Conformal invariance: from Weyl to SO(2,d)
TL;DR: In this article, the authors show that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to a two-dimensional SO(2,4) invariant field in arbitrary conformally flat spaces.
Abstract: The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in $d>2$ dimensional spaces and SO(2,d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of String Theory but in the particular case of $d=2$ spaces. Indeed, the Polyakov formalism describes world-sheets in terms of two-dimensional conformal field theory. On the other hand, B. Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS). This allows us to assert that a classical $SO(2,d)$-invariant field does not distinguish, at least locally, between two different $d$-dimensional CFSs.
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TL;DR: In this article, the relationship between nonlinear p-form electrodynamics with conformal-invariant weak-field and strong-field limits is clarified, with a focus on duality invariant (2n − 1)-form electdynamics and chiral 2n-forms in Minkowski spacetime.
Abstract: Relations between the various formulations of nonlinear p-form electrodynamics with conformal-invariant weak-field and strong-field limits are clarified, with a focus on duality invariant (2n − 1)-form electrodynamics and chiral 2n-form electrodynamics in Minkowski spacetime of dimension D = 4n and D = 4n + 2, respectively. We exhibit a new family of chiral 2-form electrodynamics in D = 6 for which these limits exhaust the possibilities for conformal invariance; the weak-field limit is related by dimensional reduction to the recently discovered ModMax generalisation of Maxwell’s equations. For n > 1 we show that the chiral ‘strong-field’ 2n-form electrodynamics is related by dimensional reduction to a new Sl(2; ℝ)-duality invariant theory of (2n − 1)-form electrodynamics.
46 citations
TL;DR: In this paper, a bottom-up argument to derive some known results from holographic renormalization using the classical bulk-bulk equivalence of General Relativity and Shape Dynamics, a theory with spatial conformal invariance, was provided.
Abstract: We provide a bottom-up argument to derive some known results from holographic renormalization using the classical bulk–bulk equivalence of General Relativity and Shape Dynamics, a theory with spatial conformal (Weyl) invariance. The purpose of this paper is twofold: (1) to advertise the simple classical mechanism, trading off gauge symmetries, that underlies the bulk–bulk equivalence of General Relativity and Shape Dynamics to readers interested in dualities of the type of AdS/conformal field theory (CFT); and (2) to highlight that this mechanism can be used to explain certain results of holographic renormalization, providing an alternative to the AdS/CFT conjecture for these cases. To make contact with the usual semiclassical AdS/CFT correspondence, we provide, in addition, a heuristic argument that makes it plausible that the classical equivalence between General Relativity and Shape Dynamics turns into a duality between radial evolution in gravity and the renormalization group flow of a CFT. We believe that Shape Dynamics provides a new perspective on gravity by giving conformal structure a primary role within the theory. It is hoped that this work provides the first steps toward understanding what this new perspective may be able to teach us about holographic dualities.
27 citations
TL;DR: Weyl-to-Riemann as mentioned in this paper is a method to construct conformally invariant equations in arbitrary Riemann spaces based on two features of Weyl geometry, i.e., a Weyl space is defined by the metric tensor and the Weyl vector $W$, it becomes equivalent to a riemann space when $W$ is gradient.
Abstract: We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces.
This method that we call "Weyl-to-Riemann" is based on two features of Weyl geometry. i) A Weyl space is defined by the metric tensor and the Weyl vector $W$, it becomes equivalent to a Riemann space when $W$ is gradient. ii) Any homogeneous differential equation written in a Weyl space by means of the Weyl connection is conformally invariant. The Weyl-to-Riemann method selects those equations whose conformal invariance is preserved when reducing to a Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrates the efficiency of the present method. In particular, a new conformally invariant spin-2 field equation is exhibited. This equation extends Grishchuk-Yudin's equation and fixes its limitations since it does not require the Lorenz gauge. Moreover this equation reduces to the Drew-Gegenberg and Deser-Nepomechie equations in respectively Minkowski and de Sitter spaces.
15 citations
TL;DR: In this paper, a conformally invariant theory of gravitation in the context of metric measure space is studied, which is invariant under both diffeomorphism and conformal transformations.
Abstract: In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a generalization of the Einstein equation is obtained, wherein the conventional tensors are replaced by their conformally invariant counterparts, living in metric measure space. The invariance of the geometrical part of the action under a diffeomorphism leads to a generalized contracted second Bianchi identity. In metric measure space, the covariant derivative is the same as it is in the Riemannian space. Hence, in contrast to the Weyl space, the metricity and integrability are maintained. However, it is worth noting that in metric measure space the divergence of a tensor is not simply the contraction of the covariant derivative operator with the tensor that it acts on. Despite the fact that metric measure space and integrable Weyl space, are constructed based on different assumptions, it is shown that some relations in these spaces, such as the contracted second Bianchi identity, are completely similar.
4 citations
TL;DR: In this paper, the authors presented an invariant quantization of the free electromagnetic field in conformally flat spaces (CFSs), where the quantum structure is given by a vacuum state and creators/annihilators acting on some Hilbert space.
