Energy conditions in quantum field theory

Abstract: In this talk I consider regimes of Quantum Mechanics that can be described in classical terms. Such regimes constitute what I call the “Classical Periphery/Skin of Quantum Mechanics.” I won't develop the general theory, but illustrate it in a study of tracks traced out by quantum-mechanical particles propagating in detectors. These tracks are close to classical particle Trajectories. I begin my talk with comments on the notion of ”events” in QM and on “state reduction”, as manifested in measurements and observations, that is, with a short review of the “ETH-Approach to QM”.

Abstract: In this talk I will present recent results obtained in collaboration with Romeo Brunetti, Michael Duetsch and Klaus Fredenhagen. This concerns the novel approach to describe interacting QFT model using nets of C*-algebras generated by local S-matrices. In a recent paper, we have introduced the action of symmetries and the renormalization group into that model. In this talk, I will present the resulting structure and also explain how it is linked to the BV formalism and the Wess-Zumino consistency condition.

Abstract: Quantum Field Theory on curved spacetimes is one of the most basic formalisms of theoretical physics. It might be expected that one cannot say anything new about its foundations. Surprisingly, it seems that some of its basic mathematical structure has been clarified only recently, in our work with Daniel Siemssen. One of such elements of this structure is the notion of the (in-out) Feynman propagator. I will explain what it is in asymptotically stationary and stable spacetimes. I will also explain how the Feynman propagator is related to the essential self-adjointness of the Klein-Gordon operator.

Abstract: The existence of the stress-energy tensor has long been postulated as a condition characterizing QFT with meaningful particle interpretation within the Haag-Kastler setting. In this talk I will outline some old and new results and ideas pointing in this direction. On the rigorous side, I will discuss the existence of particles in the sense of the theory of particle weights. A more heuristic part of the discussion will address the problem of conventional asymptotic completeness.

Abstract: I will give a general overview of the classical energy conditions and the extent to which they may be replaced by quantum energy inequalities (QEIs) in QFT. As well as discussing the known QEIs and their properties, I will also present recent work on establishing singularity theorems and related results under weakened energy conditions.

Abstract: We show a correspondence between two rigorous mathematical descriptions of chiral superconformal field theory, namely vertex operator superalgebras on the one hand and graded-local conformal nets on the other hand. We prove some general consequences, discuss classifications, and illustrate our construction with a number of well-known examples of superconformal field theories.

Abstract: Random phenomena are often modelled in terms of nonlinear stochastic partial differential equations (SPDEs). On the one hand the goal is to prove existence of solutions and in the past years, several new successful frameworks, such as regularity structures and paracontrolled calculus, have been devised. On the other one is also interested in computing the associated expectation values and correlation functions. Yet it turns out that the analysis of this class of systems shares several similarities with interacting quantum field theories, first and foremost the necessity of introducing a suitable renormalization scheme aimed at controlling unwanted divergences originating due to the underlying randomness. From this perspective it turns out that the viewpoint of algebraic quantum field theory is especially useful since it can be adapted to study also nonlinear SPDEs. We show in particular that it is possible to develop an algorithmic scheme which allows to compute at a perturbative level both the expectation values and the correlation functions, while taking into account intrinsically the underlying renormalization freedoms. As an example we apply this scheme to the nonlinear stochastic Schroedinger equation.

Abstract: I will talk about recent results on the modular Hamiltonian in QFT. In particular, the notion of entropy of a vector in a complex Hilbert space H with respect to a real linear subspace of H plays a key role.

Abstract: The classical singularity theorems predict the existence of singularities, defined using incomplete geodesics, under a set of general assumptions. One of those assumptions, namely the energy condition, is always violated by quantum fields and thus the realm of semiclassical gravity is outside the scope of these theorems. However, quantum fields do obey weaker conditions which can also be used to predict singularities. In this talk, I will present such semiclassical singularity theorems both in the timelike and the null case. Then I will argue for the need of singularity theorems with worldvolume averaged energy conditions and discuss the challenges and open questions for each case.

Abstract: We give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound that is optimal in a certain sense, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. Using this, we show that the vertex operator algebra given by a unitary vacuum representation of the W_3-algebra yields a new conformal net.

Abstract: 4D Quantum field theories are somehow squeezed between triviality and the Millennium Prize challenge. To get a little insight we relax a key condition on QFT – Poincaré or Euclidian invariance -- and study scalar quantum fields on noncommutative geometries. This gives the possibility of a topological expansion. The problem turns out to be tractable in every topological sector, and for the spherical sector the triviality problem disappears in 4D. In dimension 0 the control of all topologies is within reach, and remarkable connections to complex algebraic geometry and to enumerative geometry are found.

