arXiv subjects

Stefan Hollands

Publications and source records attributed to Stefan Hollands.

At least 55 records · Page 3Linked to original sources

arXiv2017

Action principle for OPE

We formulate an "action principle" for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings "In memoriam Rudolf Haag" in September 2016 and the "Wolfhart Zimmermann memorial symposium" in May 2017.

hep-th↗

arXiv2017

Relative entanglement entropy for widely separated regions in curved spacetime

We give an upper bound of the relative entanglement entropy of the ground state of a massive Dirac-Majorana field across two widely separated regions $A$ and $B$ in a static slice of an ultrastatic Lorentzian spacetime. Our bound decays exponentially in $dist (A, B)$, at a rate set by the Compton wavelength and the spatial scalar curvature. The physical interpretation our result is that, on a manifold with positive spatial scalar curvature, one cannot use the entanglement of the vacuum state to teleport one classical bit from $A$ to $B$ if their distance is of the order of the maximum of the curvature radius and the Compton wave length or greater.

math-ph↗

arXiv2017

The Memory Effect for Particle Scattering in Even Spacetime Dimensions

We explicitly calculate the gravitational wave memory effect for classical point particle sources in linearized gravity off of an even dimensional Minkowski background. We show that there is no memory effect in $d>4$ dimensions, in agreement with the general analysis of Hollands, Ishibashi, and Wald (2016).

gr-qc↗

arXiv2017

SO(d,1)-invariant Yang-Baxter operators and the dS/CFT correspondence

We propose a model for the dS/CFT correspondence. The model is constructed in terms of a "Yang-Baxter operator" $R$ for unitary representations of the deSitter group $SO(d,1)$. This $R$-operator is shown to satisfy the Yang-Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: a) A chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of $SO(d,1)$. By analogy with the $O(N)$ non-linear sigma model, this chiral CFT can be viewed as propagating in a deSitter spacetime. b) A (non-unitary) Euclidean conformal quantum field theory on ${\mathbb R}^{d-1}$, where $SO(d,1)$ now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret ${\mathbb R}^{d-1}$ as conformal infinity of deSitter spacetime. Our constructions use semi-local generator fields defined in terms of $R$ and abstract methods from operator algebras.

gr-qc↗

arXiv2016

All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory

We give a complete, self-contained, and mathematically rigorous proof that Euclidean Yang-Mills theories are perturbatively renormalisable, in the sense that all correlation functions of arbitrary composite local operators fulfil suitable Ward identities. Our proof treats rigorously both all ultraviolet and infrared problems of the theory and provides, in the end, detailed analytical bounds on the correlation functions of an arbitrary number of composite local operators. These bounds are formulated in terms of certain weighted spanning trees extending between the insertion points of these operators. Our proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, combined with estimation techniques based on tree structures. Compared with previous mathematical treatments of massless theories without local gauge invariance [R. Guida and Ch. Kopper, arXiv:1103.5692; J. Holland, S. Hollands, and Ch. Kopper, Commun. Math. Phys. 342 (2016) 385] our constructions require several technical advances; in particular, we need to fully control the BRST invariance of our correlation functions.

math-ph↗

arXiv2016

BMS Supertranslations and Memory in Four and Higher Dimensions

We consider the memory effect in even dimensional spacetimes of dimension $d \ge 4$ arising from a burst of gravitational radiation. When $d=4$, the natural frames in the stationary eras before and after the burst differ by the composition of a boost and supertranslation, and this supertranslation characterizes the "memory effect," i.e., the permanent displacement of test particles near infinity produced by the radiation burst. However, we show that when $d > 4$, this supertranslation and the corresponding memory effect vanish. Consequently, when $d >4$, it is natural to impose stronger asymptotic conditions at null infinity that reduce the asymptotic symmetry group to the Poincare group. Conversely, when $d=4$, the asymptotic symmetry group at null infinity must be taken to be the BMS group.

gr-qc↗

arXiv2016

Non-Equilibrium Thermodynamics and Conformal Field Theory

We present a model independent, operator algebraic approach to non-equilibrium quantum thermodynamics within the framework of two-dimensional Conformal Field Theory. Two infinite reservoirs in equilibrium at their own temperatures and chemical potentials are put in contact through a defect line, possibly by inserting a probe. As time evolves, the composite system then approaches a non-equilibrium steady state that we describe. In particular, we re-obtain recent formulas of Bernard and Doyon.

