arXiv subjects
Publications and source records attributed to Silvio Paolo Sorella.
We elaborate on the recent proposal of employing unitary operators in Quantum Mechanics. The Bell and Mermin inequalities for entangled coherent states are scrutinized by making use of the unitary displacement operators. A violation of the Mermin inequality close to the maximum allowed value is reported, in agreement with the existing literature.
The violation of the Bell-CHSH inequality for bipartite systems is discussed by making use of the pseudospin operators which enable us to group all modes of the Hilbert space of the system in pairs. We point out that a single pair can be already employed to perform a test of the Bell-CHSH inequality in order to check out its violation. The mechanism is illustrated with the help of $N00N$ states as well as with coherent and squeezed states.
We discuss the possibility to obtain a massive Landau gauge, based on the local composite operator (LCO) effective action framework combined with the Zimmerman reduction of couplings prescription. As a way to deal with the gauge ambiguity, we check that the ghost propagator remains positive, a necessary condition for gluon field configurations beyond the Gribov region to be negligible. We pay attention to the BRST invariance of the construction, allowing for a future generalization to a class of massive linear covariant gauges. As a litmus test, we compare our predictions to the lattice data for the two-point functions in Landau gauge introducing the "Dynamically Infrared-Safe" renormalization scheme, including the renormalization group optimization of both the gap equation and the two-point functions. We also discuss the relation to and differences with the Curci-Ferrari model, the usefulness of which in providing an effective perturbative description of non-perturbative Yang-Mills theories became clear during recent years.
A relativistic Quantum Field Theory framework is devised for Mermin's inequalities. By employing smeared Dirac spinor fields, we are able to introduce unitary operators which create, out of the Minkowski vacuum $| 0 \rangle$, GHZ-type states. In this way, we are able to obtain a relation between the expectation value of Mermin's operators in the vacuum and in the GHZ-type states. We show that Mermin's inequalities turn out to be maximally violated when evaluated on these states.
We present a set of BRST invariant composite operators in the $SU\left(2\right)\times U\left(1\right)$ Higgs model which exhibit an overlap with the observable scalar and vector particle states of the theory. Some of these operators are non-local in the standard formulation of the Higgs model, however, we show that they can be localized using the Stuckelberg trick, this without changing the physical content. These operators provide thus a BRST invariant resolution of the physical spectrum of the theory, thereby giving the next step in constructing a manifestly BRST invariant formulation of the particle spectrum of the Electroweak and Standard model.
We point out that, when the dimension of the Hilbert space is greater than two, Bell's operators entering the Bell-CHSH inequality exhibit unitarily inequivalent representations. Although the Bell-CHSH inequality turns out to be violated, the size of the violation is different for different representations, the maximum violation being given by Tsirelson's bound. The feature relies on a pairing mechanism between the modes of the Hilbert space of the system.
By employing a free scalar Quantum Field Theory model previously introduced \cite{Peruzzo:2022pwv}, we attempt at formulating the Bell-CHSH inequality within the Feynman path integral. This possibility relies on the observation that the Bell-CHSH inequality exhibits a natural extension to Quantum Field Theory in such a way that it is compatible with the time ordering $T$. By treating the Feynman propagator as a distribution and by introducing a suitable localizing set of compact support smooth test functions, we work out the path integral setup for the Bell-CHSH inequality, recovering the same results of the canonical quantization.
We study the CHSH inequality for a system of two spin $j$ particles, for generic $j$. The CHSH operator is constructed using a set of unitary, Hermitian operators $\left\{ A_{1},A_{2},B_{1},B_{2}\right\} $. The expectation value of the CHSH operator is analyzed for the singlet state $\left|ψ_{s}\right\rangle $. Being $\left|ψ_{s}\right\rangle $ an entangled state, a violation of the CHSH inequality compatible with Tsirelson's bound is found. Although the construction employed here differs from that of [1], full agreement is recovered.
We present an investigation of the $CHSH$ inequality within a relativistic quantum field theory model built up with a pair of free massive scalar fields $(φ_A, φ_B)$ where, as it is customary, the indices $(A,B)$ refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A $CHSH$ type correlator is constructed and evaluated in the Fock vacuum by means of the canonical quantization. Although the observed violation of the $CHSH$ inequality turns out to be rather small as compared to Tsirelson's bound of Quantum Mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as: the Higgs models, for which local $BRST$ invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to $BRST$ invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant $BRST$ environment.
The renormalization properties of two local gauge invariant composite operators $(O,R^a_μ)$ corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the $SU(2)$ Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean space-time. The present analysis generalizes earlier results presented in the case of the $U(1)$ Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators $R^a_μ$ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization $Z$-factors of the corresponding external sources, needed to define the operators $R^a_μ$ at the quantum level, do not receive any quantum corrections. Another consistency feature of our analysis is that the longitudinal component of the two-point correlation function $\langle R^a_μ(p) R^b_ν(-p) \rangle$ exhibits only a tree level non-vanishing contribution which, moreover, is momentum independent, being thus not associated to any physical propagating mode. Finally, we point out that the renowned non-renormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in presence of the Higgs field.