Abstract: The Tomita-Takesaki modular operator for local algebras plays an important structural role in quantum field theory, and more recently in relativistic quantum information. However, beyond the case of wedge algebras, describing this operator more concretely is a longstanding open problem. For the case of linear quantum fields, we present a numerical approach to this end. Specifically, we approximate the modular generator corresponding to a double cone region for a massive scalar free field in (1+1)- and (3+1)-dimensional Minkowski space. We highlight similarities and differences to the wedge case, in particular when the mass of the field is varied.

Abstract: Since their introduction in the early 40s, Feynman path integrals have always been a powerful tool for theoretical physics on the one hand and a mathematical challenge on the other. Despite decades of effort, a definitive mathematical theory of Feynman path integration is still missing and while some steps have been taken in this direction, there are fundamental issues that still deserve further investigation. Remarkably, even rather simple quantum systems, such as a non-relativistic particle moving in an external magnetic field, lead to non trivial problems appearing in the Feynman path-integral construction of quantum dynamics. In this talk I shall give an overview of this topic with a historical perspective, highlighting recent developments and some open problems.

Abstract: Quantum energy inequalities can be used to express the stability of a quantum field theory in analogy to the classical energy conditions in GR. In this talk I will argue that they also have another important application: they can be used to estimate the high energy behaviour of quantum fields in analogy to the Hamiltonian bounds for quantum fields in Minkowski space. Such bounds potentially open up the way to manipulate pointwise quantum fields and operator product expansions in a rigorous way, also for quantum fields in curved spacetimes. However, this new application does require a new form of quantum energy inequality. Focusing on a massive minimally coupled free scalar field I will describe the techniques needed to prove such a quantum inequality.

Abstract: I will discuss what types of traversable wormholes can be supported by allowed violations of the Null Energy Condition. I will give a concrete construction of an eternal traversable wormhole in asymptotically Anti-de Sitter spacetime which can be probed using holography.

Abstract: The diffeomorphism invariance of general relativity implies that there are no local (i.e. at a point) observables because diffeomorphism transformations move spacetime points around. This is particularly true for the energy, momentum and stress carried by the gravitational field: any attempt to build locally conserved densities leads to diffeomorphism-variant quantities. In this talk I put forward a proposal to build stress tensors for gravity as relational observables. The idea is to define the stress tensor as a field measured with respect to dynamical fields in the system. These dynamical fields form a reference frame that depends non-locally on the metric in general. I will present concrete examples of these gravitational stress tensors in the context of perturbative gravity and discuss their potential physical significance.

Abstract: We present a complete list of cosmological de Sitter solutions for the semiclassical Einstein equation (SCE) with a scalar field in the (pullback) Bunch-Davies state. In this setting, the SCE may be viewed as a (non-dynamic) consistency equation for the parameters of the model. Our approach allows to distinguish parameter settings in which there exist multiple de Sitter solutions with expansion rates $H$ differing by several magnitudes. By these observations a quantum scalar field is, in principle, capable to drive both an inflationary phase (approximated by the large-$H$ solution) and a dark energy-dominated late time expansion (approximated by the small-$H$ solution).

Abstract: During this talk we shall discuss the construction of the massive Sine-Gordon field in the ultraviolet finite regime when the background is a two-dimensional Minkowski spacetime. The correlation functions of the model in the adiabatic limit will be obtained combining recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive quantum field theory over Euclidean spacetimes. More precisely, perturbation theory is used to represent interacting fields as power series in the coupling constant over the free theory. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian signature, we shall see that these power series converge if the interaction Lagrangian has generic compact support. The latter observation implies also convergence in the strong operator topology in the GNS representations of states in which the system is analyzed. Finally, adapting the cluster expansion technique to the Lorentzian case, we shall see that the adiabatic limit of the correlation functions of the interacting equilibrium state at finite temperature (KMS state) is finite. The talk is based on a joint work with D. Bahns and K. Rejzner [arxiv.org:2103.09328]

Abstract: None

Abstract: I will derive some bounds on the trace T of the stress tensor in a 2 dimensional QFT. The first bound (inspired by the QNEC) applies to any quantum state and places an upper bound on T proportional to the Laplacian of the entanglement entropy. The other bound applies to compact 2d Riemann surfaces---or at least to genus-0 surfaces of rotation---and bounds T by the trace anomaly of the UV fixed point. (In all of the derivations there is a loophole which allows for an arbitrary cosmological constant to exist.) I will comment on some connections to c-theorems, and if time permits to off-shell string theory.

Abstract: The string-localized potential of the mass-zero and helicity-s field strength tensor lives on the Hilbert space of the latter. One might thus assume that positivity is intrinsic to string-localized quantum field theories. In reality, the situation is more complicated and maintaining positivity is a non-trivial task. I will present recent results on this matter at the examples of QED and a graviton coupling.

