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Valter Moretti

Publications and source records attributed to Valter Moretti.

At least 55 records · Page 3Linked to original sources
arXiv2011

State independence for tunneling processes through black hole horizons and Hawking radiation

Tunneling processes through black hole horizons have recently been investigated in the framework of WKB theory discovering interesting interplay with the Hawking radiation. In this paper we instead adopt the point of view proper of QFT in curved spacetime, namely, we use a suitable scaling limit technique to obtain the leading order of the correlation function related with some tunneling process through a Killing horizon. The computation is done for certain large class of reference quantum states for scalar fields. In the limit of sharp localization either on the external side or on opposite sides of the horizon, the quantum correlation functions appear to have thermal nature, where in both cases the characteristic temperature is the Hawking one. Our approach is valid for every stationary charged rotating non extremal black hole, however, since the computation is completely local, it covers the case of a Killing horizon which just temporarily exists in some finite region too. These results give a strong support to the idea that the Hawking radiation, which is detected at future infinity and needs some global structures to be defined, is actually related to a local phenomenon taking place even for local geometric structures (local Killing horizons) existing just for a while.

gr-qc↗
arXiv2010

Modular dynamics in diamonds

We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem.

math-ph↗
arXiv2009

Topological features of massive bosons on two dimensional Einstein space-time

In this paper we tackle the problem of constructing explicit examples of topological cocycles of Roberts' net cohomology, as defined abstractly by Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum field theory on the two dimensional Einstein cylinder. After deriving some crucial results of the algebraic framework of quantization, we address the problem of the construction of the topological cocycles. All constructed cocycles lead to unitarily equivalent representations of the fundamental group of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces). The construction is carried out using only Cauchy data and related net of local algebras on the circle.

gr-qc↗
arXiv2008

Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property

In a recent paper, we proved that a large class of spacetimes, not necessarily homogeneous or isotropous and relevant at a cosmological level, possesses a preferred codimension one submanifold, i.e., the past cosmological horizon, on which it is possible to encode the information of a scalar field theory living in the bulk. Such bulk-to-boundary reconstruction procedure entails the identification of a preferred quasifree algebraic state for the bulk theory, enjoying remarkable properties concerning invariance under isometries (if any) of the bulk and energy positivity, and reducing to well-known vacua in standard situations. In this paper, specialising to open FRW models, we extend previously obtained results and we prove that the preferred state is of Hadamard form, hence the backreaction on the metric is finite and the state can be used as a starting point for renormalisation procedures. That state could play a distinguished role in the discussion of the evolution of scalar fluctuations of the metric, an analysis often performed in the development of any model describing the dynamic of an early Universe which undergoes an inflationary phase of rapid expansion in the past.

gr-qc↗
arXiv2006

Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence

Scalar BMS-invariant QFT defined on the causal boundary $\scri$ of an asymptotically flat spacetime is discussed. (a)(i) It is noticed that the natural $BMS$ invariant pure quasifree state $λ$ on $\cW(\scri)$, recently introduced by Dappiaggi, Moretti an Pinamonti, enjoys positivity of the self-adjoint generator of $u$-translations with respect to {\em every} Bondi coordinate frame $(u,\z,\bz)$ on $\scri$, $u\in \bR$ being the affine parameter of the null geodesics forming $\scri$. This fact may be interpreted as a remnant of spectral condition inherited from Minkowski spacetime. (ii) It is proved cluster property under $u$-displacements holds for $u$-invariant pure state on $\cW(\scri)$. (iii) It is proved that there is a unique algebraic pure quasifree state invariant under $u$-displacements (of a fixed Bondi frame) having positive self-adjoint generator of $u$-displacements. It coincides with the GNS-invariant state $λ$.(iv) It is showed that in the folium of a pure $u$-invariant state $ω$ (not necessarily quasifree) on $\cW(\scri)$, $ω$ is the only state invariant under $u$-displacement. (b) It is proved that the theory can formulated for spacetimes asymptotically flat at null infinity which admit future time completion. In this case a $*$-isomorphism $\imath$ exists which identifies the (Weyl) algebra of observables of linear fields in the bulk with a sub algebra of $\cW(\scri)$. A preferred state on the field algebra in the bulk is induced by the $BMS$-invariant state $λ$.

