arXiv subjects

Leonardo Castellani

Publications and source records attributed to Leonardo Castellani.

At least 19 recordsLinked to original sources

arXiv2024

Space and time correlations in quantum histories

The formalism of generalized quantum histories allows a symmetrical treatment of space and time correlations, by taking different traces of the same history density matrix. We recall how to characterize spatial and temporal entanglement in this framework. An operative protocol is presented, to map a history state into the ket of a static composite system. We show, by examples, how the Leggett-Garg and the temporal CHSH inequalities can be violated in our approach.

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arXiv2023

Noncommutative hamiltonian formalism for noncommutative gravity

We present a covariant canonical formalism for noncommutative gravity, and in general for noncommutative geometric theories defined via a twisted $\star$-wedge product between forms. Noether theorems are generalized to the noncommutative setting, and gauge generators are constructed in a twisted phase space with $\star$-deformed Poisson bracket. This formalism is applied to noncommutative $d=4$ vierbein gravity, and allows to find the canonical generators of the tangent space $\star$-gauge group.

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arXiv2023

Hodge Duality and Supergravity

The Hodge dual operator, recently introduced for supermanifolds, is used to reformulate super Yang-Mills and supergravity in $D=4$. We first recall the definition of the Hodge dual operator for flat and curved supermanifolds. Then we show how to recover the usual super-Yang-Mills equations of motion for $N=1,2$ supersymmetry, and the obstacles (as seen from Hodge dual point of view) in the case $N \geq 3$. We reconsider several ingredients of supergeometry, relevant for a superspace formulation of supergravity, in terms of the Hodge dual operator. Finally we discuss how $D=4$ and $N=1$ supergravity is obtained in this framework.

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arXiv2022

Group manifold approach to supergravity

We present a short review of the group-geometric approach to supergravity theories, from the point of view of recent developments. The central idea is the unification of usual diffeomorphisms, gauge symmetries and supersymmetries into superdiffeomorphisms in a supergroup manifold. The example of $N=1$ supergravity in $d=4$ is discussed in detail, and used to illustrate all the steps in the construction of a group manifold action. In the Appendices we summarize basic notions of group manifold geometry, and of integration on supermanifolds.

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arXiv2022

Noncommutative gauge and gravity theories and geometric Seiberg-Witten map

We give a pedagogical account of noncommutative gauge and gravity theories, where the exterior product between forms is deformed into a $\star$-product via an abelian twist (e.g. the Groenewold-Moyal twist). The Seiberg-Witten map between commutative and noncommutative gauge theories is introduced. It allows to express the action of noncommutative Einstein gravity coupled to spinor fields in terms of the usual commutative action with commutative fields plus extra interaction terms dependent on the noncommutativity parameter.

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arXiv2022

Entropy of temporal entanglement

A recently proposed history formalism is used to define temporal entanglement in quantum systems, and compute its entropy. The procedure is based on the time-reduction of the history density operator, and allows a symmetrical treatment of space and time correlations. Temporal entanglement entropy is explicitly calculated in two simple quantum computation circuits.

quant-ph↗

arXiv2022

All quantum mixtures are proper

It is argued that proper and improper quantum mixed states have no observable differences, and hence should not be distinguished. This has implications for subjective approaches to quantum mechanics, and invalidates one of the main motivations for relational interpretations of QM.

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arXiv2021

History entanglement entropy

A formalism is proposed to describe entangled quantum histories, and their entanglement entropy. We define a history vector, living in a tensor space with basis elements corresponding to the allowed histories, i.e. histories with nonvanishing amplitudes. The amplitudes are the components of the history vector, and contain the dynamical information. Probabilities of measurement sequences, and resulting collapse, are given by generalized Born rules: they are all expressed by means of projections and scalar products involving the history vector. Entangled history states are introduced, and a history density matrix is defined in terms of ensembles of history vectors. The corresponding history entropies (and history entanglement entropies for composite systems) are explicitly computed in two examples taken from quantum computation circuits.

