Implications of Conformal Invariance in Field Theories for General Dimensions

Abstract

Recently obtained results for two and three point functions for quasi-primary operators in conformally invariant theories in arbitrary dimensions d are described. As a consequence the three point function for the energy momentum tensor has three linearly independent forms for general d compatible with conformal invariance. The corresponding coefficients may be regarded as possible generalisations of the Virasoro central charge to d larger than 2. Ward identities which link two linear combinations of the coefficients to terms appearing in the energy momentum tensor trace anomaly on curved space are discussed. The requirement of positivity for expectation values of the energy density is also shown to lead to positivity conditions which are simple for a particular choice of the three coefficients. Renormalisation group like equations which express the constraints of broken conformal invariance for quantum field theories away from critical points are postulated and applied to two point functions. Talk presented at the XXVII Ahrenshoop International Symposium.

Universal bounds on operator dimensions from the average null energy condition

“…by conformal symmetry, and can be produced by a variety of techniques [14][15][16][17]. For the specific case of chiral operators of interest to us we follow [18].…”

Scale invariance, conformality, and generalized free fields

“…[15]). For example, we can make the replacement x 2 → x 2 + a 2 and taking the limit a → 0 at the end of the calculation.…”

“…We can also obtain eq. (4.14) from the general momentum-space formula [17] 15) using the asymptotic expansion of the Bessel functions. In general, Fourier transforms for high-dimension operators must be defined by analytic continuation, but as we have seen, in spite of this, the short distance and high momentum limits may fail to be identical.…”

Comments on Jacobson’s “entanglement equilibrium and the Einstein equation”

“…Finally, it is useful to consider the central charge C T which appears in the two-point function of the stress tensor [32,33], 19) for the boundary CFT dual to eq. (2.1).…”