Localization of Negative Energy and the Bekenstein Bound

A simple argument shows that negative energy cannot be isolated far away from positive energy in a conformal field theory and strongly constrains its possible dispersal. This is also required by consistency with the Bekenstein bound written in terms of the positivity of relative entropy. We prove a new form of the Bekenstein bound based on the monotonicity of the relative entropy, involving a "free" entropy enclosed in a region which is highly insensitive to space-time entanglement, and show that it further improves the negative energy localization bound.Comment: 5 pages, 1 figur

Mutual information challenges entropy bounds

We consider some formulations of the entropy bounds at the semiclassical level. The entropy S(V) localized in a region V is divergent in quantum field theory (QFT). Instead of it we focus on the mutual information I(V,W)=S(V)+S(W)-S(V\cup W) between two different non-intersecting sets V and W. This is a low energy quantity, independent of the regularization scheme. In addition, the mutual information is bounded above by twice the entropy corresponding to the sets involved. Calculations of I(V,W) in QFT show that the entropy in empty space cannot be renormalized to zero, and must be actually very large. We find that this entropy due to the vacuum fluctuations violates the FMW bound in Minkowski space. The mutual information also gives a precise, cutoff independent meaning to the statement that the number of degrees of freedom increases with the volume in QFT. If the holographic bound holds, this points to the essential non locality of the physical cutoff. Violations of the Bousso bound would require conformal theories and large distances. We speculate that the presence of a small cosmological constant might prevent such a violation.Comment: 10 pages, 2 figures, minor change

Area terms in entanglement entropy

We discuss area terms in entanglement entropy and show that a recent formula by Rosenhaus and Smolkin is equivalent to the term involving a correlator of traces of the stress tensor in Adler-Zee formula for the renormalization of the Newton constant. We elaborate on how to fix the ambiguities in these formulas: Improving terms for the stress tensor of free fields, boundary terms in the modular Hamiltonian, and contact terms in the Euclidean correlation functions. We make computations for free fields and show how to apply these calculations to understand some results for interacting theories which have been studied in the literature. We also discuss an application to the F-theorem.Comment: 26 pages, no figures, references adde

Entanglement and alpha entropies for a massive scalar field in two dimensions

We find the analytic expression of the trace of powers of the reduced density matrix on an interval of length L, for a massive boson field in 1+1 dimensions. This is given exactly (except for a non universal factor) in terms of a finite sum of solutions of non linear differential equations of the Painlev\'e V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the Sine-Gordon model in agreement with a result by Delfino et al. We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c-function interpolates between the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two dimensional entropy functions.Comment: 13 pages, 2 figure

Relative Entropy and Holography

Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation $\Delta S=\Delta H$ for the first order variation of the entanglement entropy $\Delta S$ and the expectation value of the \modu Hamiltonian $\Delta H$. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation $\Delta S=\Delta H$ for vacuum state tomography and obtain modified versions of the Bekenstein bound.Comment: 75 pages, 3 figures, added reference

The neuropeptide systems and their potential role in the treatment of mammalian retinal ischemia: a developing story

The multiplicity of peptidergic receptors and of the transduction pathways they activate offers the possibility of important advances in the development of specific drugs for clinical treatment of central nervous system disorders. Among them, retinal ischemia is a common clinical entity and, due to relatively ineffective treatment, remains a common cause of visual impairment and blindness. Ischemia is a primary cause of neuronal death, and it can be considered as a sort of final common pathway in retinal diseases leading to irreversible morphological damage and vision loss. Neuropeptides and their receptors are widely expressed in mammalian retinas, where they exert multifaceted functions both during development and in the mature animal. In particular, in recent years somatostatin and pituitary adenylate cyclase activating peptide have been reported to be highly protective against retinal cell death caused by ischemia, while data on opioid peptides, angiotensin II, and other peptides have also been published. This review provides a rationale for harnessing the peptidergic receptors as a potential target against retinal neuronal damages which occur during ischemic retinopathies

Geometric modular action for disjoint intervals and boundary conformal field theory

In suitable states, the modular group of local algebras associated with unions of disjoint intervals in chiral conformal quantum field theory acts geometrically. We translate this result into the setting of boundary conformal QFT and interpret it as a relation between temperature and acceleration. We also discuss aspects ("mixing" and "charge splitting") of geometric modular action for unions of disjoint intervals in the vacuum state.Comment: Dedicated to John E. Roberts on the occasion of his 70th birthday; 24 pages, 3 figure

Relative entropy and the Bekenstein bound

Elaborating on a previous work by Marolf et al, we relate some exact results in quantum field theory and statistical mechanics to the Bekenstein universal bound on entropy. Specifically, we consider the relative entropy between the vacuum and another state, both reduced to a local region. We propose that, with the adequate interpretation, the positivity of the relative entropy in this case constitutes a well defined statement of the bound in flat space. We show that this version arises naturally from the original derivation of the bound from the generalized second law when quantum effects are taken into account. In this formulation the bound holds automatically, and in particular it does not suffer from the proliferation of the species problem. The results suggest that while the bound is relevant at the classical level, it does not introduce new physical constraints semiclassically.Comment: 12 pages, 1 figure, minor changes and references adde

Recent advances in cellular and molecular aspects of mammalian retinal ischemia

Retinal ischemia is a common clinical entity and, due to relatively ineffective treatment, remains a common cause of visual impairment and blindness. Generally, ischemic syndromes are initially characterized by low homeostatic responses which, with time, induce injury to the tissue due to cell loss by apoptosis. In this respect, retinal ischemia is a primary cause of neuronal death. It can be considered as a sort of final common pathway in retinal diseases and results in irreversible morphological and functional changes. This review summarizes the recent knowledge on the effects of ischemia in retinal tissue and points out experimental strategies/models performed to gain better comprehension of retinal ischemia diseases. In particular, the nature of the mechanisms leading to neuronal damage (i.e., excess of glutamate release, oxidative stress and inflammation) will be outlined as well as the potential and most intriguing retinoprotective approaches and the possible therapeutic use of naturally occurring molecules such as neuropeptides. There is a general agreement that a better understanding of the fundamental pathophysiology of retinal ischemia will lead to better management and improved clinical outcome. In this respect, to contrast this pathological state, specific pharmacological strategies need to be developed aimed at the many putative cascades generated during ischemia