arXiv subjects

Fabio Ciolli

Publications and source records attributed to Fabio Ciolli.

18 recordsLinked to original sources

arXiv2025

Charges in light cones and quenched infrared radiation

The creation of electrically charged states and the resulting electromagnetic fields are considered in space-time regions in which such experiments can actually be carried out, namely in future-directed light cones. Under the simplifying assumption of external charges, charged states are formed from neutral pairs of opposite charges, with one charge being shifted to light-like infinity. It thereby escapes observation. Despite the fact that this charge moves asymptotically at the speed of light, the resulting electromagnetic field has a well-defined energy operator that is bounded from below. Moreover, due to the spatiotemporal restrictions, the transverse electromagnetic field (the radiation) has no infrared singularities in the light cone. They are quenched and the observed radiation can be described by states in the Fock space of photons. The longitudinal field between the charges (giving rise to Gauss's law) disappears for inertial observers in an instant. This is consistent with the fact that there is no evidence for the existence of longitudinal photons. The results show that the restrictions of operations and observations to light cones, which are dictated by the arrow of time, amount to a Lorentz-invariant infrared cutoff.

math-ph↗

arXiv2024

Relative entropy and curved spacetimes

Given any half-sided modular inclusion of standard subspaces, we show that the entropy function associated with the decreasing one-parameter family of translated standard subspaces is convex for any given (not necessarily smooth) vector in the underlying Hilbert space. In second quantisation, this infers the convexity of the vacuum relative entropy with respect to the translation parameter of the modular tunnel of von Neumann algebras. This result allows us to study the QNEC inequality for coherent states in a free Quantum Field Theory on a stationary curved spacetime, given a KMS state. To this end, we define wedge regions and appropriate (deformed) subregions. Examples are given by the Schwarzschild spacetime and null translated subregions with respect to the time translation Killing flow. More generally, we define wedge and stripe regions on a globally hyperbolic spacetime, so to have non trivial modular inclusions of von Neumann algebras, and make our analysis in this context.

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arXiv2023

Where charged sectors are localizable: a viewpoint from covariant cohomology

Given a Haag-Kastler net on a globally hyperbolic spacetime, one can consider a family of regions where quantum charges are supposed to be localized. Assuming that the net fulfils certain minimal properties (factoriality of the global observable algebra and relative Haag duality), we give a geometric criterion that the given family must fulfil to have a superselection structure with charges localized on its regions. Our criterion is fulfilled by all the families used in the theory of sectors (double cones, spacelike cones, diamonds, hypercones). In order to take account of eventual spacetime symmetries, our superselection structures are constructed in terms of covariant charge transporters, a novel cohomological approach generalizing that introduced by J.E. Roberts. In the case of hypercones, with the forward light cone as an ambient spacetime, we obtain a superselection structure with Bose-Fermi parastatistics and particle-antiparticle conjugation. It could constitute a candidate for a different description of the charged sectors introduced by Buchholz and Roberts for theories including massless particles.

math-ph↗

arXiv2023

The information in a wave

We provide the notion of entropy for a classical Klein-Gordon real wave, that we derive as particular case of a notion entropy for a vector in a Hilbert space with respect to a real linear subspace. We then consider a localised automorphism on the Rindler spacetime, in the context of a free neutral Quantum Field Theory, that is associated with a second quantised wave, and we explicitly compute its entropy $S$, that turns out to be given by the entropy of the associated classical wave. Here $S$ is defined as the relative entropy between the Rindler vacuum state and the corresponding sector state (coherent state). By $λ$-translating the Rindler spacetime into itself along the upper null horizon, we study the behaviour of the corresponding entropy $S(λ)$. In particular, we show that the QNEC inequality in the form $\frac{d^2}{dλ^2}S(λ)\geq 0$ holds true for coherent states, because $\frac{d^2}{dλ^2}S(λ)$ is the integral along the space horizon of a manifestly non-negative quantity, the component of the stress-energy tensor in the null upper horizon direction.

math-ph↗

arXiv2023

On the Thermodynamics of Particles Obeying Monotone Statistics

The aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor $n!$ (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of "exclusion principle" analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected.

