Finite energy solutions for interacting Yang-Mills and Dirac fields on Minkowski space

The Lagrangian for a SU(2) Yang-Mills field interacting with a massless isospin 1/2 Dirac field is conformally invariant. Finite energy solutions are obtained by a conformal mapping of Minkowski space onto the compact manifoldS ¹×S³ with pseudo-Riemannian metric. They are symmetric with respect to the isometric group SO(2)×SO(4) ofS ¹×S³.     

Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories

The operator content of unitary conformally invariant theories with c<1 is further analysed by deriving the spectrum of the transfer matrix for finite width strips and a variety of boundary conditions: antiperiodic, cyclic, twisted, free, fixed and a mixture of the last two. Complete results are obtained for the Ising model and for the three-state Potts model, as illustrations of the method. They demonstrate how the internal symmetries of these theories are tied in with their conformal properties.     

Nonlocal theory of the electromagnetic and weak interactions with W-boson

The perturbation theory of the electromagnetic and weak interactions is considered in the framework of nonlocal theory. A hypothesis is proposed that the photon and neutrino fields are connected with the charged local fields of the electrons, muons, and W bosons in the nonlocal way.The definite intermediate regularization procedure is introduced that the S matrix is finite, unitary, causal, gauge invariant in perturbation theory when regularization is moved off. The interaction Lagrangian contains no infinite counter terms and the S matrix is finite without any infinite renormalizations.     

Relativistic Effects in Neutron�CDeuteron Elastic Scattering and Breakup

We solved the Faddeev equation in a Poincaré invariant model of the three-nucleon system. Two-body interactions are generated so that when they are added to the two-nucleon invariant mass operator (rest energy) the two-nucleon S matrix is identical to the experimental S matrix modeled with a given nucleon?Cnucleon interaction. Cluster properties of the three-nucleon S-matrix determine how these two-nucleon interactions are embedded in the three-nucleon mass operator. Differences in the predictions of the relativistic and corresponding non-relativistic models for elastic and breakup processes are investigated. Of special interest are effects of relativity on the elastic scattering angular distribution and total cross sections, the lowering of the A _y maximum in elastic nucleon-deuteron (Nd) scattering below ??25?MeV caused by the Wigner spin rotations and the significant changes of the breakup cross sections in certain regions of the phase-space.     

Bosonization and the O(3) σ-model

The rules of vector bosonization with respect to the O(4) σ-model with a Wess—Zumino term, and the O(3) σ-model with an instanton term are derived. The critical points at which the theories become conformally invariant are found.     

Conformal flatness, non-Abelian Kaluza-Klein reduction and quaternions

The non-Abelian Kaluza-Klein reduction of conformally flat spaces is considered for arbitrary dimensions and signatures. The corresponding equations are particularly elegant when the internal space supports a global Killing parallelization. Assuming this imposes the generalized ‘spacetime’ to be maximally symmetric with holonomy in the unitary quaternionic group Sp(d/4). Recalling an analogous result for the complex case, we conclude that all special manifolds with constant properly ‘holonomy-related’ sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza-Klein spaces.     

Anomaly-induced effective actions in even dimensions and reliability of s-wave approximation

The reduction of 4d matter-gravity theory to S₂, H₂ or R₂ leads to effective 2d dilatonic gravity with dilaton coupled matter. Spinors give the exceptional example of the theory which is conformally invariant in 4d as well as in 2d, after reduction. We find 4d and 2d conformal anomaly induced effective action (EA) for Majorana spinor. It is expected for some time that s-wave EA (i.e. the one for dilaton coupled 2d matter) is some (s-wave) approximation to 4d EA. We compare such 2d and 4d spinor EAs on the same gravitational background and argue that s-wave EA indeed qualitatively corresponds to no higher derivatives approximation for 4d EA.     

Operator products in 2-dimensional critical theories

A noteworthy feature of certain conformally invariant 2-dimensional theories, such as the Ising and 3-state Potts models at the critical point, is the existence of “degenerate primary fields” associated with nullvectors of the Virasoro algebra. Such fields are endowed with a remarkably simple multiplication table under the operator product expansion, known as the fusion rules. In addition, correlation functions made up of these fields satisfy a system of linear homogeneous partial differential equations. We show here that these two properties are intimately related: for any n-point function, the number of conformally invariant solutions to the system of equations equals the number of times that the identity operator appears in the fusion of all n fields in the correlator. This theorem permits the calculation of some apparently intractable correlation functions. Finally, we generalize these ideas to the Neveu-Schwarz sector of superconformal theories.     

Determination of all Petrov type-N space-times on which the conformally invariant scalar wave equation satisfies Huygen's principle

It is shown that the conformally invariant wave equation on a Petrov type-N space-time satisfies Huygens' principle if and only if the space-time is conformally related to a plane wave space-time.     