gr-qc↗
arXiv2006

Bose-Einstein Condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons II

In a previous paper (hep-th/0407256) local scalar QFT (in Weyl algebraic approach) has been constructed on degenerate semi-Riemannian manifolds $\bS^1\times Σ$ corresponding to the extension of Killing horizons by adding points at infinity to the null geodesic forming the horizon. It has been proved that the theory admits a natural representation of $PSL(2,\bR)$ in terms of $*$-automorphisms and this representation is unitarily implementable if referring to a certain invariant state $λ$. Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states $\{λ_ζ\}$, with $ζ\in L^2(Σ)$, which break part of the symmetry, in the sense that each of them is not invariant under the full group $PSL(2,\bR)$ and so there is no unitary representation of whole group $PSL(2,\bR)$ which leaves fixed the cyclic GNS vector. These states, if restricted to suitable portions of $\bM$ are invariant and extremal KMS states with respect a surviving one-parameter group symmetry. In this paper we clarify the nature of symmetry breakdown. We show that, in fact, {\em spontaneous} symmetry breaking occurs in the natural sense of algebraic quantum field theory: if $ζ\neq 0$, there is no unitary representation of whole group $PSL(2,\bR)$ which implements the $*$-automorphism representation of $PSL(2,\bR)$ itself in the GNS representation of $λ_ζ$ (leaving fixed or not the state).

gr-qc↗
arXiv2002

The interplay of the polar decomposition theorem and the Lorentz group

It is shown that the polar decomposition theorem of operators in (real) Hilbert spaces gives rise to the known decomposition in boost and spatial rotation part of any matrix of the orthochronous proper Lorentz group $SO(1,3)\uparrow$. This result is not trivial because the polar decomposition theorem is referred to a positive defined scalar product while the Lorentz-group decomposition theorem deals with the indefinite Lorentz metric. A generalization to infinite dimensional spaces can be given. It is finally shown that the polar decomposition of $SL(2,\bC)$ is preserved by the covering homomorphism of $SL(2,\bC)$ onto $SO(1,3)\spa\uparrow$

math-ph↗
arXiv2002

Comments on the Stress-Energy Tensor Operator in Curved Spacetime

Hollands and Wald's technique based on *-algebras of Wick products of field operators is strightforwardly generalized to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. The proposed stress-energy tensor operator is conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. They are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of field equation. The averaged stress-energy tensor with respect to Hadamard quantum states can be obtained by an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is completely determined by the local geometry and the parameters which appear in the Klein-Gordon operator. The averaged stress-energy tensor also coincides with that found by employing the local $ζ$-function approach.

gr-qc↗
arXiv2000

Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in $C^{\infty}$ Lorentzian manifolds by a local Wick rotation

Completing the results obtained in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth $D$-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of ``local Wick rotation'' of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates are ``physical''. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point, or, more generally, in a neighborhood of a space-like (Cauchy) hypersurface, into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or Kählerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to $C^\infty$ non analytic Lorentzian manifolds by approximating Lorentzian $C^{\infty}$ metrics by analytic metrics in common geodesically convex neighborhoods. The symmetry requirement plays a central rôle in the point-splitting renormalization procedure of the one-loop stress-energy tensor in curved spacetimes for Hadamard quantum states.

gr-qc↗
arXiv1999

Proof of the symmetry of the off-diagonal heat-kernel and Hadamard's expansion coefficients in general $C^{\infty}$ Riemannian manifolds

We consider the problem of the symmetry of the off-diagonal heat-kernel coefficients as well as the coefficients corresponding to the short-distance-divergent part of the Hadamard expansion in general smooth (analytic or not) manifolds. The requirement of such a symmetry played a central rôle in the theory of the point-splitting one-loop renormalization of the stress tensor in either Riemannian or Lorentzian manifolds. Actually, the symmetry of these coefficients has been assumed as a hypothesis in several papers concerning these issues without an explicit proof. The difficulty of a direct proof is related to the fact that the considered off-diagonal heat-kernel expansion, also in the Riemannian case, in principle, may be not a proper asymptotic expansion. On the other hand, direct computations of the off-diagonal heat-kernel coefficients are impossibly difficult in nontrivial cases and thus no case is known in the literature where the symmetry does not hold. By approximating $C^\infty$ metrics with analytic metrics in common (totally normal) geodesically convex neighborhoods, it is rigorously proven that, in general $C^\infty$ Riemannian manifolds, any point admits a geodesically convex neighborhood where the off-diagonal heat-kernel coefficients, as well as the relevant Hadamard's expansion coefficients, are symmetric functions of the two arguments.