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arXiv2022

Gauss's law, the manifestations of gauge fields, and their impact on local observables

Within the framework of the universal algebra of the electromagnetic field, the impact of globally neutral configurations of external charges on the field is analyzed. External charges are not affected by the field, but they induce localized automorphisms of the universal algebra. Gauss's law implies that these automorphisms cannot be implemented by unitary operators involving only the electromagnetic field, they are outer automorphisms. The missing degrees of freedom can be incorporated in an enlargement of the universal algebra, which can concretely be represented by exponential functions of gauge fields and an abelian algebra describing the external charges. In this manner, gauge fields manifest themselves in the framework of gauge invariant observables. The action of the automorphisms on the vacuum state gives rise to representations of the electromagnetic field with vanishing global charge, which are locally disjoint from the vacuum representation. This feature disappears in the enlarged universal algebra of the electromagnetic field. The energy content of the states is well defined in both cases and bounded from below. The passage from these globally neutral states to charged states and the determination of their energy content are also being discussed.

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arXiv2022

Spectral actions for q-particles and their asymptotics

For spectral actions consisting of the average number of particles and arising from open systems made of general free $q$-particles (including Bose, Fermi and classical ones corresponding to $q=\pm 1$ and $0$, respectively) in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off. We treat both relevant situations relative to massless and non relativistic massive particles, where the natural cut-off is $1/β=k_{\rm B}T$ and $1/\sqrtβ$, respectively. We show that the massless situation enjoys less regularity properties than the massive one. We also treat in some detail the relativistic massive case for which the natural cut-off is again $1/β$. We then consider the passage to the continuum describing infinitely extended open systems in thermal equilibrium, by also discussing the appearance of condensation phenomena occurring for Bose-like $q$-particles, $q\in(0,1]$. We then compare the arising results for the finite volume situation (discrete spectrum) with the corresponding infinite volume one (continuous spectrum).

math-ph↗

arXiv2022

On the Thermodynamics of the q-Particles

Since the grand partition function $Z_q$ for the so-called $q$-particles (i.e.,\ quons), $q\in(-1,1)$, cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for $q=0$, and its $q$-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to $q\in [-1,1]$. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e.,\ $q=0$) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor $1/n!$ in the $n$ term of the power series expansion with respect to the fugacity $z$. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons $q\in(0,1)$.

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arXiv2022

Modular spectral triples and deformed Fredholm modules

In the setting of non-type $\ty{II_1}$ representations, we propose a definition of {\it deformed Fredholm module} $\big[D_\ct|D_\ct|^{-1}\,,\,{\bf\cdot}\,\big]_\ct$ for a modular spectral triple $\ct$, where $D_\ct$ is the deformed Dirac operator. $D_\ct$ is assumed to be invertible for the sake of simplicity, and its domain is an "essential" operator system $\ce_\ct$. According to such a definition, we obtain $\big[D_\ct|D_\ct|^{-1}\,,\,{\bf\cdot}\,\big]_\ct=|D_\ct|^{-1}d_\ct(\,{\bf\cdot}\,)+d_\ct(\,{\bf\cdot}\,)|D_\ct|^{-1}$, where $d_\ct$ is the deformed derivation associated to $D_\ct$. Since the "quantum differential" $1/|D_\ct|$ appears in a symmetric position, such a definition of Fredholm module differs from the usual one even in the undeformed case, that is in the tracial case. Therefore, it seems to be more suitable for the investigation of noncommutative manifolds in which the nontrivial modular structure might play a crucial role. We show that all models in \cite{FS} of non-type $\ty{II_1}$ representations of noncommutative 2-tori indeed provide modular spectral triples, and in addition deformed Fredholm modules according to the definition proposed in the present paper. Since the detailed knowledge of the spectrum of the Dirac operator plays a fundamental role in spectral geometry, we provide a characterisation of eigenvalues and eigenvectors of the deformed Dirac operator $D_\ct$ in terms of the periodic solutions of a particular class of eigenvalue Hill equations.

math.OA↗

arXiv2021

The universal algebra of the electromagnetic field III. Static charges and emergence of gauge fields

A universal C*-algebra of gauge invariant operators is presented, describing the electromagnetic field as well as operations creating pairs of static electric charges having opposite signs. Making use of Gauss' law, it is shown that the string-localized operators, which necessarily connect the charges, induce outer automorphisms of the algebra of the electromagnetic field. Thus they carry additional degrees of freedom which cannot be created by the field. It reveals the fact that gauge invariant operators encode information about the presence of non-observable gauge fields underlying the theory. Using the Gupta-Bleuler formalism, concrete implementations of the outer automorphisms by exponential functions of the gauge fields are presented. These fields also appear in unitary operators inducing the time translations in the resulting representations of the universal algebra.

