Energy-momentum Tensor Operators Research Articles

Note on holographic torus stress tensor correlators in AdS3 gravity

In the AdS3/CFT2 framework, the Euclidean BTZ black hole corresponds to the dominant high-temperature phase of its dual field theory. We initially employ perturbative methods to solve the Einstein equations as boundary value problems, providing correlators for the energy-momentum tensor operator at low points. Utilizing operator equations established in our previous work, we further compute arbitrary high-point correlators for the energy-momentum tensor operator in the high-temperature phase and recursive relations for these high-point functions. Concurrently, we employ the Chern-Simons formalism to derive consistent results. Further, using the cut-off AdS/TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}-deformed CFT duality, we calculate the energy-momentum tensor correlators, contributing to the comprehensive understanding of the system’s dynamics. Finally, stress tensor correlators enable us to ascertain the corresponding KdV operator correlators at low-temperature.

Intrinsic approach to 1 + 1D Carrollian Conformal Field Theory

The 3D Bondi-Metzner-Sachs (BMS3) algebra that is the asymptotic symmetry algebra at null infinity of the 1 + 2D asymptotically flat space-time is isomorphic to the 1 + 1D Carrollian conformal algebra. Building on this connection, various preexisting results in the BMS3-invariant field theories are reconsidered in light of a purely Carrollian perspective in this paper. In direct analogy to the covariant transformation laws of the Lorentzian tensors, the flat Carrollian multiplets are defined and their conformal transformation properties are established. A first-principle derivation of the Ward identities in a 1 + 1D Carrollian conformal field theory (CCFT) is presented. This derivation introduces the use of the complex contour-integrals (over the space-variable) that provide a strong analytic handle to CCFT. The temporal step-function factors appearing in these Ward identities enable the translation of the operator product expansions (OPEs) into the language of the operator commutation relations and vice versa, via a contour-integral prescription. Motivated by the properties of these step-functions, the iϵ-forms of the Ward identities and OPEs are proposed that permit for the hassle-free use of the algebraic properties of the latter. Finally, utilizing the computational techniques developed, it is shown that the modes of the quantum energy-momentum tensor operator generate the centrally extended version of the infinite-dimensional 1 + 1D Carrollian conformal algebra.

Gauge-invariant renormalization scheme in QCD: Application to fermion bilinears and the energy-momentum tensor

We consider a gauge-invariant, mass-independent prescription for renormalizing composite operators, regularized on the lattice, in the spirit of the coordinate space (X-space) renormalization scheme. The prescription involves only Green's functions of products of gauge-invariant operators, situated at distinct space-time points, in a way as to avoid potential contact singularities. Such Green's functions can be computed nonperturbatively in numerical simulations, with no need to fix a gauge: thus, renormalization to this "intermediate" scheme can be carried out in a completely nonperturbative manner. Expressing renormalized operators in the $\overline{\rm MS}$ scheme requires the calculation of corresponding conversion factors. The latter can only be computed in perturbation theory, by the very nature of the $\overline{\rm MS}$; however, the computations are greatly simplified by virtue of the following attributes: i) In the absense of operator mixing, they involve only massless, two-point functions; such quantities are calculable to very high perturbative order. ii) They are gauge invariant; thus, they may be computed in a convenient gauge. iii) Where operator mixing may occur, only gauge-invariant operators will appear in the mixing pattern: Unlike other schemes, involving mixing with gauge-variant operators (which may contain ghost fields), the mixing matrices in the present scheme are greatly reduced. Still, computation of some three-point functions may not be altogether avoidable. We exemplify the procedure by computing, to lowest order, the conversion factors for fermion bilinear operators of the form $\bar\psi\Gamma\psi$ in QCD. We also employ the gauge-invariant scheme in the study of mixing between gluon and quark energy-momentum tensor operators: We compute to one loop the conversion factors relating the nonperturbative mixing matrix to the $\overline{\rm MS}$ scheme.