gr-qc↗
arXiv1999

One-loop stress-tensor renormalization in curved background: the relation between $ζ$-function and point-splitting approaches, and an improved point-splitting procedure

We conclude the rigorous analysis of a previous paper concerning the relation between the (Euclidean) point-splitting approach and the local $ζ$-function procedure to renormalize physical quantities at one-loop in (Euclidean) QFT in curved spacetime. The stress tensor is now considered in general $D$-dimensional closed manifolds for positive scalar operators $-Δ+ V(x)$. Results obtained in previous works (in the case D=4 and $V(x) =ξR(x) + m^2$) are rigorously proven and generalized. It is also proven that, in static Euclidean manifolds, the method is compatible with Lorentzian-time analytic continuations. It is found that, for $D>1$, the result of the $ζ$ function procedure is the same obtained from an improved version of the point-splitting method which uses a particular choice of the term $w_0(x,y)$ in the Hadamard expansion of the Green function. This point-splitting procedure works for any value of the field mass $m$. Furthermore, in the case D=4 and $V(x) = ξR(x)+ m^2$, the given procedure generalizes the Euclidean version of Wald's improved point-splitting procedure. The found point-splitting method should work generally, also dropping the hypothesis of a closed manifold, and not depending on the $ζ$-function procedure. This fact is checked in the Euclidean section of Minkowski spacetime for $A = -Δ+ m^2$ where the method gives rise to the correct stress tensor for $m^2 \geq 0$ automatically.

gr-qc↗
arXiv1998

Local $ζ$-function techniques vs point-splitting procedure: a few rigorous results

Some general properties of local $ζ$-function procedures to renormalize some quantities in $D$-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators $-Δ+ V(x)$ in general closed $D$-manifolds, and a few comments are given for nonclosed manifolds too. A general comparison is carried out with respect to the more known point-splitting procedure concerning the effective Lagrangian and the field fluctuations. It is proven that, for $D>1$, the local $ζ$-function and point-splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the Hadamard divergences. It is found that the $ζ$ function procedure picks out a particular term $w_0(x,y)$ in the Hadamard expansion. Also the presence of an untrivial kernel of the operator $-Δ+V(x)$ may produce some differences between the two analyzed approaches. Finally, a formal identity concerning the field fluctuations, used by physicists, is discussed and proven within the local $ζ$-function approach. This is done also to reply to recent criticism against $ζ$-function techniques.

gr-qc↗
arXiv1997

Direct $ζ$-function approach and renormalization of one-loop stress tensors in curved spacetimes

A method which uses a generalized tensorial $ζ$-function to compute the renormalized stress tensor of a quantum field propagating in a (static) curved background is presented. The starting point of the method is the direct computation of the functional derivatives of the Euclidean one-loop effective action with respect to the background metric. This method, when available, gives rise to a conserved stress tensor and produces the conformal anomaly formula directly. It is proven that the obtained stress tensor agrees with statistical mechanics in the case of a finite temperature theory. The renormalization procedure is controlled by the structure of the poles of the stress-tensor $ζ$ function. The infinite renormalization is automatic and is due to a ``magic'' cancellation of two poles. The remaining finite renormalization involves conserved geometrical terms arising by a certain residue. Such terms renormalize coupling constants of the geometric part of Einstein's equations (customary generalized through high-order curvature terms). The method is checked on particular cases (closed and open Einstein`s universe) finding agreement with other approaches. The method is also checked considering a massless scalar field in the presence of a conical singularity in the Euclidean manifold (i.e. Rindler spacetimes/large mass black hole manifold/cosmic string manifold). There, the method gives rise to the stress tensor already got by the point-splitting approach for every coupling with the curvature regardless of the presence of the singular curvature. Comments on the measure employed in the path integral, the use of the optical manifold and different approaches to renormalize the Hamiltonian are made.

hep-th↗
arXiv1997

zeta-function regularization and one-loop renormalization of field fluctuations in curved space-times

A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the $ζ$-function approach is presented. The method produces finite quantities directly and finite scale-parametrized counterterms at most. These finite couterterms are related to the presence of a particular pole of the effective-action $ζ$ function as well as to the heat kernel coefficients. The method is checked in several examples obtaining known or reasonable results. Finally, comments are given for as it concerns the recent proposal by Frolov et al. to get the finite Bekenstein-Hawking entropy from Sakharov's induced gravity theory.

gr-qc↗