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arXiv2020

No relation for Wigner's friend

We argue that Wigner's friend thought experiment does not support observer dependence of quantum states. An analysis in terms of history vectors suggests that quantum collapse is to be understood as collapse of histories rather than collapse of states.

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arXiv2020

Covariant hamiltonian for supergravity in $d=3$ and $d=4$

We extend the covariant canonical formalism recently discussed in ref. [1] to geometric theories coupled to both bosonic and fermionic $p$-forms. This allows a covariant hamiltonian treatment of supergravity theories. As examples we present the covariant hamiltonian formulation for $d=3$ anti-De Sitter supergravity and for the "new minimal" $d=4$, $N=1$ supergravity (with $1$-form and $2$-form auxiliary fields). Form-Poisson brackets and form-Dirac brackets are defined, and used to find the form-canonical generators of all gauge symmetries via an algorithmic procedure.

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arXiv2019

Supergravity in the group-geometric framework: a primer

We review the group-geometric approach to supergravity theories, in the perspective of recent developments and applications. Usual diffeomorphisms, gauge symmetries and supersymmetries are unified as superdiffeomorphisms in a supergroup manifold. Integration on supermanifolds is briefly revisited, and used as a tool to provide a bridge between component and superspace actions. As an illustration of the constructive techniques, the cases of $d=3,4$ off-shell supergravities and $d=5$ Chern-Simons supergravity are discussed in detail. A cursory account of $d=10+2$ supergravity is also included. We recall a covariant canonical formalism, well adapted to theories described by Lagrangians $d$-forms, that allows to define a form hamiltonian and to recast constrained hamiltonian systems in a covariant form language. Finally, group geometry and properties of spinors and gamma matrices in $d=s+t$ dimensions are summarized in Appendices.

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arXiv2019

History operators in quantum mechanics

It is convenient to describe a quantum system at all times by means of a "history operator" $C$, encoding measurements and unitary time evolution between measurements. These operators naturally arise when computing the probability of measurement sequences, and generalize the "sum over position histories " of the Feynman path-integral. As we argue in the present note, this description has some computational advantages over the usual state vector description, and may help to clarify some issues regarding nonlocality of quantum correlations and collapse. A measurement on a system described by $C$ modifies the history operator, $C \rightarrow PC$, where $P$ is the projector corresponding to the measurement. We refer to this modification as "history operator collapse". Thus $C$ keeps track of the succession of measurements on a system, and contains all histories compatible with the results of these measurements. The collapse modifies the history content of $C$, and therefore modifies also the past (relative to the measurement), but never in a way to violate causality. Probabilities of outcomes are obtained as $Tr(C^\dagger P C)/Tr(C^\dagger C)$. A similar formula yields probabilities for intermediate measurements, and reproduces the result of the two-vector formalism in the case of given initial and final states. We apply the history operator formalism to a few examples: entangler circuit, Mach-Zehnder interferometer, teleportation circuit, three-box experiment. Not surprisingly, the propagation of coordinate eigenstates $|q\rangle$ is described by a history operator $C$ containing the Feynman path-integral.

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arXiv2019

Covariant hamiltonian for gravity coupled to $p$-forms

We review the covariant canonical formalism initiated by D'Adda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $ϕ$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $dϕ$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, both for global and local symmetries. We apply the formalism to vielbein gravity in $d=3$ and $d=4$. In the $d=3$ theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In $d=4$ the canonical analysis is carried out using FPB, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A "doubly covariant" hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as "velocities" in the definition of momenta.

