiond=2, 3 (and for related hypothetical theories in dimension 4). Methods used are more convenient and more general than those used previously (with more limited results) forP()₂ theories.     

Irreducible kernels and nonperturbative expansions in a theory with purem →m interaction

Recent results on the structure of theS matrix at them-particle threshold (m≧2) in a simplifiedm→m scattering theory with no subchannel interaction are extended to the Green functionF on the basis of off-shell unitarity, through an adequate mathematical extension of some results of Fredholm theory: local two-sheeted or infinite-sheeted structure ofF arounds=(mμ)² depending on the parity of (m?1)(ν?1) (where μ>0 is the mass and ν is the dimension of space-time), off-shell definition of the irreducible kernelU which is the analogue of theK matrix in the two different parity cases (m?1)(ν?1) odd or even, and related local expansion ofF, for (m?1)(ν?1) even, in powers of σ_β ln σ(σ=(mμ)²?s). It is shown that each term in this expansion is the dominant contribution to a Feynman-type integral in which each vertex is a kernelU. The links between the kernelU and Bethe-Salpeter type kernelsG of the theory are exhibited in both parity cases, as also the links between the above expansion ofF and local expansions, in the Bethe-Salpeter type framework, ofF _λ in terms of Feynman-type integrals in which each vertex is a kernelG and which include both dominant and subdominant contributions.     

Irreducible Lie algebra extensions of the Poincaré algebra

We analyse the extensions of the Poincaré algebraP with arbitrary kernels. The main tool is a reduction theorem which generalizes the Hochschild-Serre theorem forn=2. This reduction theorem is proved and used to investigate the structure of the Lie algebras obtained by extension.We look particularly for the irreducible and -irreducible extensions ofP and we classify the types of irreducible extensions with arbitrary kernels.     

Irreducible Lie algebra extensions of the Poincaré algebra

We use cohomology of Lie algebras to analyse the abelian extensions of the Poincaré algebraP. We study particularly the irreducible and truly irreducible extensions: some irreducibility criteria are proved and applied to obtain a classification of types of irreducible abelian extensions ofP. We give a characterization of the minimal essential extensions in terms of truly irreducible extensions.     

RenormalizedG-convolution ofN-point functions in quantum field theory: Convergence in the Euclidean case I

The notion of Feynman amplitude associated with a graphG in perturbative quantum field theory admits a generalized version in which each vertexv ofG is associated with ageneral (non-perturbative)n _v-point functionH ⁿ _v,n _v denoting the number of lines which are incident tov inG. In the case where no ultraviolet divergence occurs, this has been performed directly in complex momentum space through Bros-Lassalle'sG-convolution procedure.In the present work we propose a generalization ofG-convolution which includes the case when the functionsH ⁿ _v arenot integrable at infinity but belong to a suitable class of slowly increasing functions. A finite part of theG-convolution integral is then defined through an algorithm which closely follows Zimmermann's renormalization scheme. In this work, we only treat the case of Euclideanr-momentum configurations.The first part which is presented here contains together with a general introduction, the necessary mathematical material of this work, i.e., Sect. 1 and appendices A and B.The second part, which will be published in a further issue, will contain the Sects. 2, 3 and 4 which are devoted to the statement and to the proof of the main result, i.e., the convergence of the renormalizedG-convolution product.The table of references will be given in both parts.     

The representations of the oscillator group

Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie groupG generated by the canonical variables and a positive definite quadratic hamiltonian are found. These are shown to be in a one to one correspondence with the orbits underG in the dual spaceG to the Lie algebraG ofG, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that ofKirillov andBernat gives all the representations of this group.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research, United States Air Force.     

Conformal gauges and renormalized equations of motion in massless quantum electrodynamics

A formulation of massless QED is studied with a non-singular Lagrangian and conformal invariant equations of motion. It makes use of non-decomposable representations of the conformal groupG and involves two dimensionless scalar fields (in addition to the conventional charged field and electromagnetic potential) but gauge invariant Green functions are shown to coincide with those of standard (massless) QED. Assuming that the (non-elementary) representation ofG for the 5-potential which leaves the equations of motion invariant and leads to the free photon propagator of Johnson-Baker-Adler (JBA) conformal QED remains unaltered by renormalization, we prove that consistency requirements for conformal invariant 2-, 3-, and 4-point Green functions satisfying (renormalized) equations of motion and standard Ward identities lead to either a trivial solution (withe=0) or to a subcanonical dimensiond=1/2 for the charged field.To the memory of Kurt Symanzik     

Self-avoiding walks and trees in spread-out lattices

LetG _R be the graph obtained by joining all sites ofZ ^d which are separated by a distance of at mostR. Let (G _R ) denote the connective constant for counting the self-avoiding walks in this graph. Let (G _R ) denote the coprresponding constant for counting the trees embedded inG _R . Then asR, (G _R ) is asymptotic to the coordination numberk _R ofG _R , while (G _R ) is asymptotic toek _R. However, ifd is 1 or 2, then (G _R )-k _R diverges to –.Dedicated to Oliver Penrose on this occasion of his 65th birthday.     

