All unitary ray representations of the conformal group SU(2,2) with positive energy

Mack, G.

doi:10.1007/bf01613145

Cited by 524 publications

(513 citation statements)

References 15 publications

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“…We demonstrate that, for spinning primary operators in very chiral representations of the Lorentz group, there are universal lower bounds on scaling dimensions that are strictly stronger than those implied by the more elementary unitarity bounds of [2,3]. Based on our calculations we conjecture general formulas for these new bounds.…”

Section: Introductionmentioning

confidence: 68%

“…a representation of the Lorentz group. In four-dimensional CFTs (which are our focus here) we specify the spin of an operator h by its transformation properties under su(2) ⊕ su (2). This is a pair of integers (k,k) specifying the number of chiral and antichiral spinor indices carried by the operator h = h (α 1 α 2 ···α k ),( .…”

Section: Conformal Field Theories and The Unitarity Boundmentioning

confidence: 99%

“…Therefore although we have two operators h and h † , only one Fourier transform is required. 7 Here we are using the fact that in our spinor conventions the index names indicate the transformation properties under this SO (2). See appendix A.…”

Section: Jhep02(2018)131mentioning

confidence: 99%

“…All other states live in two-by-two blocks. We write the state with minimal SO (2) + . The files themselves are Wolfram Language expressions that can be imported using the Get function, e.g.…”

Section: Conformal Collider Inequalities For (K 1) Operatorsmentioning

confidence: 99%

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Universal bounds on operator dimensions from the average null energy condition

Córdova

Diab

2018

J. High Energ. Phys.

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We show that the average null energy condition implies novel lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories. Denoting the spin of an operator by a pair of integers (k,k) specifying the transformations under chiral su(2) rotations, we explicitly demonstrate these new bounds for operators transforming in (k, 0) and (k, 1) representations for sufficiently large k. Based on these calculations, along with intuition from free field theory, we conjecture that in any unitary conformal field theory, primary local operators of spin (k,k) and scaling dimension ∆ satisfy ∆ ≥ max{k,k}. If |k −k| > 4, this is stronger than the unitarity bound.

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Section: Introductionmentioning

confidence: 68%

Section: Conformal Field Theories and The Unitarity Boundmentioning

confidence: 99%

Section: Jhep02(2018)131mentioning

confidence: 99%

Section: Conformal Collider Inequalities For (K 1) Operatorsmentioning

confidence: 99%

See 2 more Smart Citations

Universal bounds on operator dimensions from the average null energy condition

Córdova

Diab

2018

J. High Energ. Phys.

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“…However, repeated lowering of scale dimension must stop at some point because there is a lower bound on how small scaling dimensions can be in a unitary CFT, known as the "unitarity bound" [25], which depends on the Lorentz representation of the scaling operator. Here, we settle for a crude argument, based on scale symmetry, for why scale dimensions are bounded below.…”

Section: Lightning Derivationmentioning

confidence: 99%

From fixed points to the fifth dimension

Sundrum

2012

Phys. Rev. D

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Abstract4D Lorentzian conformal field theory (CFT) is mapped into 5D anti-de Sitter spacetime (AdS), from the viewpoint of "geometrizing" conformal current algebra. A large-N expansion of the CFT is shown to lead to (infinitely many) weakly coupled AdS particles, in one-to-one correspondence with minimal-color-singlet CFT primary operators. If all but a finite number of "protected" primary operators have very large scaling dimensions, it is shown that there exists a low-AdS-curvature effective field theory regime for the corresponding finite set of AdS particles. Effective 5D gauge theory and General Relativity on AdS are derived in this way from the most robust examples of protected CFT primaries, Noether currents of global symmetries and the energy-momentum tensor. Witten's prescription for computing CFT local operator correlators within the AdS dual is derived. The main new contribution is the derivation of 5D locality of AdS couplings. This is accomplished by studying a confining IR-deformation of the CFT in the large-N "planar" approximation, where the discrete spectrum and existence of an S-matrix allow the constraints of unitarity and crossing symmetry to be solved (in standard fashion) by a tree-level expansion in terms of 4D local "glueball" couplings. When the deformation is carefully removed, this 4D locality (with plausible assumptions specifying its precise nature) combines with the restored conformal symmetry to yield 5D AdS locality. The sense in which AdS/CFT duality illustrates the possibility of emergent relativity, and the special role of strong coupling, are briefly discussed. Care is taken to conclude each step with well-defined mathematical expressions and convergent integrals. *

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On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics

Kastrup¹

2008

Ann. Phys.

113

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Dedicated to the memory of Julius Wess (1934Wess ( -2007, colleague and friend for many years.The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic role of transformations by reciprocal radii in dimensions higher than two and their linearization in terms of polyspherical coordinates by Darboux, Weyl's attempt to extend General Relativity, the slow rise of finite dimensional conformal transformations in classical field theories and the problem of their interpretation, then since about 1970 the rapid spread of their acceptance for asymptotic and structural problems in quantum field theories and beyond, up to the current AdS/CFT conjecture. The occasion for the present article: hundred years ago Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal space-time transformations.

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All unitary ray representations of the conformal group SU(2,2) with positive energy

Cited by 524 publications

References 15 publications

Universal bounds on operator dimensions from the average null energy condition

Universal bounds on operator dimensions from the average null energy condition

From fixed points to the fifth dimension

On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics

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