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arXiv2018

A locally supersymmetric $SO(10,2)$ invariant action for $D=12$ supergravity

We present an action for $N=1$ supergravity in $10+2$ dimensions, containing the gauge fields of the $OSp(1|64)$ superalgebra, i.e. one-forms $B^{(n)}$ with $n$=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino $ψ$. The vielbein and spin connection correspond to $B^{(1)}$ and $B^{(2)}$ respectively. The action is not gauge invariant under the full $OSp(1|64)$ superalgebra, but only under a subalgebra ${\tilde F}$ (containing the $F$ algebra $OSp(1|32)$), whose gauge fields are $B^{(2)}$, $B^{(6)}$, $B^{(10)}$ and the Weyl projected Majorana gravitino ${1 \over 2} (1+Γ_{13}) ψ$. Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply $\int STr ({\bf R}^6 {\bf Γ})$ where ${\bf R}$ is the $OSp(1|64)$ curvature supermatrix two-form, and ${\bf Γ}$ is a constant supermatrix involving $Γ_{13}$ and breaking $OSp(1|64)$ to its ${\tilde F}$ subalgebra. The action includes the usual Einstein-Hilbert term.

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arXiv2017

Gauge supergravity in $D=2+2$

We present an action for chiral $N=(1,0)$ supergravity in $2+2$ dimensions. The fields of the theory are organized into an $OSp(1|4)$ connection supermatrix, and are given by the usual vierbein $V^a$, spin connection $ω^{ab}$, and Majorana gravitino $ψ$. In analogy with a construction used for $D=10+2$ gauge supergravity, the action is given by $\int STr ({\bf R}^2 {\bf Γ})$, where ${\bf R}$ is the $OSp(1|4)$ curvature supermatrix two-form, and ${\bf Γ}$ a constant supermatrix containing $γ_5$. It is similar, but not identical to the MacDowell-Mansouri action for $D=2+2$ supergravity. The constant supermatrix breaks $OSp(1|4)$ gauge invariance to a subalgebra $OSp(1|2) \oplus Sp(2)$, including a Majorana-Weyl supercharge. Thus half of the $OSp(1|4)$ gauge supersymmetry survives. The gauge fields are the selfdual part of $ω^{ab}$ and the Weyl projection of $ψ$ for $OSp(1|2)$, and the antiselfdual part of $ω^{ab}$ for $Sp(2)$. Supersymmetry transformations, being part of a gauge superalgebra, close off-shell. The selfduality condition on the spin connection can be consistently imposed, and the resulting "projected" action is $OSp(1|2)$ gauge invariant.

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arXiv2015

Hodge Dualities on Supermanifolds

We discuss the cohomology of superforms and integral forms from a new perspective based on a recently proposed Hodge dual operator. We show how the superspace constraints (a.k.a. rheonomic parametrisation) are translated from the space of superforms $Ω^{(p|0)}$ to the space of integral forms $Ω^{(p|m)}$ where $0 \leq p \leq n$, $n$ is the bosonic dimension of the supermanifold and $m$ its fermionic dimension. We dwell on the relation between supermanifolds with non-trivial curvature and Ramond-Ramond fields, for which the Laplace-Beltrami differential, constructed with our Hodge dual, is an essential ingredient. We discuss the definition of Picture Lowering and Picture Raising Operators (acting on the space of superforms and on the space of integral forms) and their relation with the cohomology. We construct non-abelian curvatures for gauge connections in the space $Ω^{(1|m)}$ and finally discuss Hodge dual fields within the present framework.

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arXiv2014

Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map

We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $θ$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $θ$ dependent terms are present for $D\geq 5$ and give a higher derivative theory on commutative space reducing to classical CS theory for $θ\to 0$. These terms depend on the field strength and not on the bare gauge potential. In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension. The Seiberg-Witten map on the $D=5$ noncommutative CS theory is explored in more detail, and we give its second order $θ$-expansion for any gauge group. The example of extended $D=5$ CS gravity, where the gauge group is $SU(2,2)$, is treated explicitly.

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arXiv2014

Higher form gauge fields and their nonassociative symmetry algebras

We show that geometric theories with $p$-form gauge fields have a nonassociative symmetry structure, extending an underlying Lie algebra. This nonassociativity is controlled by the same Chevalley-Eilenberg cohomology that classifies free differential algebras, $p$-form generalizations of Cartan-Maurer equations. A possible relation with flux backgrounds of closed string theory is pointed out.

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