math-ph↗

arXiv2019

On string-localized potentials and gauge fields

A recent idea, put forward by Mund, Rehren and Schroer, is discussed; it suggests that in gauge quantum field theory one can replace the point-localized gauge fields by string-localized vector potentials built from gauge invariant observables and a principle of string-independence. Based on a kinematical model, describing unmovable (static) fields carrying opposite charges, it is shown that these string-localized potentials cannot be used for the description of the gauge bridges between electrically charged fields. These bridges are needed in order to ensure the validity of Gauss' law. This observation does not preclude the existence of Poincaré invariant theories, describing the coupling of string-localized gauge invariant potentials to matter fields. But these potentials are not a full-fledged substitute for the gauge fields in ``usual'' quantum electrodynamics.

hep-th↗

arXiv2018

Linking numbers in local quantum field theory

Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article.

math-ph↗

arXiv2016

The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields

Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of "spacelike linearity". Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.

math-ph↗

arXiv2015

The universal C*-algebra of the electromagnetic field

A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of this field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.

math-ph↗

arXiv2013

QED Representation for the Net of Causal Loops

The present work tackles the existence of local gauge symmetries in the setting of Algebraic Quantum Field Theory (AQFT). The net of causal loops, previously introduced by the authors, is a model independent construction of a covariant net of local C*-algebras on any 4-dimensional globally hyperbolic spacetime, aimed to capture some structural properties of any reasonable quantum gauge theory. In fact, representations of this net can be described by causal and covariant connection systems, and the local gauge transformations arise as maps between equivalent connection systems. The present paper completes these abstract results, realizing QED as a representation of the net of causal loops in Minkowski spacetime. More precisely, we map the quantum electromagnetic field Fμν, not free in general, into a representation of the net of causal loops and show that the corresponding connection system and local gauge transformations find a counterpart in terms of Fμν.

math-ph↗

arXiv2011

Causal posets, loops and the construction of nets of local algebras for QFT

We provide a model independent construction of a net of C*-algebras satisfying the Haag-Kastler axioms over any spacetime manifold. Such a net, called the net of causal loops, is constructed by selecting a suitable base K encoding causal and symmetry properties of the spacetime. Considering K as a partially ordered set (poset) with respect to the inclusion order relation, we define groups of closed paths (loops) formed by the elements of K. These groups come equipped with a causal disjointness relation and an action of the symmetry group of the spacetime. In this way the local algebras of the net are the group C*-algebras of the groups of loops, quotiented by the causal disjointness relation. We also provide a geometric interpretation of a class of representations of this net in terms of causal and covariant connections of the poset K. In the case of the Minkowski spacetime, we prove the existence of Poincaré covariant representations satisfying the spectrum condition. This is obtained by virtue of a remarkable feature of our construction: any Hermitian scalar quantum field defines causal and covariant connections of K. Similar results hold for the chiral spacetime $S^1$ with conformal symmetry.

math-ph↗

arXiv2010

Massless scalar free Field in 1+1 dimensions, II: Net Cohomology and Completeness of Superselection Sectors

As an application of Roberts' cohomology (net cohomology), we prove the completeness of the DHR sectors of the local observables of the model in the title, detailed in [8]. This result is achieved via the triviality of the net 1-cohomology, with values in the local fields, enhancing the Roberts' methods to the case of anyonic Weyl nets, not satisfying the split property. We take advantage of using different causal index sets for the nets involved. The presence of anyonic commutation relations is treated introducing the notion of nets graded by a generic group, and the related properties of graded locality and graded duality. As a further result, we obtain the description of twisted and untwisted sectors of the model as two symmetric subcategories of a W*-braided category, whose objects are the same as the dual category of the compact Abelian group of the gauge symmetry. The work also furnish some hints for the analysis of the sector structure of generic models on different spacetimes.

math-ph↗

arXiv2009

Massless scalar free Field in 1+1 dimensions I: Weyl algebras Products and Superselection Sectors

This is the first of two papers on the superselection sectors of the conformal model in the title, in a time zero formulation. A classification of the sectors of the net of observables as restrictions of solitonic (twisted) and non-solitonic (untwisted) sector automorphisms of proper extensions of the observable net is given. All of them are implemented by the elements of a field net in a non-regular vacuum representation and the existence of a global compact Abelian gauge group is proved. A non-trivial center in the fixed-point net of this gauge group appears, but in an unphysical representation and reducing to the identity in the physical one. The completeness of the described superselection structure, to which the second paper is devoted, is shown in terms of Roberts' net cohomology. Some general features of physical field models defined by twisted cross products of Weyl algebras in non-regular representations are also presented.

math-ph↗