Correlation Functions of the Energy Momentum Tensor on Spaces of Constant Curvature

Abstract

An analysis of one and two point functions of the energy momentum tensor on homogeneous spaces of constant curvature is undertaken. The possibility of proving a $c$-theorem in this framework is discussed, in particular in relation to the coefficients $c,a$, which appear in the energy momentum tensor trace on general curved backgrounds in four dimensions. Ward identities relating the correlation functions are derived and explicit expressions are obtained for free scalar, spinor field theories in general dimensions and also free vector fields in dimension four. A natural geometric formalism which is independent of any choice of coordinates is used and the role of conformal symmetries on such constant curvature spaces is analysed. The results are shown to be constrained by the operator product expansion. For negative curvature the spectral representation, involving unitary positive energy representations of $O(d-1,2)$, for two point functions of vector currents is derived in detail and extended to the energy momentum tensor by analogy. It is demonstrated that, at non coincident points, the two point functions are not related to $a$ in any direct fashion and there is no straightforward demonstration obtainable in this framework of irreversibility under renormalisation group flow of any function of the couplings for four dimensional field theories which reduces to $a$ at fixed points.

72 pages, plain TeX file using harvmac, final revision for publication

References in corpus (1)

• Anomalies, Unitarity and Quantum Irreversibility

Cited by in corpus (43)

• Aspects of Holographic Entanglement Entropy
• Supersymmetric Gauge Theories and the AdS/CFT Correspondence
• Brane New World
• Trace anomaly driven inflation
• The S-matrix Bootstrap I: QFT in AdS
• Stochastic Gravity: Theory and Applications
• Stochastic Gravity: Theory and Applications
• Evidence for the Strongest Version of the 4d a-Theorem, via a-Maximization Along RG Flows
• A pedagogical review on solvable irrelevant deformations of 2d quantum field theory
• Constraints on RG Flow for Four Dimensional Quantum Field Theories
• Stochastic Gravity: A Primer with Applications
• Riemann correlator in de Sitter including loop corrections from conformal fields
• Stress tensor fluctuations in de Sitter spacetime
• Can Renormalization Group Flow End in a Big Mess?
• Expectation value of $\mathrm{T}\overline{\mathrm{T}}$ operator in curved spacetimes
• Exact Consequences of the Trace Anomaly in Four Dimensions
• Spontaneously Broken Boosts in CFTs
• Cutoff $\mathrm{Ad}{\mathrm{S}}_3$ versus $\mathrm{T}\bar{\mathrm{T}}$ $\mathrm{CF}{\mathrm{T}}_2$ in the large central charge sector: correlators of energy-momentum tensor
• Stress-Tensor Commutator in CFT near the Lightcone
• Renormalization group flow of entanglement entropy on spheres
• Stress-energy Tensor Correlators of a Quantum Field in Euclidean $R^N$ and $Ad{S}^N$ spaces via the generalized zeta-function method
• Chern-Simons theory from M5-branes and calibrated M2-branes
• Thermodynamics and transport of holographic nodal line semimetals
• The structure of the trace anomaly of higher spin conformal currents in the bulk of $Ad{S}_4$
• Quantum gravitational corrections for spinning particles
• Graviton correlator and metric perturbations in de Sitter brane-world
• Gravity, Quantum Fields and Quantum Information: Problems with classical channel and stochastic theories
• Quadratic gravity in first order formalism
• Cosmological bounds on TeV-scale physics and beyond
• Renormalization group irreversible functions in more than two dimensions
• RG flows on $S^d$ and Hamiltonian truncation
• CFT in Conformally Flat Spacetimes
• Scale and Conformal Invariance on (A)dS
• The C-Theorem and Chiral Symmetry Breaking in Asymptotically Free Vectorlike Gauge Theories
• Stochastic formulation of the renormalization group: supersymmetric structure and topology of the space of couplings
• Quantum information and the C-theorem in de Sitter
• Holographic 2-Point Functions in the Pseudo-Conformal Universe
• Patterns of chiral symmetry breaking and a candidate for a C-theorem in four dimensions
• Correlations of the stress-energy tensor in AdS spaces via the generalized zeta-function method
• Universal thermalization dynamics in (1+1)d QFTs
• Massive Bosons and the dS/CFT correspondence
• Gradient flows for $\beta$ functions via multi-scale renormalization group equations
• Effective action in elliptic and hyperbolic spacetimes