Landau-Khalatnikov-Fradkin transformations (LKFTs) yield the gauge dependence of correlation functions within the class of linear covariant gauges. We derive the LKFT for the quark propagator and explicitly evaluate it up to the two loop level in the chiral limit. Although the number of diagrams to be evaluated is significantly larger than with the conventional computational scheme, the diagrams are simpler in nature, thereby leading to a considerably faster evaluation of the gauge dependent part than naively expected. Finally, we also resum the LKFT generated terms and compare our results with earlier work in the literature.
The Landau-Khalatnikov-Fradkin transformations (LKFTs) represent an important tool for probing the gauge dependence of the correlation functions within the class of linear covariant gauges. Recently these transformations have been derived from first principles in the context of non-Abelian gauge theory (QCD) introducing a gauge invariant transverse gauge field expressible as an infinite power series in a Stueckelberg field. In this work we explicitly calculate the transformation for the gluon propagator, reproducing its dependence on the gauge parameter at the one loop level and elucidating the role of the extra fields involved in this theoretical framework. Later on, employing a unifying scheme based upon the BRST symmetry and a resulting generalized Slavnov-Taylor identity, we establish the equivalence between the LKFTs and the Nielsen identities which are also known to connect results in different gauges.
The spectral properties of a set of local gauge-invariant composite operators are investigated in the $U(1)$ Higgs model quantized in the 't Hooft $R_ξ$ gauge. These operators enable us to give a gauge-invariant description of the spectrum of the theory, thereby surpassing certain incommodities when using the standard elementary fields. The corresponding two-point correlation functions are evaluated at one-loop order and their spectral functions are obtained explicitly. As expected, the above mentioned correlation functions are independent from the gauge parameter $ξ$, while exhibiting positive spectral densities as well as gauge-invariant pole masses corresponding to the massive photon and Higgs physical excitations.
The two-point gauge correlation function in Yang--Mills--Chern--Simons theory in three dimensional Euclidean space is analysed by taking into account the non-perturbative effects of the Gribov horizon. In this way, we are able to describe the confinement and de-confinement regimes, which naturally depend on the topological mass and on the gauge coupling constant of the theory.
An equivalent formulation of the Gribov-Zwanziger theory accounting for the gauge fixing ambiguity in the Landau gauge is presented. The resulting action is constrained by a Slavnov-Taylor identity stemming from a nilpotent exact BRST invariance which is spontaneously broken due to the presence of the Gribov horizon. This spontaneous symmetry breaking can be described in a purely algebraic way through the introduction of a pair of auxiliary fields which give rise to a set of linearly broken Ward identities. The Goldstone sector turns out to be decoupled. The underlying exact nilpotent BRST invariance allows to employ BRST cohomology tools within the Gribov horizon to identify renormalizable extensions of gauge invariant operators. Using a simple toy model and appropriate Dirac bracket quantization, we discuss the time-evolution invariance of the operator cohomology. We further comment on the unitarity issue in a confining theory, and stress that BRST cohomology alone is not sufficient to ensure unitarity, a fact, although well known, frequently ignored.
A few issues on gluon confinement are addressed with the help of a renormalizable gauge model obtained by introducing a replica of the Faddeev-Popov action and a soft breaking of the BRST symmetry. Confinement turns out to be encoded in the spectral properties of the corresponding correlation functions. While the propagators of the elementary fields have no interpretation in terms of physical excitations, examples of local composite operators whose two-point correlation functions possess a spectral representation with positive spectral density can be introduced. These composite operators turn out to be left invariant by the BRST transformations, a feature which has strong consequences on their renormalizability properties. Moreover, they display a direct interpretation in terms of i-particles [1], which are the unphysical modes corresponding to a confining propagator of the Gribov type. A possible way to take into account the effects of the Gribov copies is also outlined.
This paper presents a complete algebraic analysis of the renormalizability of the $d=4$ operator $F^2_{μν}$ in the Gribov-Zwanziger (GZ) formalism as well as in the Refined Gribov-Zwanziger (RGZ) version. The GZ formalism offers a way to deal with gauge copies in the Landau gauge. We explicitly show that $F^2_{μν}$ mixes with other $d=4$ gauge variant operators, and we determine the mixing matrix $Z$ to all orders, thereby only using algebraic arguments. The mixing matrix allows us to uncover a renormalization group invariant including the operator $F^2_{μν}$. With this renormalization group invariant, we have paved the way for the study of the lightest scalar glueball in the GZ formalism. We discuss how the soft breaking of the BRST symmetry of the GZ action can influence the glueball correlation function. We expect non-trivial mass scales, inherent to the GZ approach, to enter the pole structure of this correlation function.
Recent lattice data have reported an infrared suppressed, positivity violating gluon propagator which is nonvanishing at zero momentum and a ghost propagator which is no longer enhanced. This paper discusses how to obtain analytical results which are in qualitative agreement with these lattice data within the Gribov-Zwanziger framework. This framework allows one to take into account effects related to the existence of gauge copies, by restricting the domain of integration in the path integral to the Gribov region. We elaborate to great extent on a previous short paper by presenting additional results, also confirmed by the numerical simulations. A detailed discussion on the soft breaking of the BRST symmetry arising in the Gribov-Zwanziger approach is provided.