Scale
and Conformal Invariance in Physics| 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.
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11 oct 2009