Scale and Conformal Invariance in Physics  

Scale Invariance / Symmetry > s.a. potentials [anomaly].
* Vs conformal symmetry: There is a conjecture that in unitary field theories scale invariance implies conformality, and a proof by Zamolodchikov and Polchinski for 2D theories, that is not valid in higher dimensions.
@ Vs conformal symmetry: Awad & Johnson PRD(00)ht, IJMPA(01)ht/00-in [from AdS-cft]; Riva and Cardy PLB(05)ht [in 2D elasticity]; Dorigoni & Rychkov a0910 [conjecture that scale invariance and unitarity imply conformal invariance].
@ Other topics: Camblong et al PRL(01) [breaking, in molecule + electron]; Belitz et al RMP(05) [and phase transitions]; Hill ht/05-in [and dimension, cosmological constant, physical scales]; Sochichiu JHEP(09) [3D dilatation operator, perturbative]; Westman a0910 [spatial scale invariance as a local gauge symmetry].
@ Generalizations: Gozzi & Mauro JPA(06)qp/05 [mechanical similarity as generalization].

Conformal Invariance / Symmetry > s.a. conformal structures [including conformal Killing fields]; mass [origin].
* History: 1908, Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal space-time transformations; The reasons for conformal invariance were originally pointed out by H Weyl.
* Conditions: In the absence of dimensional parameters, conformal invariance requires vanishing of Taa.
@ General references: Flato et al AP(78) [covariance of field equations]; Boyanovsky & Naon RNC(90) [in quantum field theory / statistical mechanics]; Zuber Rech(93)feb; Nikolov & Todorov IJMPA(04) [and rationality of correlation functions]; Jackiw gq/05-in [examples in 3D]; Kastrup AdP(08)-a0808-in [historical developments].
@ Spontaneous breaking: Smolin PLB(80) [and general relativity as low-energy limit]; Nieh PLA(82), Venturi gq/06-in [and generation of gravitational constant G]; Edery et al CQG(06)ht [Weyl theory + matter]; Kaplan et al a0905 [as phase transition].
@ In electromagnetism: Rosen AJP(72)jul [conformal invariance of Maxwell's equations].
@ In standard model: Meissner & Nicolai PLB(07).
@ In other theories: Smirnov in(06)-a0708 [2D lattice models], a0708 [Ising model].
> In other theories: see Conformal Field Theory; higher-spin theories; quintessence.
> Related topics: see anomaly; Biconformal Space.

Conformal Structure / Invariance in Gravity > s.a. formulations of general relativity.
* Idea: The volume element, or determinant of the metric, |g|1/2.
@ General references: in Mukhanov et al PRP(92) [and degrees of freedom]; Schmidt PRD(95)gq/01; Garfinkle PRD(97)gq/96 [and Choptuik scaling]; Faraoni et al FCP(99)gq/98; Forte & Laciana CQG(99); Wei & Cai JCAP(07)ap/06 [Cheng-Weyl vector field]; Dabrowski et al AdP(09)-a0806 [rev].
@ Conformal factor in cosmology: Barbashov et al ht/04-in [as time].
@ Conformal structure: Barut et al FP(94) [conformal spacetimes]; > s.a. types of metrics and spacetimes [Brinkman's theorem].
@ Evolving conformal geometries: Barbour & O'Murchadha gq/99; Anderson et al CQG(05)gq/04 [evolving conformal geometry].
@ Conformally invariant theories: Deser AP(70); Bicknell JPA(76); Suggett JPA(79), JPA(79); Kelleher CQG(04)gq/03, CQG(04) [and the cosmological constant]; Cadoni PLB(06) [as broken by matter coupling, and the cosmological constant]; 't Hooft a0909-in [conformal transformations in quantum gravity]; > s.a. Conformal Gravity; theories of gravity; Weyl Invariance.
@ Quantization: in Narlikar & Padmanabhan PRP(83); Padmanabhan PRD(83); Hu PLA(89); > s.a. quantum gravity and approaches.


main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 11 oct 2009