hep-th↗

arXiv2015

Superradiant instabilities of asymptotically anti-de Sitter black holes

We study the linear stability of asymptotically anti-de Sitter black holes in general relativity in spacetime dimension $d\ge4$. Our approach is an adaptation of the general framework of Hollands and Wald, which gives a stability criterion in terms of the sign of the canonical energy, $\mathcal{E}$. The general framework was originally formulated for static or stationary and axisymmetric black holes in the asymptotically flat case, and the stability analysis for that case applies only to axisymmetric perturbations. However, in the asymptotically anti-de Sitter case, the stability analysis requires only that the black hole have a single Killing field normal to the horizon and there are no restrictions on the perturbations (apart from smoothness and appropriate behavior at infinity). For an asymptotically anti-de Sitter black hole, we define an ergoregion to be a region where the horizon Killing field is spacelike; such a region, if present, would normally occur near infinity. We show that for black holes with ergoregions, initial data can be constructed such that $\mathcal{E}<0$, so all such black holes are unstable. To obtain such initial data, we first construct an approximate solution to the constraint equations using the WKB method, and then we use the Corvino-Schoen technique to obtain an exact solution. We also discuss the case of charged asymptotically anti-de Sitter black holes with generalized ergoregions.

gr-qc↗

arXiv2015

Associativity of the operator product expansion

We consider a recursive scheme for defining the coefficients in the operator product expansion (OPE) of an arbitrary number of composite operators in the context of perturbative, Euclidean quantum field theory in four dimensions. Our iterative scheme is consistent with previous definitions of OPE coefficients via the flow equation method, or methods based on Feynman diagrams. It allows us to prove that a strong version of the "associativity" condition holds for the OPE to arbitrary orders in perturbation theory. Such a condition was previously proposed in an axiomatic setting in [1] and has interesting conceptual consequences: 1) One can characterise perturbations of quantum field theories abstractly in a sort of "Hochschild-like" cohomology setting, 2) one can prove a "coherence theorem" analogous to that in an ordinary algebra: The OPE coefficients for a product of two composite operators uniquely determine those for $n$ composite operators. We concretely prove our main results for the Euclidean $φ^4_4$ quantum field theory, covering also the massless case. Our methods are rather general, however, and would also apply to other, more involved, theories such as Yang-Mills theories.

math-ph↗

arXiv2015

Instabilities of extremal rotating black holes in higher dimensions

Recently, Durkee and Reall have conjectured a criterion for linear instability of rotating, extremal, asymptotically Minkowskian black holes in $d\ge 4$ dimensions, such as the Myers-Perry black holes. They considered a certain elliptic operator, $\cA$, acting on symmetric trace-free tensors intrinsic to the horizon. Based in part on numerical evidence, they suggested that if the lowest eigenvalue of this operator is less than the critical value $-1/4$ ( called "effective BF-bound"), then the black hole is linearly unstable. In this paper, we prove an extended version of their conjecture. Our proof uses a combination of methods such as (i) the "canonical energy method" of Hollands-Wald, (ii) algebraically special properties of the near horizon geometries associated with the black hole, (iii) the Corvino-Schoen technique, and (iv) semiclassical analysis. Our method of proof is also applicable to rotating, extremal asymptotically Anti-deSitter black holes. In that case, we find additional instabilities for ultra-spinning black holes. Although we explicitly discuss in this paper only extremal black holes, we argue that our results can be generalized to {\em near} extremal black holes.

hep-th↗

arXiv2014

The operator product expansion converges in massless $φ_4^4$-theory

It has been shown recently that the mathematical status of the operator product expansion (OPE) is better than was expected before: namely considering massive Euclidean $φ^4$-theory in the perturbative loop expansion, the OPE converges at any loop order when considering (as is usually done) composite operator insertions into correlation functions. In the present paper we prove the same result for the massless theory. While the short-distance properties of massive and massless theories may be expected to be similar on physical grounds, the proof in the massless case requires entirely new techniques. In our inductive construction we have to control with sufficient precision the exceptional momentum singularities of the massless correlation functions. In fact the bounds we state are expressed in terms of weight factors associated to certain tree graphs. Our proof is again based on the flow equations of the renormalisation group.

hep-th↗

arXiv2014

Quantum fields in curved spacetime

We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as time-ordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity.

gr-qc↗

arXiv2014

Superconformal quantum field theory in curved spacetime

By conformally coupling vector and hyper multiplets in Minkowski space, we obtain a class of field theories with extended rigid conformal supersymmetry on any Lorentzian four-manifold admitting twistor spinors. We construct the conformal symmetry superalgebras which describe classical symmetries of these theories and derive an appropriate BRST operator in curved spacetime. In the process, we elucidate the general framework of cohomological algebra which underpins the construction. We then consider the corresponding perturbative quantum field theories. In particular, we examine the conditions necessary for conformal supersymmetries to be preserved at the quantum level, i.e. when the BRST operator commutes with the perturbatively defined S-matrix, which ensures superconformal invariance of amplitudes. To this end, we prescribe a renormalization scheme for time-ordered products that enter the perturbative S-matrix and show that such products obey certain Ward identities in curved spacetime. These identities allow us to recast the problem in terms of the cohomology of the BRST operator. Through a careful analysis of this cohomology, and of the renormalization group in curved spacetime, we establish precise criteria which ensure that all conformal supersymmetries are preserved at the quantum level. As a by-product, we provide a rigorous proof that the beta-function for such theories is one-loop exact. We also briefly discuss the construction of chiral rings and the role of non-perturbative effects in curved spacetime.

