Conformal invariance and the critical behavior of the tripletXY quantum chain

We introduce and study the critical properties of the tripletXY quantum chain. This system is described in terms of three-spin interactions and is the generalization of the standardXY quantum chain. We show that this model, with periodic boundaries, has a local gauge invariance and can be described by the composition of two triplet Ising models, with general toroidal boundary conditions. From this composition the phase diagram as well the conformai anomaly and critical exponents are determined by exploring their relations with the mass gap amplitudes predicted by conformai invariance.     

Conformal invariance,current algebra and modular invariance

A large class of conformally invariant models in two dimensions is realised by constraining free fermion theories. The Fock spaces of the constrained theories are described, using the representation theory of affine Kac-Moody algebras. The results are extended to superconformally invariant theories. Projections of the models, producing consistent two-dimensional field theories, are discussed.     

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.     

Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.     

Conformal invariance and spontaneous symmetry breaking

We study the spontaneous symmetry breaking in a conformally invariant gravitational theory. We particularly emphasize on the nonminimal coupling of matter fields to gravity. By the nonminimal coupling we consider a local distinction between the conformal frames of metric of matter fieldsand the metric explicitly entering the vacuum sector. We suppose that these two frames are conformally related by a dilaton field. We show that the imposition of a condition on the variable mass term of a scalar field may lead to the spontaneous symmetry breaking. In this way the scalar field may imitate the Higgs field behavior. Attributing a constant configuration to the ground state of the Higgs field, a Higgs conformal frame is specified. We define the Higgs conformal frame as a cosmological frame which describes the large scale characteristics of the observed universe. In the cosmological frame the gravitational coupling acquires a correct value and one no longer deals with the vacuum energy problem. We then study a more general case by considering a variable configuration for the ground state of Higgs field. In this case we introduce a cosmological solution of themodel.     

Conformal invariance and surface critical behavior

Conformal invariance constrains the form of correlation functions near a free surface. In two dimensions, for a wide class of models, it completely determines the correlation functions at the critical point, and yields the exact values of the surface critical exponents. They are related to the bulk exponents in a non-trivial way. For the Q-state Potts model (0 Q 4) we find η_<|; = 2/(3v − 1), and for the O(N) model (−2 N 2), η_<|; = (2v − 1)/(4v − 1).     

Conformal invariance in quantum gravity

Trace anomalies in a conformal invariant theory do not arise when its conformal invariance in four dimensions is extended to an arbitrary number n of space-time dimensions: the theory can be made finite in any order of perturbation theory by conformal invariant counterterms in n dimensions. Such an extension of conformal invariance is possible provided one works in the framework of spontaneously broken conformal invariance. This is shown explicitly by working out several examples at the one-loop level and by examining the Ward identities which lead to a general proof.We speculate upon possible consequences of these results on the nature of gravitation and other fundamental interactions.     

Conformal invariance in perturbation theory

Asymptotic conformal invariance is proved to be true at all orders in perturbation theory. The correct Ward Identities for broken conformal invariance are derived: they are the extension of the Callan Symanzik equation from scale to conformal transformations.     

The classical su(2) invariance of the su(2) q -invariant XXZ spin chain

The XXZ spin-chain Hamiltonian has been constructed to be su(2) _q -invariant, but naively does not appear to be su(2)-invariant. However, using recently discovered deforming maps between representations of su(2) _q and corresponding representations of su(2), we prove a theorem which states that if a Hamiltonian is su(2) _q -invariant, it is also su(2)-invariant. The theorem generalizes to any quantized Lie algebra.     

Conformal invariance in Weyl gravity

A conformal-invariant model of Weyl gravity, based on a nondecomposable representation of the conformal group, allows one to have a conformal-invariant propagator in an arbitrary gauge, as well as a conformal-invariant gauge-fixing term in the Lagrangian approach. It is shown that in the gauge-invariant sector this theory coincides with ordinary Weyl gravity (with conformal-noninvariant gauge fixing). The corresponding BRST transformations are found and are used for derivation of the Slavnov-Taylor identities.     

