Narrow Results

Let $H:O\mapsto H(O)$ be a local net of real Hilbert subspaces of a complex Hilbert space ${\mathscr{H}}$ on the family $𝒪$ of double cones of the spacetime ${ℝ}^{d+1}$ ($d$ odd), covariant with respect to a positive energy, unitary representation $U$ of the Poincaré group ${{𝒫}}_{+}^{\uparrow }$, with the Bisognano–Wichmann property for the wedge modular group. We set an upper bound on the local entropy $S_H(\varphi ||C)$ of a vector $\varphi \in {\mathscr{H}}$ in a given region $C\subset {ℝ}^{d+1}$ that depends only on $U$ and the PCT anti-unitary canonically associated with $H$. A similar result holds for local, Möbius covariant nets of standard subspaces on the circle. We compute the entropy increase with respect to the dual net and illustrate this bound for the nets associated with the $U(1)$-current derivatives.

Scalar QFT on the boundary ℑ⁺ at future null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMS-invariant symplectic form. The constructed theory turns out to be invariant under a suitable strongly-continuous unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to ℑ⁺ of massless minimally coupled fields propagating in the bulk. The group theoretical analysis of the found unitary BMS representation proves that such a field on ℑ⁺ coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group Δ, the semidirect product between SO(2) and the two-dimensional translations group. This wave function is massless with respect to the notion of mass for BMS representation theory. The presented result proposes a natural criterion to solve the long-standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually, some theorems towards a holographic description on ℑ⁺ of QFT in the bulk are established at level of C*-algebras of fields for asymptotically flat at null infinity spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective *-homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary ℑ⁺. Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincaré invariance in the bulk and BMS invariance on the boundary at null infinity is established at the level of QFT. It arises that, in this case, the *-homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on ℑ⁺.

A noncommutative spacetime admitting dilation symmetry was briefly mentioned in the seminal work [8] of Doplicher, Fredenhagen and Roberts. In this paper, we explicitly construct the model in detail and carry out an indepth analysis. The C*-algebra that describes this quantum spacetime is determined, and it is shown that it admits an action by _*-automorphisms of the dilation group, along with the expected Poincaré covariance. In order to study the main physical properties of this scale-covariant model, a free scalar neutral field is introduced as an investigation tool. Our key results are then the loss of locality and the irreducibility, or triviality, of special field algebras associated with regions of the ordinary Minkowski spacetime. It turns out, in the conclusions, that this analysis allows also to argue on viable ways of constructing a full conformally covariant model for quantum spacetime.

The prototypes of mutually independent systems are systems which are localized in spacelike separated regions. In the framework of locally covariant quantum field theory, we show that the commutativity of observables in spacelike separated regions can be encoded in the tensorial structure of the functor which associates unital C*-algebras (the local observable algebras) to globally hyperbolic spacetimes. This holds under the assumption that the local algebras satisfy the split property and involves the minimal tensor product of C*-algebras.

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the $\mathrm{\text{CCR}}$-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of $C^{\ast }$-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet–Lie group structure on differential cohomology groups.

In the bootstrap approach to integrable quantum field theories in the $(1+1)$-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. We find a family of two-particle S-matrices parametrized by two positive numbers, which are separated from the free field or any other known S-matrix. We propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense.

The sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather–breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather–breather S-matrix have wrong signs and cannot be considered as a separate model. Our S-matrices differ from the breather–breather S-matrix in the sine-Gordon model by CDD factors which adjust the signs, so that this sector alone satisfies reasonable assumptions.

On a conformal net ${𝒜}$, one can consider collections of unital completely positive maps on each local algebra ${𝒜}(I)$, subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on ${𝒜}$ the subset of extreme such maps. The usual automorphisms of ${𝒜}$ (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of ${𝒜}$ under all quantum operations is the Virasoro net generated by the stress-energy tensor of ${𝒜}$. Furthermore, we show that every irreducible conformal subnet ${ℬ}\subset {𝒜}$ is the fixed points under a subset of quantum operations.

When ${ℬ}\subset {𝒜}$ is discrete (or with finite Jones index), we show that the set of quantum operations on ${𝒜}$ that leave ${ℬ}$ elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003) 7–30].

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore, a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell $p$-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory (pAQFT), in D’Angelo et al. [Ann. Henri Poinc.25 (2024) 2295–2352] it has been derived a Lorentzian version of a flow equation à la Wetterich, which can be used to study nonlinear, quantum scalar field theories on a globally hyperbolic spacetime. In this work, we show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half-Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we are able to provide strong evidences that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half-Minkowski spacetime.

An approach to renormalization of scalar fields on the Doplicher–Fredenhagen–Roberts (DFR) quantum spacetime is presented. The effective nonlocal theory obtained through the use of states of optimal localization for the quantum spacetime is reformulated in the language of (perturbative) Algebraic Quantum Field Theory. The structure of the singularities associated to the nonlocal kernel that codifies the effects of non-commutativity is analyzed using the tools of microlocal analysis.

Goal of this paper is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to construct models of quantum field theories (QFTs) on a general class of Lorentzian manifolds. Recently, this idea has been applied also to perturbative quantum gravity (QG), treated as an effective theory. The difficulty was to find the right notion of observables that would in an appropriate sense be diffeomorphism invariant. In this paper, I will outline a general framework that allows to quantize theories with local symmetries (this includes infinitesimal diffeomorphism transformations) with the use of the Batalin–Vilkovisky (BV) formalism. This approach has been successfully applied to effective QG in a recent paper by Brunetti, Fredenhagen and myself. In the same paper, we also proved perturbative background independence of the quantized theory, which is going to be discussed in the present work as well.

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions such as time-slicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen’s universal algebra.

We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations (PDEs), which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic ${\mathrm{\Phi }}_d^3$ model.

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to construct models of quantum field theories (QFTs) on a general class of Lorentzian manifolds. Recently, this idea has been applied also to perturbative quantum gravity (QG), treated as an effective theory. The difficulty was to find the right notion of observables that would in an appropriate sense be diffeomorphism invariant. In this paper, I will outline a general framework that allows to quantize theories with local symmetries (this includes infinitesimal diffeomorphism transformations) with the use of the Batalin–Vilkovisky (BV) formalism. This approach has been successfully applied to effective QG in a recent paper by Brunetti, Fredenhagen and myself. In the same paper, we also proved perturbative background independence of the quantized theory, which is going to be discussed in the present work as well.

In this paper it is given an explicit construction of the algebraic quantization of a massless scalar field on two prototypical examples of spacetime with boundary, usually related to the Casimir effect. This paper is based on Ref. 1.

The Bisognano-Wichmann property for local, Poincaré covariant nets of standard subspaces is discussed. We present a sufficient algebraic condition on the covariant representation ensuring Bisognano-Wichmann and Duality properties without further assumptions on the net. Our “modularity” condition holds for direct integrals of scalar massive and masselss representations. We conclude that in these cases the Bisognano-Wichmann property is much weaker than the Split property. Furthermore, we present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property.

Within the general setting of algebraic quantum field theory, a new approach to the analysis of the physical state space of a theory is presented; it covers theories with long range forces, such as quantum electrodynamics. Making use of the notion of charge class, which generalizes the concept of superselection sector, infrared problems are avoided. In fact, on this basis one can determine and classify in a systematic manner the proper charge content of a theory, the statistics of the corresponding states and their spectral properties. A key ingredient in this approach is the fact that in real experiments the arrow of time gives rise to a Lorentz invariant infrared cutoff of a purely geometric nature.