Two-loop renormalization and mixing of gluon and quark energy-momentum tensor operators

In this paper, we present one- and two-loop results for the renormalization of the gluon and quark gauge-invariant operators which appear in the definition of the QCD energy-momentum tensor, in dimensional regularization. To this end, we consider a variety of Green's functions with different incoming momenta. We identify the set of twist-2 symmetric traceless and flavor singlet operators which mix among themselves and we calculate the corresponding mixing coefficients for the nondiagonal components. We also provide results for some appropriate regularization-independent (RI')-like schemes, which address this mixing, and we discuss their application to nonperturbative studies via lattice simulations. Finally, we extract the one- and two-loop expressions of the conversion factors between the proposed RI' and the MSbar schemes. From our results regarding the MSbar-renormalized Green's functions, one can easily derive conversion factors relating numerous variants of RI'-like schemes to MSbar. To make our results easily accessible, we also provide them as Supplemental Material, in the form of a Mathematica input file and, also, an equivalent text file.

Horizon constraints on holographic Green\u2019s functions

We explore a new class of general properties of thermal holographic Green’s functions that can be deduced from the near-horizon behaviour of classical perturbations in asymptotically anti-de Sitter spacetimes. We show that at negative imaginary Matsubara frequencies and appropriate complex values of the wavenumber the retarded Green’s functions of generic operators are not uniquely defined, due to the lack of a unique ingoing solution for the bulk perturbations. From a boundary perspective these ‘pole-skipping’ points correspond to locations in the complex frequency and momentum planes at which a line of poles of the retarded Green’s function intersects with a line of zeroes. As a consequence the dispersion relations of collective modes in the boundary theory at energy scales ω ∼ T are directly constrained by the bulk dynamics near the black-brane horizon. For the case of conserved U (1) current and energy-momentum tensor operators we give examples where the dispersion relations of hydrodynamic modes pass through a succession of pole- skipping points as real wavenumber is increased. We discuss implications of our results for transport, hydrodynamics and quantum chaos in holographic systems.

Holography as a highly efficient renormalization group flow. I. Rephrasing gravity

We investigate how the holographic correspondence can be reformulated as a generalisation of Wilsonian RG flow in a strongly interacting large $N$ quantum field theory. We firstly define a \textit{highly efficient RG flow} as one in which the Ward identities related to local conservation of energy, momentum and charges preserve the same form at each scale -- to achieve this it is necessary to redefine the background metric and external sources at each scale as functionals of the effective single trace operators. These redefinitions also absorb the contributions of the multi-trace operators to these effective Ward identities. Thus the background metric and external sources become effectively dynamical reproducing the dual classical gravity equations in one higher dimension. Here, we focus on reconstructing the pure gravity sector as a highly efficient RG flow of the energy-momentum tensor operator, leaving the explicit constructive field theory approach for generating such RG flows to the second part of the work. We show that special symmetries of the highly efficient RG flows carry information through which we can decode the gauge fixing of bulk diffeomorphisms in the corresponding gravity equations. We also show that the highly efficient RG flow which reproduces a given classical gravity theory in a given gauge is \textit{unique} provided the endpoint can be transformed to a non-relativistic fixed point with a finite number of parameters under a universal rescaling. The results obtained here are used in the second part of this work, where we do an explicit field-theoretic construction of the RG flow, and obtain the dual classical gravity theory.

Vacuum interaction of conic singularities

In the (tr log)-formalism, we consider the problem of the vacuum interaction of conic singularities in a D-dimensional (D ≥ 3) space–time. We show that the interaction energy regularized by dimensional regularization contains neither ultraviolet divergences nor divergences associated with the nonintegrable nature of the vacuum mean of the energy–momentum tensor operator. In the case of four space–time dimensions, the result coincides with those obtained previously in a local approach.

Stability of the pion and the pattern of chiral symmetry breaking

We investigate the pressure of the pion, which should be equal to zero to ensure the stability of the pion, within the framework of the chiral quark model beyond the chiral limit. The pressure of the pion vanishes nontrivially by the Gell-Mann-Oakes-Renner relation within the present framework. This implies that the stability of the pion might be deeply rooted in spontaneous chiral symmetry breaking. We also discuss physical quantities relevant to the energy-momentum tensor operator.

