Conformal Coupling and Invariance in Different Dimensions

Physical Review D, 2000

We discuss the possibility of realizing the infinite dimensional symmetries of conformal mechanics as time reparametrizations, generalizing the realization of the SL(2,R) symmetry of the de Alfaro, Fubini, Furlan model in terms of quasi--primary fields. We find that this is possible using an appropriate generalization of the transformation law for the quasi-primary fields.

Annalen der Physik, 2009

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and the Einstein relativity. 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 R 2 , R ab R ab , R abcd R 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.

arXiv: General Relativity and Quantum Cosmology, 2018

Here we follow the mainstream of thinking about physical equivalence of different representations of a theory, regarded as the consequence of invariance of the laws of physics -- represented by an action principle and the derived motion equations -- under given transformations; be it coordinate, gauge or conformal transformations. Accordingly the conformal transformations' issue is discussed by invoking the assumed invariance of the laws of physics -- in particular the laws of gravity -- under conformal transformations of the metric. It is shown that Brans-Dicke and scalar-tensor theories are not well-suited to address physical equivalence of the conformal frames since the corresponding laws of gravity are not invariant under the conformal transformations or Weyl rescalings. The search for conformal symmetry leads us to explore the physical consequences of Weyl-invariant theories of gravity, that represent a natural arena where to discuss on physical equivalence of the conformal...

EPJ Web of Conferences

A gravity model based on the conformal symmetry is presented. To specify the structure of the general coordinate transformations the Ogievetsky theorem is applied. The nonlinear symmetry realization approach is used. Canonical quantization is performed with the use of reparameterizationinvariant time and the Arnowitt-Deser-Misner foliation. Renormalizability of the constructed quantum gravity model is discussed.

Classical and Quantum Gravity, 2016

In this paper we consider conformal symmetry in the context of manifolds with general affine connection. We extend the conformal transformation law of the metric to a general metric compatible affine connection, and find that it is a symmetry of both the geodesic equation and the Riemann tensor. We derive the generalised Jacobi equation and Raychaudhuri equation and show that they are both conformally invariant. Using the geodesic deviation (Jacobi) equation we analyse the behaviour of geodesics in different conformal frames. Since we find that our version of conformal symmetry is exact in classical pure Einstein's gravity, we ask whether one can extend it to the standard model. We find that it is possible to write conformal invariant lagrangians in any dimensions for vector, fermion and scalar fields, but that such lagrangians are only gauge invariant in four dimensions. Provided one introduces a dilaton field, gravity can be conformally coupled to matter.

This short note provides a general introduction to conformal symmetry in arbitrary dimension. 1 The Conformal Group We denote by g µν the metric tensor in a space-time of dimension d. By definition, a conformal transformation of the coordinates is an invertible mapping x → x which leaves the metric tensor invariant up to a scale g µν (x) = Λ(x)g µν (x). In other words, a conformal transformation is locally equivalent to a pseudo-rotation and a dilation. The set of conformal transformation manifestly forms a group and it is obviously has the Poincaré group as a subgroup since the latter corresponds to the special case Λ(x) = 1. The epithet conformal derives from the property that the transformation does not affect the angle between two arbitrary curves crossing each other at some point, despite a local dilation: the conformal group preserves angles. We investigate the consequences of the definition g µν (x) = Λ(x)g µν (x) on an infinitesimal transformation x µ → x µ = x µ + µ (x). The metric, at first order in , changes as follows g µν → g µν − (∂ µ ν + ∂ ν µ). The requirement that the transformation be conformal implies that ∂ µ ν + ∂ ν µ = f (x)g µν. The factor f (x) is determined by taking the trace on both sides f (x) = 2 d ∂ ρ ρ. For simplicity, we assume that the conformal transformation is an infinitesimal deformation of the standard Cartesian metric g µν = η µν where η µν = diag(1, 1, ..., 1). By applying an extra derivative ∂ ρ on ∂ µ ν + ∂ ν µ = f (x)g µν permuting the indices and taking a linear combination we arrive at 2∂ µ ∂ ν ρ = η µρ ∂ ν f + η νρ ∂ µ f − η µν ∂ ρ f. Upon contracting with η µν gives 2∂ 2 µ = (2 − d)∂ µ f.

2016

The conformal transformations corresponding to N-Galilean conformal symmetries, previously defined as canonical symmetry transformations on phase space, are constructed as point transformations in coordinate space.

2011

It is demonstrated that, unless the meaning of conformal transformations for the underlying geometrical structure is discussed on a same footing as it is done for the equations of the given gravity theory, the notion of "conformal equivalence", as it has been mostly used in the bibliography to date, lacks physical and mathematical content. A principle of conformal equivalence is

Progress of Theoretical Physics Supplement, 2011

As we shall briefly recall, Nordström's theory of gravity is observationally ruled out. It is however an interesting example of non-minimal coupling of matter to gravity and of the role of conformal transformations. We show in particular that they could be useful to extend manifolds through curvature singularities.

1995

We analyze the constraints of general coordinate invariance for quantum theories possessing conformal symmetry in four dimensions. The character of these constraints simplifies enormously on the Einstein universe $R \times S^3$. The $SO(4,2)$ global conformal symmetry algebra of this space determines uniquely a finite shift in the Hamiltonian constraint from its classical value. In other words, the global Wheeler-De Witt equation is {\it modified} at the quantum level in a well-defined way in this case. We argue that the higher moments of $T^{00}$ should not be imposed on the physical states {\it a priori} either, but only the weaker condition $\langle \dot T^{00} \rangle = 0$. We present an explicit example of the quantization and diffeomorphism constraints on $R \times S^3$ for a free conformal scalar field.