Conformal invariance in incommensurate phases

We study finite-size corrections to the free energy of free-fermion models on a torus with periodic, twisted, and fixed boundary conditions. Inside the critical (striped-incommensurate) phase, the free energy densityf(N, M) on anN×M square lattice with periodic (or twisted) boundary conditions scales asf(N, M)=f _–A(s)/(NM)+.... We derive exactly the finite-size-scaling (FSS) amplitudesA(s) as a function of the aspect ratios=M/N. These amplitudes are universal because they do not depend on details of the free-fermion Hamiltonian. We establish an equivalence between the FSS amplitudes of the free-fermion model and the Coulomb gas system with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while electric defect lines are generated by competition between domain wall separation and system size. The FSS behavior of the free-fermion model is consistent with predictions of the theory of conformal invariance with the conformal chargec=l. For instance, the FSS amplitude on an infinite cylinder with fixed boundary conditions is found to be one-quarter of that with periodic boundary conditions. Finally, we conjecture the exact form of the FSS amplitudes for an interacting-fermion model on a torus. Numerical calculations employing the Bethe Ansatz confirm our conjecture in the infinite-cylinder limit.     

Thermodynamics of the quantum spin-S XXZ chain

The thermodynamics of the spin-S anisotropic quantum XXZ chain with arbitrary value of S and unitary norm, in the high-temperature regime, is reported. The single-ion anisotropy term and the interaction with an external magnetic field in the z-direction are taken into account. We obtain, for arbitrary value of S, the β-expansion of the Helmholtz free energy of the model up to order β⁶ and show that it actually depends on . Its classical limit is obtained by simply taking S→∞. At h=0 and D=0, our high temperature expansion of the classical model coincides with Joyce’s exact solution [11]. We study, in the high temperature region, some thermodynamic quantities such as the specific heat and the magnetic susceptibility as functions of spin and verify for which values of S those thermodynamic functions behave classically. Their finite temperature behavior is inferred from interpolation of their high- and low-temperature behavior, and shown to be in good agreement with numerical results. The finite temperature behavior is shown for higher values of spin.     

XXZ chain with a boundary

The XXZ spin chain with a boundary magnetic field h is considered, using the vertex operator approach to diagonalize the hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when h ⩾ 0, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary S-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the XXX model.     

Conformal invariance,supersymmetry and string theory

Covariant quantization of string theories is developed in the context of conformal field theory and the BRST quantization procedure. The BRST method is used to covariantly quantize superstrings, and in particular to construct the vertex operators for string emission as well as the supersymmetry charge. The calculation of string loop diagrams is sketched. We discuss how conformal methods can be used to study string compactification and dynamics.     

Conformal transformations and conformal invariance in gravitation

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and Einstein's general relativity theory. In this paper we discuss the rules of these transformations for geometric quantities as well as for the matter energy‐momentum tensor. We show the subtlety of the matter energy‐momentum conservation law which refers to the fact that the conformal transformation “creates” an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is “created” due to work done by the conformal transformation to bend the spacetime which was originally flat. We discuss how to construct the conformally invariant gravity theories and also find the conformal transformation rules for the curvature invariants R², R_abR^ab, R_abcdR^abcd and the Gauss‐Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present the conformal transformation rules in the fashion of the duality transformations of the superstring theory. In such a case the transitions between conformal frames reduce to a simple change of the sign of a redefined conformal factor.     

Conformal invariance and torsion in general relativity

We study a theory for gravity in which the linear connections are assumed to be arbitrary, except that they are restricted to satisfy the metric condition g =0. A scalar field is added to the theory, and a conformally invariant action integral, linear in the curvature tensor, is defined. The linear connections emerging from the variational principle contain torsion that is related to a propagating spin-1 vector field, identified as the electromagnetic gauge potential. We obtain a set of conformally invariant equations for the metric field, and conclude that Einstein's equations arise from a particular choice of gauge. Finally, spin-1/2 fields are introduced by means of the vierbein formalism, and the qualitative features of the theory are maintained.