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Patrizia Vitale

Publications and source records attributed to Patrizia Vitale.

At least 55 records · Page 3Linked to original sources
arXiv2018

Fisher Metric, Geometric Entanglement and Spin Networks

Starting from recent results on the geometric formulation of quantum mechanics, we propose a new information geometric characterization of entanglement for spin network states in the context of quantum gravity. For the simple case of a single-link fixed graph (Wilson line), we detail the construction of a Riemannian Fisher metric tensor and a symplectic structure on the graph Hilbert space, showing how these encode the whole information about separability and entanglement. In particular, the Fisher metric defines an entanglement monotone which provides a notion of distance among states in the Hilbert space. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We further extend such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network graph characterized by the bipartite unfolding of an Intertwiner state. Our analysis confirms the interpretation of spin network bonds as a result of entanglement and to regard the same spin network graph as an information graph, whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.

gr-qc↗
arXiv2017

Tomographic Reconstruction of Quantum Metrics

In the framework of quantum information geometry we investigate the relationship between monotone metric tensors uniquely defined on the space of quantum tomograms, once the tomographic scheme chosen, and monotone quantum metrics on the space of quantum states, classified by operator monotone functions, according to Petz classification theorem. We show that different metrics can be related through a change of the tomographic map and prove that there exists a bijective relation between monotone quantum metrics associated with different operator monotone functions. Such bijective relation is uniquely defined in terms of solutions of a first order second degree differential equation for the parameters of the involved tomographic maps. We first exhibit an example of a non-linear tomographic map which connects a monotone metric with a new one which is not monotone. Then we provide a second example where two monotone metrics are uniquely related through their tomographic parameters.

math-ph↗
arXiv2017

Metric on the space of quantum states from relative entropy. Tomographic reconstruction

In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher- Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.

quant-ph↗
arXiv2016

The Gribov problem in Noncommutative QED

It is shown that in the noncommutative version of QED (NCQED) Gribov copies induced by the noncommutativity of space-time appear in the Landau gauge. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.

hep-th↗
arXiv2015

$κ$-Minkowski star product in any dimension from symplectic realization

We derive an explicit expression for the star product reproducing the $κ$-Minkowski Lie algebra in any dimension $n$. The result is obtained by suitably reducing the Wick-Voros star product defined on $\mathbb{C}^{d}_θ$ with $n=d+1$. It is thus shown that the new star product can be obtained from a Jordanian twist.

math-ph↗
arXiv2015

Green's Functions for Translation Invariant Star Products

We calculate the Green functions for a scalar field theory with quartic interactions for which the fields are multiplied with a generic translation invariant star product. Our analysis involves both noncommutative products, for which there is the canonical commutation relation among coordinates, and nonlocal commutative products. We give explicit expressions for the one-loop corrections to the two and four point functions. We find that the phenomenon of ultraviolet/infrared mixing is always a consequence of the presence of noncommuting variables. The commutative part of the product does not have the mixing.

hep-th↗
arXiv2014

Self-dual road to noncommutative gravity with twist: a new analysis

The field equations of noncommutative gravity can be obtained by replacing all exterior products by twist-deformed exterior products in the action functional of general relativity, and are here studied by requiring that the torsion 2-form should vanish, and that the Lorentz-Lie-algebra- valued part of the full connection 1-form should be self-dual. Other two conditions, expressing self-duality of a pair 2-forms occurring in the full curvature 2-form, are also imposed. This leads to a systematic solution strategy, here displayed for the first time, where all parts of the connection 1-form are first evaluated, hence the full curvature 2-form, and eventually all parts of the tetrad 1-form, when expanded on the basis of γ-matrices. By assuming asymptotic expansions which hold up to first order in the noncommutativity matrix in the neighbourhood of the vanishing value for noncommutativity, we find a family of self-dual solutions of the field equations. This is generated by solving first a inhomogeneous wave equation on 1-forms in a classical curved spacetime (which is itself self-dual and solves the vacuum Einstein equations), subject to the Lorenz gauge condition. In particular, when the classical undeformed geometry is Kasner spacetime, the above scheme is fully computable out of solutions of the scalar wave equation in such a Kasner model.

hep-th↗
arXiv2014

Matrix Bases for Star Products: a Review

We review the matrix bases for a family of noncommutative $\star$ products based on a Weyl map. These products include the Moyal product, as well as the Wick-Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity.

