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Romeo Brunetti

Publications and source records attributed to Romeo Brunetti.

At least 19 recordsLinked to original sources
arXiv2025

Perturbative algebraic quantum field theory and beyond

In this review, we summarize the main ideas of perturbative algebraic quantum field theory, which is a rigorous framework combining some of the Haag-Kastler axioms with perturbative methods involving formal power series. It allows for the construction of interacting QFT models in four spacetime dimensions and works on arbitrary globally hyperbolic manifolds. This approach has also led to the development of a non-perturbative construction of local nets of C*-algebras for interacting theories, which will also be discussed at the end of this review.

math-ph↗
arXiv2025

Thermodynamical aspects of fermions in external electromagnetic fields

The thermodynamics of Dirac fields under the influence of external electromagnetic fields is studied. For perturbations which act only for finite time, the influence of the perturbation can be described by an automorphism which can be unitarily implemented in the GNS representations of KMS states, a result long known for the Fock representation. For time-independent perturbations, however, the time evolution cannot be implemented in typical cases, so the standard methods of quantum statistical mechanics do not apply. Instead we show that a smooth switching on of the external potential allows a comparison of the free and the perturbed time evolution, and approach to equilibrium, a possible existence of non-equilibrium stationary states (NESS) and Araki's relative entropy can be investigated. As a byproduct, we find an explicit formula for the relative entropy of gauge invariant quasi-free states.

math-ph↗
arXiv2024

M{\o}ller maps for Dirac fields in external backgrounds

In this paper we study the foundations of the algebraic treatment of classical and quantum field theories for Dirac fermions under external backgrounds following the initial contributions made by several collegues. The treatment is restricted to contractible spacetimes of globally hyperbolic nature in dimensions $d\ge 4$. In particular, we construct the classical M{\o}ller maps intertwining the configuration spaces of charged and uncharged fermions. In the last part, as a first step towards a quantization of the theory, we explore the combination of the classical M{\o}ller maps with Hadamard bidistributions and prove that they are involutive isomorphisms (algebraically and topologically) between suitable (formal) algebras of functionals (observables) over the configuration spaces of charged and uncharged Dirac fields.

math-ph↗
arXiv2022

Locally covariant approach to effective quantum gravity

Despite the fact that quantum gravity is non-renormalisable, a consistent and mathematically rigorous construction of a perturbation series is possible. This is based on the use of the Batalin-Vilkovisky-Becchi-Rouet-Stora-Tyutin formalism for gauge theories, the methods of perturbative algebraic quantum field theory and the principle of local covariance. The truncation of the series can be interpreted as an effective quantum field theory which provides predictions for observations at sufficiently small energy scales. Quantum cosmology can be seen as its lowest order expansion, and precision measurements on the cosmic microwave background yield the first empirical test of this approach to quantum gravity.

gr-qc↗
arXiv2022

Non-trivial bundles and Algebraic Classical Field Theory

Inspired by the recent algebraic approach to classical field theory, we propose a more general setting based on the manifold of smooth sections of a non-trivial fiber bundle. Central is the notion of observables over such sections, i.e. appropriate smooth functions on them. The kinematic will be further specified by means of the Peierls brackets, which in turn are defined via the causal propagators of linearized field equations. We shall compare the formalism we use with the more traditional ones.

math-ph↗
arXiv2022

Unitary, anomalous Master Ward Identity and its connections to the Wess-Zumino condition, BV formalism and $L_\infty$-algebras

The C*-algebraic construction of QFT by Buchholz and one of us relies on the causal structure of spacetime and a classical Lagrangian. In one of our previous papers we have introduced additional structure into this construction, namely an action of symmetries, which is related to fixing renormalisation conditions. This action characterizes anomalies and satisfies a cocycle condition which is summarized in the unitary anomalous Master Ward identity. Here (using perturbation theory) we show how this cocycle condition is related to the Wess-Zumino consistency relation and the consistency relation for the anomaly in the BV formalism, where the latter is the generalized Jacobi identity for the associated $L_\infty$-algebra.

math-ph↗
arXiv2022

The unitary Master Ward Identity: Time slice axiom, Noether's Theorem and Anomalies

The C*-algebraic formulation of generic interacting quantum field theories, recently presented by Detlev Buchholz and one of the authors (KF), is enriched by a unitary version of the Master Ward Identity, which was postulated some time ago by Franz Marc Boas, Ferdinand Brennecke and two of us (MD,KF). It is shown that the corresponding axiom implies the validity of the time slice axiom. Moreover, it opens the way for a new approach to Noether's Theorem where it yields directly the unitaries implementing the symmetries. It also unravels interesting aspects of the role of anomalies in quantum field theory.

math-ph↗
arXiv2020

Algebraic approach to Bose-Einstein Condensation in relativistic Quantum Field Theory. Spontaneous symmetry breaking and the Goldstone Theorem

