Introduction to Conformal Invariance

Journal of Physics A: Mathematical and Theoretical, 2009

This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational "dynamical exponent", z. The Schrödinger-Virasoro algebra of Henkel et al. corresponds to z = 2. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which z = 2. For lightlike geodesics, they yield, in turn, the Conformal Galilean Algebra (CGA) of Henkel, and Lukierski, Stichel and Zakrzewski, with z = 1. Physical systems realizing these symmetries include, e.g., classical systems of massive, and massless non-relativistic particles, and also hydrodynamics, as well as Galilean electromagnetism.

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...

2013

This is the first of three papers on Conformal General Relativity (CGR), which differs from Einstein's General Relativity (GR) in that it requires action-integral invariance under local scale transformations in addition to general coordinate transformations. The theory is here introduced in the semiclassical approximation as a preliminary approach to a quantum theoretical implementation. The idea of a conformal-invariant extension of GR was introduced by Weyl in 1919. For several decades it had little impact, as CGR implies that all fields are massless. Today this does not appear to be an unsurmountable difficulty since nonzero mass parameters may result from the spontaneous breakdown of conformal symmetry. The theory leads to very interesting results and predictions: 1) the spontaneous breakdown of conformal symmetry is only possible in a 4D-spacetime with small negative curvature; 2) CGR requires the introduction of a ghost scalar field σ(x) invested with geometric meaning and a physical scalar field ϕ(x) of zero mass, both of which have nonzero vacuum expectation values; 3) in order to preserve S-matrix unitarity, σ(x) and ϕ(x) must interact in such a way that the total energy density is bounded from below; 4) this interaction makes ϕ(x) behave like a Higgs field of varying mass, which is capable of promoting a huge energy transfer from geometry to matter identifiable as the big bang; 5) in the course of time, the Higgs boson mass becomes a constant and CGR converges to GR.

This is the first of three papers on Conformal General Relativity (CGR), which differs from Einstein's General Relativity (GR) in that it requires action-integral invariance under local scale transformations in addition to general coordinate transformations. The theory is here introduced in the semiclassical approximation as a preliminary approach to a quantum theoretical implementation. The idea of a conformal-invariant extension of GR was introduced by Weyl in 1919. For several decades it had little impact, as CGR implies that all fields are massless. Today this does not appear to be an unsurmountable difficulty since nonzero mass parameters may result from the spontaneous breakdown of conformal symmetry. The theory leads to very interesting results and predictions: 1) the spontaneous breakdown of conformal symmetry is only possible in a 4D-spacetime with small negative curvature; 2) CGR requires the introduction of a ghost scalar field σ(x) invested with geometric meaning and a physical scalar field ϕ(x) of zero mass, both of which have nonzero vacuum expectation values; 3) in order to preserve S-matrix unitarity, σ(x) and ϕ(x) must interact in such a way that the total energy density is bounded from below; 4) this interaction makes ϕ(x) behave like a Higgs field of varying mass, which is capable of promoting a huge energy transfer from geometry to matter identifiable as the big bang; 5) in the course of time, the Higgs boson mass becomes a constant and CGR converges to GR.

Il Nuovo Cimento B, 1977

A 'minimal' change in the four-dimensional theory of relativity is performed by extending the theory to six-dimensional conformal space (flat or curved) with a given metric tensor for which all components of the 6-vector are considered as independent physical degrees of freedom. All the basic equations of both special and general relativity in flat or curved six-dimensional conformal space are found to have the same form as the corresponding equations in four-dimensional space. A special feature of the extended theory (referred to as 'conformal relativity') is the introduction of a scalar degree of freedom (k), which may differ from unity and may vary along the world line of a particle. It is shown that the conformal theory of relativity reduces to the usual four-dimensional theory if k = 1 and that geodesics in curved six-dimensional conformal space correspond to the motion of electrically charged test particles in a gravitational field, an electromagnetic field, or both. It is noted that the Einstein equations generalized to six-dimensional conformal space constitute the field equations for the metric tensor; these equations describe both gravitational and electromagnetic fields simultaneously.

Physical Review D, 2001

General Relativity and Gravitation, 2013

Arguments are given which point out that the notion of "conformal equivalence" -as it has been used in the bibliography on the issue to date -is lacking in physical and mathematical content. Unless the meaning of conformal transformations for the underlying spacetime geometry is discussed on a same footing as it is done for the relevant equations of the given gravity theory, the issue is no more than a semantic one. To put the discussion on concrete grounds a principle of conformal equivalence is formulated and an alternative geometric interpretation of the conformal transformations is given. This allows demonstrating that, contrary to previous claims in the literature, vacuum Brans-Dicke (BD) theory of gravity (ω = −3/2) is not conformally invariant and, consequently, the different conformal frames in which the theory can be formulated are just different -conformally related -representations of a same phenomenon. These can not be compared: either all of them are physically correct, or none is it. To illustrate our arguments, we explore a modification of BD theory of gravitation which is indeed conformally invariant. The different points of view existing in the bibliography about the (non)equivalence of the Jordan's and Einstein's conformal frames of Brans-Dicke theory are critically analyzed on the light of the formulated principle of conformal equivalence and of the alternative geometric interpretation of the conformal transformations given here. Our results can be straightforwardly applied to scalar-tensor theories in general.

Physical Review D, 2014

A system of relativistic Snyder particles with mutual two-body interaction that lives in a Non-Commutative Snyder geometry is studied. The underlying symplectic structure is a coupled, novel kind and extended version of Snyder algebra. In a recent work by Casalbuoni and Gomis, Phys.Rev. D90, 026001 (2014) [1], a system of interacting conventional particles (in commutative spacetime) was studied with special emphasis on it's Conformal Invariance. Succeed along the same lines [1] we have shown that our interacting Snyder particle model is also conformally invariant. Moreover the conformal Killing vectors have been constructed. Our main emphasis is on the Hamiltonian analysis of the conformal symmetry generators. We demonstrate that the Lorentz algebra remains undeformed but validity of the full conformal algebra requires further restrictions.

2010

We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called "Natural Theories" and "Gauge-Natural Theories" that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincaré-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether symmetries", energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-called ADM laws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation "à la Palatini" and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).