Casimir densities for a spherical boundary in de Sitter spacetime

Two-point functions, mean-squared fluctuations, and the vacuum expectation value of the energy-momentum tensor operator are investigated for a massive scalar field with an arbitrary curvature coupling parameter, subject to a spherical boundary in the background of de Sitter spacetime. The field is prepared in the Bunch-Davies vacuum state and is constrained to satisfy Robin boundary conditions on the sphere. Both the interior and exterior regions are considered. For the calculation in the interior region, a mode-summation method is employed, supplemented with a variant of the generalized Abel-Plana formula. This allows us to explicitly extract the contributions to the expectation values which come from de Sitter spacetime without boundaries. We show that the vacuum energy-momentum tensor is nondiagonal, with the off-diagonal component corresponding to the energy flux along the radial direction. With dependence on the boundary condition and the mass of the field, this flux can be either positive or negative. Several limiting cases of interest are then studied. In terms of the curvature coupling parameter and the mass of the field, two very different regimes are realized, which exhibit monotonic and oscillatory behavior of the vacuum expectation values, respectively, far from the sphere. The decay of the boundary-induced expectation values at large distances from the sphere is shown to be a power-law decay (monotonic or oscillating), independent of the value of the field mass. The expressions for the Casimir densities in the exterior region are generalized for a more general class of spherically symmetric spacetimes inside the sphere.

Long-range rapidity correlations in heavy ion collisions at strong coupling from AdS/CFT

We use AdS/CFT correspondence to study two-particle correlations in heavy ion collisions at strong coupling. Modeling the colliding heavy ions by shock waves on the gravity side, we observe that at early times after the collision there are long-range rapidity correlations present in the two-point functions for the glueball and the energy-momentum tensor operators. We estimate rapidity correlations at later times by assuming that the evolution of the system is governed by ideal Bjorken hydrodynamics, and find that glueball correlations in this state are suppressed at large rapidity intervals, suggesting that late-time medium dynamics can not “wash out” the long-range rapidity correlations that were formed at early times. These results may provide an insight on the nature of the “ridge” correlations observed in heavy ion collision experiments at RHIC and LHC, and in proton-proton collisions at LHC.

Vacuum polarization in the Schwarzschild spacetime and dimensional reduction

A massless scalar field minimally coupled to gravity and propagating in Schwarzschild spacetime is considered. After dimensional reduction under spherical symmetry the resulting 2D field theory is canonically quantized and the renormalized expectation values 〈Tab〉 of the relevant energy-momentum tensor operator are investigated. Asymptotic behaviors and analytical approximations are given for 〈Tab〉 in the Boulware, Unruh and Hartle-Hawking states. Special attention is devoted to the black-hole horizon region where the WKB approximation breaks down.

Modified weak energy condition for the energy momentum tensor in quantum field theory

The weak energy condition is known to fail in general when applied to expectation values of the energy momentum tensor in flat space quantum field theory. It is shown how the usual counter arguments against its validity are no longer applicable if the states |ψ〉 for which the expectation value is considered are restricted to a suitably defined subspace. A possible natural restriction on |ψ〉 is suggested and illustrated by two quantum mechanical examples based on a simple perturbed harmonic oscillator Hamiltonian. The proposed alternative quantum weak energy condition is applied to states formed by the action of the scalar, vector and the energy momentum tensor operators on the vacuum. We assume conformal invariance in order to determine almost uniquely three-point functions involving the energy momentum tensor in terms of a few parameters. The positivity conditions lead to non-trivial inequalities for these parameters. They are satisfied in free field theories, except in one case for dimensions close to two. Further restrictions on |ψ〉 are suggested which remove this problem. The inequalities which follow from considering the state formed by applying the energy momentum tensor to the vacuum are shown to imply that the coefficient of the topological term in the expectation value of the trace of the energy momentum tensor in an arbitrary curved space background is positive, in accord with calculations in free field theories.

Hawking radiation and back-reaction in a unitary theory of 2D quantum gravity

We construct a manifestly unitary quantum theory of the CGHS model of 2D dilaton gravity by performing a reduced phase space quantization. It is shown that physical states exist such that in the semiclassical limit the Hawking radiation is present. A new method for calculating the back-reaction corrections to the classical metric is introduced, based on the expansion of the metric operator in a power series of the matter energy-momentum tensor operator. The first order correction to a local black hole mass function is evaluated, and it agrees with the expected semiclassical behaviour. We briefly discuss the calculation of higher-order corrections and implications for the quantum fate of a 2D black hole.

