Conformal Invariance in Physics  

In General > s.a. conformal structures [including conformal Killing fields]; Scale Invariance.
* History: 1908, Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal spacetime transformations; The reasons for conformal invariance were originally pointed out by H Weyl.
* Conformal invariance and Weyl invariance: Conformal and Weyl invariance are sometimes taken to be synonymous, although it may be best to distinguish conformal invariance of a theory in flat spacetime from the Weyl invariance of a theory coupled to gravity, in curved spacetime.
* Conditions: In the absence of dimensional parameters, conformal invariance requires the vanishing of Taa.
* Special types: Restricted Weyl invariance refers to transformations with conformal factor satisfying ∇2Ω = 0.
* And quantization: The conformal invariance of a classical theory can be broken by quantization, as in the case of QCD with massless quarks.
@ 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 AIP(05)gq [examples in 3D]; Kastrup AdP(08)-a0808-in [historical developments]; Jackiw & Pi JPA(11) [in diverse dimensions]; Quirós et al GRG(13)-a1108 [formulation of principle of conformal equivalence, and applications]; László a1406-conf [without reference to a metric]; Mannheim a1506 [as an alternative to supersymmetry]; Todorov BJP-a1905 [hist].
@ Weyl vs conformal invariance: Karananas & Monin PLB(16)-a1510; Farnsworth et al JHEP(17)-a1702 [in quantum field theory]; Alvarez et al EPJC(19)-a1903 [spin-2].
@ Restricted Weyl invariance: Edery & Nakayama PRD(14)-a1406; Edery & Nakayama MPLA(15)-a1502 [and Einstein gravity with cosmological constant and Higgs mass]; Oda a2005.
@ Spontaneous breaking: Smolin PLB(80) [and general relativity as low-energy limit]; Nieh PLA(82), Venturi gq/06-MG11 [and generation of gravitational constant G]; Edery et al CQG(06)ht [Weyl theory + matter]; Kaplan et al PRD(09)-a0905 [as phase transition]; Schwimmer & Theisen NPB(11) [and trace anomaly]; 't Hooft FP(11) [and elementary particle models without free parameters]; Hinterbichler & Khoury JCAP(12) [scale invariance, and cosmology]; Guendelman et al GRG(15)-a1408 [and cosmology]; 't Hooft a1410-GRF [small-distance structure of gravity]; de Cesare et al EPJC(17)-a1612 [emergence of physical scales]; Ghilencea & Lee PRD(19)-a1809 [in model building beyond the Standard Model and inflation]; > s.a. conformal gravity; particle physics.
@ Generalizations: Pérez-Nadal EPJC(17)-a1609 [anisotropic conformal invariance]; > s.a. hořava gravity.
> Online resources: see Wikipedia page.

Conformal Invariance in Gravity \ s.a. gravity and gravitational theories [including scale invariant].
@ General references: Deser AP(70); Bicknell JPA(76); Suggett JPA(79), JPA(79); Dąbrowski et al AdP(09)-a0806 [rev]; 't Hooft a0909-conf [conformal transformations in quantum gravity]; Moon et al MPLA(10)-a0912 [in Einstein-Cartan-Weyl space]; Nobili a1201, a1201 [Conformal General Relativity]; Clark & Love MPLA(12)-a1205 [local Weyl scaling and dilatation invariance]; Quiros a1401 [physical consequences]; Rahmanpour & Shojaie GRG(16)-a1608 [metric measure spaces]; Nikolić a1702-conf [conformal non-invariance of Einstein-Hilbert action]; Hobson & Lasenby a2008 [gauging].
@ And matter: del Campo et al JCAP(10)-a1006 [with dilaton, and spontaneous breaking of symmetry]; 't Hooft a1011 [conformal constraint and the coupling to matter]; Padilla et al PRD(14) [scalar fields coupled to gravity]; Lucat & Prokopec CQG(16)-a1606 [and the standard model]; Shaposhnikov & Shkerin PLB(18)-a1803 [conformal symmetry breaking].
@ And cosmology: Kelleher CQG(04)gq/03, CQG(04) [and the cosmological constant]; Cadoni PLB(06) [as broken by matter coupling, and the cosmological constant]; Mottola a1103-proc [and dark energy]; Nguyen a1111; Bars et al PRD(14)-a1307 [lifting a non-scale-invariant theory to a Weyl-invariant one, and cosmology]; Barvinsky JCAP(14)-a1311 [ghost-free conformal extension of Einstein's theory, and dark metter]; Álvarez et al JCAP(15)-a1501; Libanov et al JETPL(15)-a1508 [scalar perturbations]; Alexeyev & Krichevskiy PPNL-a2012 [inflationary solutions].
@ Related topics: Wei & Cai JCAP(07)ap/06 [and the Cheng-Weyl vector field]; Attard & Lazzarini NPB(16)-a1607 [and the Wess-Zumino functional].
> Various theories: see conformal gravity [including spatial conformal invariance and quantum thory]; Shape Dynamics [evolving conformal geometry]; teleparallel gravity; Weyl Invariance; unified theories.

In Other Theories > s.a. Conformal Field Theory; higher-spin theories; quintessence; spin-2 fields; Stealth Fields.
@ Electromagnetism: Rosen AJP(72)jul [conformal invariance of Maxwell's equations]; Wulfman a1003/JPA [consequences]; > s.a. electromagnetism in curved spacetime.
@ Standard model: Meissner & Nicolai PLB(07); Fabbri GRG(12)-a1107; > s.a. electroweak theory.
@ Other theories: Smirnov in(06)-a0708 [2D lattice models], a0708 [Ising model]; Shaukat & Waldron NPB(10) [explicit coupling of theories to scale]; Andrzejewski & Gonera a1108 [mechanics]; Faci a1110 [constructing conformally invariant equations]; Hofman & Strominger PRL(11) [2D quantum field theory]; Casalbuoni & Gomis PRD(14)-a1404 [relativistic point particles]; Okazaki PRD(17)-a1704 [quantum mechanics]; Hammad et al a2012 [Klein-Gordon equation in curved spacetime].
> Related topics: see anomalies; Biconformal Space; mass [origin]; renormalization group.

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