About one class of representations of the Lie algebra

LetR(G) be the skewsymmetric representation of the algebraG characterized by the following main property: ifGG is some subalgebra ofG (possible noncompact) thenR(G) is integrable and reducible in the direct sum of irreducible representations of subalgebraG.The paper is devoted to the development of the elementary theory of the described representations, culminating in the proof of one version of Schur's lemma.     

Collapse transition and asymptotic scaling behavior of lattice animals: Low-temperature expansion

A low-temperature expansion for the free energy density of lattice animals is derived. Analysis of the series yields a collapse transition temperature ofT _c - 0.54, in close agreement with previous estimates. It is demonstrated that _p,k, the number ofp-particle,p-bond animals, obeys the asymptotic scaling law log _p,k pg(k/p) + o(p). The low-temperature series and numerical data are used to estimate the scaling function.     

Fields,statistics and non-Abelian gauge groups

We examine field theories with a compact groupG of exact internal gauge symmetries so that the superselection sectors are labelled by the inequivalent irreducible representations ofG. A particle in one of these sectors obeys a parastatistics of orderd if and only if the corresponding representation ofG isd-dimensional. The correspondence between representations of the observable algebra and representations ofG extends to a mapping of the intertwining operators for these representations preserving linearity, tensor products and conjugation. Although we assume no explicit commutation property between fields, the commutation relations of fields of the same irreducible tensor character underG at spacelike separations are largely determined by the statistics parameter of the corresponding sector. For fields of conjugate irreducible tensor character the observable part of the commutator (anticommutator) vanishes at spacelike separations if the corresponding sector has para-Bose (para-Fermi) statistics.     

Legendre transforms and r-particle irreducibility in quantum field theory: The formalism for r = 1, 2

We analyze the first and second Legendre transforms Γ^(r) (r = 1, 2) of the generating functional G for connected Green's functions in Euclidean boson field theories. By using Spencer's idea of t-lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that Γ^(r) generates r-irreducible vertex functions, r-irreducible expectations and r-field projectors; moreover, Γ⁽²⁾ generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.     

Lorentz covariance and kinetic charge

There is a one-to-one correspondence between inequivalent covariant displaced Fock representations of the free relativistic field and the 1-cohomology of the Poincaré group with coefficients in the 1-particle space.Representations with positive energy are obtained from cocycles with finite energy which have particle-like properties and are interpreted as condensed states of matter without a sharply defined mass.The 1-cohomology groups ofP ₊ are calculated. These are trivial in 3- or 4-dimensional space-time, or if the mass is non-zero. Non-trivial cocycles for subgroups lead to representations in whichP-invariance is spontaneously broken. We recoverP-invariance in a direct integral representation possessing a gauge group, and a superselection structure labelled by the velocities of the condensed states of matter which are the cocycles determining each irreducible component of the representation. A model in 4-dimensional space-time is constructed.     

Radiation conditions and scattering theory forN-particle Hamiltonians

The correct form of the angular part of radiation conditions is found in scattering problem forN-particle quantum systems. The estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations.     

Highest-weight representations ofsl(2, ℂ) andsl(3, ℂ) via canonical realizations

The infinite-dimensional representations of thesl(n+1, ) Lie algebras (maximal representations) constructed in our previous paper are studied on the two simplest examplesn = 1,2. The sufficient condition for irreducibility of the maximal representations is proved to be also necessary in these cases. It is further shown, that our method allows us to construct other set of infinite-dimensional highest-weight representations ofsl(3, ), so calledmixed representations which are irreducible in some cases when the maximal as well as the standard highest-weight representations (Verma modules) are reducible.Dedicated to the 25th anniversary of the Joint Institute for Nuclear Research.The authors are grateful to Prof. A. A. Kirillov, Dr. A. U. Klimyk, Dr. W. Lassner and Prof. D. P. Zhelobenko for stimulating discussions.     

On the irreducible representations of groups that contain a subgroup of index 2

The objects under consideration are a groupG containing a subgroupN of index 2 and an irreducible multiplier representationU ofG by semiunitary (=unitary or antiunitary) operators on a complex Hilbert space of arbitrary dimension. It is assumed thatU(g) is unitary for allg belonging toN. Then the following assertion is proved. The representation ofN that is obtained by restrictingU toN is either irreducible or an orthogonal sum of two irreducible representations.     

Unquenching the Quark-Antiquark Green’s Function

We propose a nonperturbative resummation scheme for the four-point connected quark-antiquark Greens function G⁴ that shows how the Bethe-Salpeter equation may be unquenched with respect to quark-antiquark loops. This mechanism allows to dynamically account for hadronic meson decays and multiquark structures whilst respecting the underlying symmetries. An initial approximation to the four-point Schwinger-Dyson equation – suitable for phenomenological application – is examined numerically in a couple of aspects. It is demonstrated that this approximation explicitly maintains the correct asymptotic limits and contains the physical resonance structures in the near timelike region in the quark-antiquark channel.This work was performed under grant no. COSY 41139452