On the Hawking radiation associated with an Oppenheimer-Snyder collapsing star

The author discusses on the basis of the geometrical optics approximation the development of the Hawking radiation in the spacetime of a black hole resulting from an Oppenheimer-Snyder collapsing star. To the extent that he can use this approximation it is shown that the asymptotic flow of radiation to future null infinity does not depend on the specific parameter describing the initial condition for the stellar collapse (in his model the initial radius). Further, at a large distance from the black hole a definition of the radial part of the radiation is given via the concept of normal ordering of the energy-momentum tensor operator with respect to a 'local ground state' of the Schwarzschild metric. It is shown that the ambiguity in this definition which results from the existence of many possible local ground states does not affect the radial part, but enters only in other parts of the radiation.

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and a Bäcklund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: ∂ μ Φ, and e nΦ . An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equivalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.

Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State

A method of self-consistent fields is used to study the equilibrium configurations of a system of self-gravitating scalar bosons or spin-\textonehalf{} fermions in the ground state without using the traditional perfect-fluid approximation or equation of state. The many-particle system is described by a second-quantized free field, which in the boson case satisfies the Klein-Gordon equation in general relativity, ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\nabla}}^{\ensuremath{\alpha}}\ensuremath{\varphi}={\ensuremath{\mu}}^{2}\ensuremath{\varphi}$, and in the fermion case the Dirac equation in general relativity ${\ensuremath{\gamma}}^{\ensuremath{\alpha}}{\ensuremath{\nabla}}_{\ensuremath{\alpha}}\ensuremath{\psi}=\ensuremath{\mu}\ensuremath{\psi}$ (where $\ensuremath{\mu}=\frac{\mathrm{mc}}{\ensuremath{\hbar}}$). The coefficients of the metric ${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$ are determined by the Einstein equations with a source term given by the mean value $〈\ensuremath{\varphi}|{T}_{\ensuremath{\mu}\ensuremath{\nu}}|\ensuremath{\varphi}〉$ of the energy-momentum tensor operator constructed from the scalar or the spinor field. The state vector $〈\ensuremath{\varphi}|$ corresponds to the ground state of the system of many particles. In both cases, for completeness, a nonrelativistic Newtonian approximation is developed, and the corrections due to special and general relativity explicitly are pointed out. For $N$ bosons, both in the region of validity of the Newtonian treatment (density from ${10}^{\ensuremath{-}80}$ to ${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, and number of particles from 10 to ${10}^{40}$) as well as in the relativistic region (density \ensuremath{\sim}${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, number of particles \ensuremath{\sim}${10}^{40}$), we obtain results completely different from those of a traditional fluid analysis. The energy-momentum tensor is anisotropic. A critical mass is found for a system of $N\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{2}\ensuremath{\sim}{10}^{40}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}25}$ g) self-gravitating bosons in the ground state, above which mass gravitational collapse occurs. For $N$ fermions, the binding energy of typical particles is ${G}^{2}{m}^{5}{N}^{\frac{4}{3}}{\ensuremath{\hbar}}^{\ensuremath{-}2}$ and reaches a value $\ensuremath{\sim}{\mathrm{mc}}^{2}$ for $N\ensuremath{\sim}{N}_{\mathrm{crit}}\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{3}\ensuremath{\sim}{10}^{57}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}24}$ g, implying mass \ensuremath{\sim}${10}^{33}$ g, radius \ensuremath{\sim}${10}^{6}$ cm, density \ensuremath{\sim}${10}^{15}$ g/${\mathrm{cm}}^{3}$). For densities of this order of magnitude and greater, we have given the full self-consistent relativistic treatment. It shows that the concept of an equation of state makes sense only up to ${10}^{42}$ g/${\mathrm{cm}}^{3}$, and it confirms the Oppenheimer-Volkoff treatment in extremely good approximation. There exists a gravitational spin-orbit coupling, but its magnitude is generally negligible. The problem of an elementary scalar particle held together only by its gravitational field is meaningless in this context.