Separation of variables and symmetry operators for the conformally invariant Klein-Gordon equation on curved spacetime

The separability of the conformally invariant Klein-Gordon equation and the Laplace-Beltrami equation are contrasted on two classes of Petrov type D curved spacetimes, showing that neither implies the other. The second-order symmetry operators corresponding to the separation of variables of the conformally invariant Klein-Gordon equation are constructed in both classes and the most general second-order symmetry operator for the conformally invariant Klein-Gordon operator on a general curved background is characterized tensorially in terms of a valence two-symmetric tensor satisfying the conformal Killing tensor equation and further constraints.     

Hyperspherical harmonics and the nuclear four-body problem

A method proposed earlier by Aguilera, Moshinsky, and Kramer, for adapting a system of translationally invariant four-particle harmonic oscillator functions to the symmetry of the permutation group S(4), is applied to hyperspherical harmonic functions depending on three relative vectors. Except for a few cases in which diagonalization of matrices is required, the method gives closed formulas for orthonormal sets of harmonic functions with good permutational symmetry. The matrix elements of S(4) permutations with respect to the harmonic functions are obtained.     

Projectively and Conformally Invariant Star-Products

We consider the Poisson algebra S(M) of smooth functions on T ^* M which are fiberwise polynomial. In the case where M is locally projectively (resp. conformally) flat, we seek the star-products on S(M) which are SL(n+1,) (resp. SO(p+1,q+1))-invariant. We prove the existence of such star-products using the projectively (resp. conformally) equivariant quantization, then prove their uniqueness, and study their main properties. We finally give an explicit formula for the canonical projectively invariant star-product.     

Gauge invariance versus masslessness in de Sitter spaces

The connection between gauge invariance, masslessness and null cone propagation is a flat space property which does not persist even in constant curvature geometries. In particular, we show that both the gauge invariant spin $\text{3}\text{2}$ and 2 fields in anti-de Sitter space have support inside the cone, whereas where are conformally invariant, but gauge variant, models which do propagate on the light cone. The Maxwell field in constant curvature spaces of dimension other than four also does not have null cone propagation; again there is a conformally invariant model which does.     

Conformally invariant σ models in 2n dimensions

The conformally invariant σ-model of Felsager and Leinaas in four dimensions (n=2) is generalised to 2n dimensions, and the unit-charge instantons are presented. This hierarchy of σ-models starts with the O(3) σ-model (n=1). The relationship of these models with the generalised YM systems for even n=2p is pointed out.     

Conformal invariance and the six-dimensional formalism

Conformal invariance is discussed assuming the equations are well defined in arbitrary coordinate systems. This assumption leads to some constraints on scale dimensions of terms, and constraints on the introduction of ‘conformally invariant massive equations’. The six-dimensional formalism is then discussed, and is generalized to project to all conformally flat spaces. Finally the imbedding of Minkowski space equations is studied.SO(4, 2) breaking is seen to enter due to the presence of a non-invariant scalar field, and a non-invariant vector field. The theorem relating invariance of the six-space equations underSO(4, 2) to the invariance of their corresponding four-space equations under the conformal group is carefully stated and proved.     

Theory of the conformally invariant mass

By interpreting the conformal transformations as space-time-dependent change of units and introducing the concept of the conformally invariant mass and charge, we develop new conformally invariant Maxwell equations with source terms and equations of motion for massive particles. Although the usual equations of motion with mass terms break the conformal symmetry, it is shown that the Minkowski space is not the most general framework to describe physical processes and there exists a wider consistent dynamics in which conformal invariance is exact. New results also include the general transformation laws of the electromagnetic fields, of currents and force densities. The theory leads naturally to an affine connection and to the 21-parameter inhomogeneous conformal group, ISO(4, 2).     

Solutions of Yang's EuclideanR-gauge equations and self-duality

Under some assumptions and transformations of variables, Yang's equations forR-gauge fields on Euclidean space lead to conformally invariant equations permitting one to obtain infinitely many other solutions from any solution of these conformally invariant equations. These conformally invariant equations closely resemble the mathematically interesting generalized Lund-Regge equations. Some exact solutions of these conformally in variant equations are obtained. Except for some singular situations, these solutions are self-dual.     

On Conformally Invariant CLE Explorations

We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles with simple loops introduced in earlier papers by Sheffield, and by Sheffield and Werner. One outcome is a conformally invariant way to measure a distance of a CLE₄ loop to the boundary “within” the CLE₄, when one identifies all points of each loop.     

Realizations of the exceptional modular invariant A(1)1 partition functions

We show that the E₆ and E₈ modular invariant combinations of A⁽¹⁾₁ characters in the classification of Cappelli, Itzykson and Zuber can be realized as partition functions of k=1 conformally invariant WZW models on the group manifolds of Sp(4) and G₂, respectively. Together with the D₄ combination, which is known to be realized by the WZW model on SU(3), these are the only such cases where the SU(2) local symmetry extends to a larger one. The E₇ combination is briefly discussed.     

Conformally flat spaces and gravitation

This article considers the theory of gravity which is defined by R ² as the free Lagrangian. The resulting equations are conformally invariant, and their equivalence to Einstein's equation is demonstrated (provided the stress tensor is traceless). The possibility of adapting this theory to massive point particles on a conformally flat background is discussed.