hep-th↗

arXiv2014

Bondi mass cannot become negative in higher dimensions

We prove that the Bondi mass of an asymptotically flat, vacuum, spacetime cannot become negative in any even dimension $d \ge 4$. The notion of Bondi mass is more subtle in $d > 4$ dimensions because radiating metrics have a slower decay than stationary ones, and those subtleties are reflected by a considerably more difficult proof of positivity. Our proof holds for the standard spherical infinities, but also extends to infinities of more general type which are $(d-2)$-dimensional manifolds admitting a real Killing spinor. Such manifolds typically have special holonomy and Sasakian structures. The main technical advance of the paper is an expansion technique based on "conformal Gaussian null coordinates". This expansion helps us to understand the consequences imposed by Einstein's equations on the asymptotic tail of the metric field. As a by-product, we derive a coordinate expression for the geometrically invariant formula for the Bondi mass originally given by Hollands and Ishibashi.

gr-qc↗

arXiv2013

Operator product expansion algebra

We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean $φ^{4}$-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of arXiv:1105.3375, that the 3-point OPE, $< O_{A_1} O_{A_2} O_{A_3} > = \sum_{C} \cal{C}_{A_1 A_2 A_3}^{C} $, usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity $\cal{C}_{A_1 A_2 A_3}^{B}=\sum_{C}\cal{C}_{A_1 A_2}^{C} \cal{C}_{C A_3}^{B}$ is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such `consistency conditions' and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.

math-ph↗

arXiv2013

Conformal symmetry superalgebras

We show how the rigid conformal supersymmetries associated with a certain class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even part of this superalgebra contains conformal isometries and constant R-symmetries. The odd part is generated by twistor spinors valued in a particular R-symmetry representation. We prove that any manifold which admits a conformal symmetry superalgebra of this type must generically have dimension less than seven. Moreover, in dimensions three, four, five and six, we provide the generic data from which the conformal symmetry superalgebra is prescribed. For conformally flat metrics in these dimensions, and compact R-symmetry, we identify each of the associated conformal symmetry superalgebras with one of the conformal superalgebras classified by Nahm. We also describe several examples for Lorentzian metrics that are not conformally flat.

hep-th↗

arXiv2013

A small cosmological constant due to non-perturbative quantum effects

We propose that the expectation value of the stress energy tensor of the Standard Model should be given by $< T_{μν} > = ρ_\vac η_{μν}$, with a vacuum energy $ρ_\vac$ that differs from the usual "dimensional analysis" result by an exponentially small factor associated with non-perturbative effects. We substantiate our proposal by a rigorous analysis of a toy model, namely the 2-dimensional Gross-Neveu model. In particular, we address, within this model, the key question of the renormalization ambiguities affecting the calculation. The stress energy operator is constructed concretely via the operator-product-expansion. The non-perturbative factor in the vacuum energy is seen as a consequence of the facts that a) the OPE-coefficients have an analytic dependence on $g$, b) the vacuum correlations have a non-analytic (=non-perturbative) dependence on $g$, which we propose to be a generic feature of QFT. Extrapolating our result from the Gross-Neveu model to the Standard Model, one would expect to find $ρ_\vac ~ Λ^4 \e^{-O(1)/g^2}$, where $Λ$ is an energy scale such as $Λ= M_{H}$, and $g$ is a gauge coupling such as $g^2/4π= α_{EW}$. The exponentially small factor due to non-perturbative effects could explain the "unnatural" smallness of this quantity.

gr-qc↗

arXiv2012

Stability of Black Holes and Black Branes

We establish a new criterion for the dynamical stability of black holes in $D \geq 4$ spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, $\E$, on a subspace, $\mathcal T$, of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon. This is shown by proving that---apart from pure gauge perturbations and perturbations towards other stationary black holes---$\E$ is nondegenerate on $\mathcal T$ and that, for axisymmetric perturbations, $\E$ has positive flux properties at both infinity and the horizon. We further show that $\E$ is related to the second order variations of mass, angular momentum, and horizon area by $\E = δ^2 M - \sum_A Ω_A δ^2 J_A - \fracκ{8π} δ^2 A$, thereby establishing a close connection between dynamical stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamical instability because the perturbations towards other members of the family will not, in general, have vanishing linearized ADM mass and/or angular momentum. However, we prove that for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $\E < 0$ and vanishing linearized ADM quantities. Thus, all black branes corresponding to thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of $\E$ on $\mathcal T$ is equivalent to the satisfaction of a "local Penrose inequality," thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability.

gr-qc↗