We construct states describing Bose Einstein condensates at finite temperature for a relativistic massive complex scalar field with $|φ|^4$-interaction. We start with the linearised theory over a classical condensate and construct interacting fields by perturbation theory. Using the concept of thermal masses, equilibrium states at finite temperature can be constructed by the methods developed in arXiv:1306.6519 and arXiv:1502.02705. Here, the principle of perturbative agreement plays a crucial role. The apparent conflict with Goldstone's Theorem is resolved by the fact that the linearised theory breaks the $U(1)$ symmetry, hence the theorem applies only to the full series but not to the truncations at finite order which therefore can be free of infrared divergences.

math-ph↗
arXiv2019

Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields

We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field configurations, given by certain spaces of functionals which are studied here in depth. The analysis of such functionals is characterized by a combination of geometric, analytic and algebraic elements which (1) make our approach closer to quantum field theory, (2) allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and (3) provide a new pathway towards understanding dynamics. Particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations.

math-ph↗
arXiv2016

Cosmological perturbation theory and quantum gravity

It is shown how cosmological perturbation theory arises from a fully quantized perturbative theory of quantum gravity. Central for the derivation is a non-perturbative concept of gauge-invariant local observables by means of which perturbative invariant expressions of arbitrary order are generated. In particular, in the linearised theory, first order gauge-invariant observables familiar from cosmological perturbation theory are recovered. Explicit expressions of second order quantities are presented as well.

gr-qc↗
arXiv2015

Quantum gravity from the point of view of locally covariant quantum field theory

We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin-Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.

math-ph↗
arXiv2012

The Locality Axiom in Quantum Field Theory and Tensor Products of $C^*$-algebras

The prototype 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.

math-ph↗
arXiv2010

Modular dynamics in diamonds

We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem.

math-ph↗
arXiv2010

Renormalization and resolution of singularities

Since the seminal work of Epstein and Glaser it is well established that perturbative renormalization of ultraviolet divergences in position space amounts to extension of distributions onto diagonals. For a general Feynman graph the relevant diagonals form a nontrivial arrangement of linear subspaces. One may therefore ask if renormalization becomes simpler if one resolves this arrangement to a normal crossing divisor. In this paper we study the extension problem of distributions onto the wonderful models of de Concini and Procesi, which generalize the Fulton-MacPherson compactification of configuration spaces. We show that a canonical extension onto the smooth model coincides with the usual Epstein-Glaser renormalization. To this end we use an analytic regularization for position space. The 't Hooft identities relating the pole coefficients may be recovered from the stratification, and Zimmermann's forest formula is encoded in the geometry of the compactification. Consequently one subtraction along each irreducible component of the divisor suffices to get a finite result using local counterterms. As a corollary, we identify the Hopf algebra of at most logarithmic Feynman graphs in position space, and discuss the case of higher degree of divergence.

hep-th↗
arXiv2009

Time in quantum physics: From an external parameter to an intrinsic observable

In the Schroedinger equation, time plays a special role as an external parameter. We show that in an enlarged system where the time variable denotes an additional degree of freedom, solutions of the Schroedinger equation give rise to weights on the enlarged algebra of observables. States in the associated GNS representation correspond to states on the original algebra composed with a completely positive unit preserving map. Application of this map to the functions of the time operator on the large system delivers the positive operator valued maps which were previously proposed by two of us as time observables. As an example we discuss the application of this formalism to the Wheeler-DeWitt theory of a scalar field on a Robertson-Walker spacetime.

math-ph↗
arXiv2009

Perturbative Algebraic Quantum Field Theory and the Renormalization Groups

A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low dimensional theories and of non-polynomial interactions. We discuss the connection between the Stueckelberg-Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales . In particular we relate the approach to renormalization in terms of Polchinski's Flow Equation to the Epstein-Glaser method. We also show that the renormalization group in the sense of Gell-Mann-Low (which characterizes the behaviour of the theory under the change of all scales) is a 1-parametric subfamily of the Stueckelberg-Petermann group and that this subfamily is in general only a cocycle. Since the algebraic structure of the Stueckelberg-Petermann group does not depend on global quantities, this group can be formulated in the (algebraic) adiabatic limit without meeting any infrared divergencies. In particular we derive an algebraic version of the Callan-Symanzik equation and define the beta-function in a state independent way.

math-ph↗
arXiv2009

Topological features of massive bosons on two dimensional Einstein space-time

In this paper we tackle the problem of constructing explicit examples of topological cocycles of Roberts' net cohomology, as defined abstractly by Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum field theory on the two dimensional Einstein cylinder. After deriving some crucial results of the algebraic framework of quantization, we address the problem of the construction of the topological cocycles. All constructed cocycles lead to unitarily equivalent representations of the fundamental group of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces). The construction is carried out using only Cauchy data and related net of local algebras on the circle.

gr-qc↗