hep-th↗
arXiv2014

Quantum gauge theories on noncommutative 3-d space

We consider a class of gauge invariant models on the noncommutative space $\mathbb{R}^3_λ$, a deformation of $\mathbb{R}^3$. Focusing on massless models with no linear $A_i$ dependence, we obtain noncommutative gauge models for which the computation of the propagator can be done in a convenient gauge. We find that the infrared singularity of the massless propagator disappears in the computation of the correlation functions. We show that massless gauge invariant models on $\mathbb{R}^3_λ$ have quantum instabilities of the vacuum, signaled by the occurrence of non vanishing 1-point functions for some but not all of the components of the gauge potential. The tadpole contribution to the effective action cannot be interpreted as a standard $σ$-term. Its global symmetry does not fit with the one of the classical action, reminiscent to an explicit global symmetry breaking term.

hep-th↗
arXiv2014

Noncommutative field theory on $\mathbb{R}^3_λ$

We consider the noncommutative space $\mathbb{R}^3_λ$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a foliation of $\mathbb{R}^3$ into fuzzy spheres. We first review the construction of a natural matrix basis adapted to $\mathbb{R}^3_λ$. We thus consider the problem of defining a new Laplacian operator for the deformed algebra. We propose an operator which is not of Jacobi type. The implication for field theory of the new Laplacian is briefly discussed.

hep-th↗
arXiv2013

The Seiberg-Witten map for non-commutative pure gravity and vacuum Maxwell theory

In this paper the Seiberg-Witten map is first analyzed for non-commutative Yang-Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg-Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space-time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, when the Seiberg-Witten map is eventually inserted. In the attempt of understanding whether or not this is a peculiar property of gravity, in the second part of the paper, the Seiberg-Witten map is considered in the simpler case of Maxwell theory in vacuum in the absence of charges and currents. Once more, no obvious solution of the non-commutative field equations is found, unless the electromagnetic potential depends in a very special way on the wave vector.

hep-th↗
arXiv2013

Noncommutative gauge theories on $\mathbb{R}^2_θ$ as matrix models

We study a class of noncommutative gauge theory models on 2-dimensional Moyal space from the viewpoint of matrix models and explore some related properties. Expanding the action around symmetric vacua generates non local matrix models with polynomial interaction terms. For a particular vacuum, we can invert the kinetic operator which is related to a Jacobi operator. The resulting propagator can be expressed in terms of Chebyschev polynomials of second kind. We show that non vanishing correlations exist at large separations. General considerations on the kinetic operators stemming from the other class of symmetric vacua, indicates that only one class of symmetric vacua should lead to fast decaying propagators. The quantum stability of the vacuum is briefly discussed.

hep-th↗
arXiv2013

Noncommutative field theories on $R^3_λ$: Towards UV/IR mixing freedom

We consider the noncommutative space $\mathbb{R}^3_λ$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a "foliation" of $\mathbb{R}^3$ into fuzzy spheres. We first construct a natural matrix base adapted to $\mathbb{R}^3_λ$. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the radial part. We find that the resulting theory is finite to all orders in perturbation expansion.

hep-th↗
arXiv2012

On the $\star$-product quantization and the Duflo map in three dimensions

We analyze the star product induced on the algebra of functions on R^3 by a suitable reduction of the Moyal product defined on F(R^4). This is obtained through the identification of R^3 with the dual of a three dimensional Lie algebra. We consider the su(2) case, exhibit a matrix basis and realize the algebra of functions on its dual in such a basis. The relation to the Duflo map is discussed. As an application to quantum mechanics we compute the spectrum of the hydrogen atom.

hep-th↗
arXiv2012

Gauge and Poincare' Invariant Regularization and Hopf Symmetries

We consider the regularization of a gauge quantum field theory following a modification of the Polchinski proof based on the introduction of a cutoff function. We work with a Poincare' invariant deformation of the ordinary point-wise product of fields introduced by Ardalan, Arfaei, Ghasemkhani and Sadooghi, and show that it yields, through a limiting procedure of the cutoff functions, to a regularized theory, preserving all symmetries at every stage. The new gauge symmetry yields a new Hopf algebra with deformed co-structures, which is inequivalent to the standard one.

hep-th↗
arXiv2012

Aspects of Group Field Theory

I review the basic ingredients of discretized gravity which motivate the introduction of Group Field Theory. Thus I describe the GFT formulation of some models and conclude with a few remarks on the emergence of noncommutative structures in such models.

hep-th↗
arXiv2010

Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models

We investigate the group field theory formulation of the EPRL/FK spin foam models. These models aim at a dynamical, i.e. non-topological formulation of 4D quantum gravity. We introduce a saddle point method for general group field theory amplitudes and compare it with existing results, in particular for a second order correction to the EPRL/FK propagator.

gr-qc↗