Abstract: We present an $SO(2,4)$-covariant quantization of the free electromagnetic field in conformally flat spaces (CFSs). A CFS is realized in a six-dimensional space as an intersection of the null cone with a given surface. The smooth move of the latter is equivalent to perform a Weyl rescaling. This allows to transport the $SO(2,4)$-invariant quantum structure of the Maxwell field from Minkowski space to any CFS. Calculations are simplified and the CFS Wightman two-point functions are given in terms of their Minkowskian counterparts. The difficulty due to gauge freedom is surpassed by introducing two auxiliary fields and using the Gupta-Bleuler quantization scheme. The quantum structure is given by a vacuum state and creators/annihilators acting on some Hilbert space. In practice, only the Hilbert space changes under Weyl rescalings. Also, the quantum $SO(2,4)$-invariant free Maxwell field does not distinguish between two CFSs.
4 citations
References
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Book•
01 Jan 1987
TL;DR: In this article, the authors synthesize the most promising approaches and ideals in field theory today, including statistical mechanics, quantum field theory and their interrelation, continuous global symmetry, non-Abelian gauge fields, instantons and the quantam theory of loops, and quantum strings and random surfaces.
Abstract: Based on his own work, the author synthesizes the most promising approaches and ideals in field theory today. He presents such subjects as statistical mechanics, quantum field theory and their interrelation, continuous global symmetry, non-Abelian gauge fields, instantons and the quantam theory of loops, and quantum strings and random surfaces. This book is aimed at postgraduate students studying field theory and statistical mechanics, and for research workers in continuous global theory.
1,909 citations
TL;DR: In this article, the authors studied the constraints imposed by the existence of a single higher spin conserved current on a three-dimensional conformal field theory and showed that the correlation functions of the stress tensor and the conserved currents are equal to those of a free field theory.
Abstract: We study the constraints imposed by the existence of a single higher spin conserved current on a three-dimensional conformal field theory (CFT). A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. This is an extension of the Coleman–Mandula theorem to CFT’s, which do not have a conventional S-matrix. We also briefly discuss the case where the higher spin symmetries are ‘slightly’ broken.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.
578 citations
Book•
01 Jan 1970
TL;DR: The conclusive volume of the Brandeis University Summer Institute lecture series of 1970 on theories of interacting elementary particles consisting of five sets of lectures as discussed by the authors is a collection of lectures on quantum field theory.
Abstract: The conclusive volume of the Brandeis University Summer Institute lecture series of 1970 on theories of interacting elementary particles consisting of five sets of lectures. The five sets of lectures are as follows:Rudolph Haag (II. Institut fur Theoretische Physik der Universitat Hamburg) on "Observables and Fields": introduction; axiomatic quantum field theory in various formulations; structure of superselection rules; charge quantum numbers; statistics; parastatistics.Maurice Jacob (CERN, European Organization for Nuclear Research) on "Regge Models and Duality": introduction; duality in a semi-local way; duality and unitary symmetry; dual models for meson-meson scattering; dual models for production proceses; from dual models to a dual theory.Henry Primakoff (University of Pennsylvania) on "Weak Interactions": introduction; lepton conversation and the implications of a possible lepton non-conversation; first-order and second-order weak collision processes; "abnormalities in the weak currents and how to discover them; conclusion.Michael C. Reed (Princeton University)on "The GNS Construction -- A Pedagogical Example": infinite tensor products of Hilbert spaces; the canonical anti-commutation relations; the example; the example -- via the GNS construction.Bruno Zumino (CERN, European Organization for Nuclear Research) on "Effective Lagrangians and Broken Symmetries": Introduction; effective action and phenomenological fields; Ward identities and the effective action; Goldstone's theorem; non-linear realizations; massive Yang-Mills fields as phenomenological fields; broken scale invariance; the fifteen parameter conformal group and the Weyl transformations; conversion identities and trace identities; invariant actions; SU(3)xSU(3)and conformalinvariance; strong gravitation; concluding remarks.
329 citations
TL;DR: In this paper, it was shown that if fields on the hypercone of a projective space approach finite limits on the boundary of the boundary, then in the conformal field theory on this boundary these limits transform with conformal dimensionality zero if they are tensors (of any rank), but with conformality dimension $1/2$ if they were spinors or spinor-tensors.
Abstract: The calculation of both spinor and tensor Green's functions in four-dimensional conformally-invariant field theories can be greatly simplified by six-dimensional methods. For this purpose, four-dimensional fields are constructed as projections of fields on the hypercone in six-dimensional projective space, satisfying certain transversality conditions. In this way some Green's functions in conformal field theories are shown to have structures more general than those commonly found by use of the inversion operator. These methods fit in well with the assumption of AdS/CFT duality. In particular, it is transparent that if fields on ${\mathrm{AdS}}_{5}$ approach finite limits on the boundary of ${\mathrm{AdS}}_{5}$, then in the conformal field theory on this boundary these limits transform with conformal dimensionality zero if they are tensors (of any rank), but with conformal dimension $1/2$ if they are spinors or spinor-tensors.
277 citations
01 Sep 1998
TL;DR: In this paper, the authors present an introduction to conformal field theory and its applications to 2-dimensional statistical mechanics of critical phenomena, including string theory, integrable systems, representations of infinite Lie algebras and automorphic functions.
Abstract: This volume contains Introductory Notes and major reprints on conformal field theory and its applications to 2-dimensional statistical mechanics of critical phenomena. The subject relates to many different areas in contemporary physics and mathematics, including string theory, integrable systems, representations of infinite Lie algebras and automorphic functions.
192 citations