arXiv subjects

Giuseppe Ruzzi

Publications and source records attributed to Giuseppe Ruzzi.

At least 19 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↗

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

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.

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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.

math-ph↗

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.

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arXiv2021

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.

math-ph↗

arXiv2019

Aharonov-Bohm superselection sectors

We show that the Aharonov-Bohm effect finds a natural description in the setting of QFT on curved spacetimes in terms of superselection sectors of local observables. The extension of the analysis of superselection sectors from Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the presence of a new quantum number labeling charged superselection sectors. In the present paper we show that this "topological" quantum number amounts to the presence of a background flat potential which rules the behaviour of charges when transported along paths as in the Aharonov-Bohm effect. To confirm these abstract results we quantize the Dirac field in presence of a background flat potential and show that the Aharonov-Bohm phase gives an irreducible representation of the fundamental group of the spacetime labeling the charged sectors of the Dirac field. We also show that non-Abelian generalizations of this effect are possible only on space-times with a non-Abelian fundamental group.

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↗

arXiv2014

The K-homology of nets of C*-algebras

Let X be a space, intended as a possibly curved spacetime, and A a precosheaf of C*-algebras on X. Motivated by algebraic quantum field theory, we study the Kasparov and Theta-summable K-homology of A interpreting them in terms of the holonomy equivariant K-homology of the associated C*-dynamical system. This yields a characteristic class for K-homology cycles of A with values in the odd cohomology of X, that we interpret as a generalized statistical dimension.

math.OA↗

arXiv2014

The C(X)-algebra of a net and index theory

Given a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra. When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.

math.OA↗

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↗

arXiv2011

Representations of nets of C*-algebras over S^1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account of the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C*-algebras over S^1 admits faithful representations, and when the net is covariant under Diff(S^1), it admits representations covariant under any amenable subgroup of Diff(S^1).

math-ph↗

arXiv2011

A new light on nets of C*-algebras and their representations

The present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems having as group the fundamental group of the poset. Any net of C*-algebras embeds into a unique C*-net bundle, the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected posets. This allows a classification of nets; in particular, we call injective those nets having a faithful embedding into the enveloping net bundle. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized Cech cocycle of the net, and this allows us to give examples of nets exhausting the above classification. Using the results of this paper we shall show, in a forthcoming paper, that any conformal net over S^1 is injective.

math.OA↗

arXiv2008

Quantum charges and spacetime topology: The emergence of new superselection sectors

In which is developed a new form of superselection sectors of topological origin. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*-algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts' cohomological analysis to the case where 1--cocycles bear non trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1-cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1-cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part resembles much what in literature are known as geometric phases. Indeed, by the very geometrical origin of the 1-cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations.

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