Abstract: We introduce an approach towards explicitely calculating the (locally covariant) stress energy tensor and other non-lineair observables for free fields on a curved background in terms of n-point functions on null-surface observables. Using this approach, we are able to formulate the semi-classical Einstein equations as a Goursat problem and are able to investigate the stress-energy tensor associated with Hawking radiation locally near a gravitationally collapsing body.

Abstract: The relationship between energy and entropy in classical thermodynamics is a close one. They are, nonetheless, conceptually and physically distinguished, with regard to their intrinsic properties, their relations to other physical quantities and their roles in dynamics and governing general principles. Several theoretical results and considerations in semi-classical gravity suggest that, when one takes account of quantum effects in strong gravitational fields, they may have a more intimate relationship. I review several of those results and discuss the role that energy conditions and entropy conditions play in them, attempting to tease out the ways that energy and entropy seem to share more of the same properties, play more of the same roles, than in classical thermodynamics. I conclude by speculating on why this may be, what a more intimate relationship between them may look like in this framework, and what it may suggest for our understanding of semi-classical gravity and possibly even for some approaches to quantum gravity.

Abstract: Quantum energy inequalities provide lower bounds for suitably averaged components of the stress tensor. Their classical counterparts ensure many structurally important properties of GR, such as the positive mass or the area theorem. In contrast, in the context of the strong cosmic censorship hypothesis, one desires the stress tensor to diverge (irrespective of the sign) at a Cauchy horizon, in order to prevent the breakdown of determinism associated to the non-unique extension beyond it. However, it has recently been established that at the Cauchy horizon of near extremal Reissner-Nordström-deSitter (RNdS) black holes the divergence of field perturbations becomes weak enough to allow for a non-unique extension (as a weak solution) beyond the Cauchy horizon, indicating a violation of strong cosmic censorship. In contrast, for quantum fields on RNdS we find that the degree of divergence of the expectation value of the renormalized stress tensor near the Cauchy horizon is state-independent, universal, and strong enough to save strong cosmic censorship and to drastically change the nature of the singularity inside the black hole. Interestingly (in particular in view of the averaged null energy condition), both signs of the divergence are possible. Based on joint work with Stefan Hollands, Christiane Klein, Bob Wald.

Abstract: Motivated by properties of tensor networks, we conjecture that an arbitrary gravitating region a can be assigned a generalized entanglement wedge E ⊃ a, such that quasi-local operators in E have a holographic representation in the full algebra generated by quasi-local operators in a. The universe need not be asymptotically flat or AdS, and a need not be asymptotic or weakly gravitating. On a static Cauchy surface Σ, we propose that E is the superset of a that minimizes the generalized entropy. We prove that E satisfies a no-cloning theorem and appropriate forms of strong subadditivity and nesting. If a lies near a portion A of the conformal boundary of AdS, our proposal reduces to the Quantum Minimal Surface prescription applied to A. We also discuss possible covariant extensions of this proposal, such as the smallest generalized entropy quantum normal superset of a. Our results are consistent with the conjecture that information in E that is spacelike to a in the semiclassical description can nevertheless be recovered from a, by microscopic operators that break that description. We thus propose that E quantifies the range of holographic encoding, an important nonlocal feature of quantum gravity, in general spacetimes.

Abstract: This talk concerns an infravacuum representation introduced by K. Kraus, L. Polley and G. Reents. It is not equivalent to the standard vacuum representation of a massless scalar free field on the Minkowski spacetime. But for subalgebras corresponding to measurements performed within double cones, restrictions of respective representations are quasi-equivalent. This means that the representation is locally normal. We give a straightforward proof of this fact which is based on the Araki-Yamagami criterion. Using this result we extend a recent approach to superselection theory, based on relative normalizers and conjugate classes, to a relativistic setting. The talk is based on arXiv:2106.02032, Lett Math Phys 112, 40 (2022).

Abstract: The usual procedure for defining Wick polynomials and time-ordered (TO) products in the contest of quantum field theories in curved spacetimes, is to give a set of axioms which, when implemented, defines uniquely, up to some classifiable ambiguities, the aforementioned quantities. Those ambiguities are known to be tightly restrained by locality, covariance and other regularity conditions. One of the additional constraints that was used, is to require continuous and analytic dependence on the metric and coupling parameters. It was recently shown that this rather strong requirement could be weakened, in the case of Wick polynomials, to the so-called parametrized microlocal spectrum condition. On the one hand, we show the existence of Wick polynomials satisfying the above condition, on the other, we extend this axiom to TO products, while reestablishing the usual uniqueness and existence results in light of the new constraint.

Abstract: In this (online) talk I will review some of the work I have been doing with many collaborators over the past 15 years. In essence, our work relates entanglement measures in integrable quantum field theories and conformal field theories with correlation functions of a certain type of non-local fields, known as branch point twist fields. I will introduce the properties of the fields and how they relate to entanglement measures. Time permitting, I will discuss how matrix elements and correlation functions may be computed.