848 research outputs found
A Special Case Of A Conjecture By Widom With Implications To Fermionic Entanglement Entropy
We prove a special case of a conjecture in asymptotic analysis by Harold
Widom. More precisely, we establish the leading and next-to-leading term of a
semi-classical expansion of the trace of the square of certain integral
operators on the Hilbert space . As already observed by Gioev and
Klich, this implies that the bi-partite entanglement entropy of the free Fermi
gas in its ground state grows at least as fast as the surface area of the
spatially bounded part times a logarithmic enhancement.Comment: 20 pages, 3 figures, improvement of the presentation, some references
added or updated, proof of Theorem 12 (formerly Lemma 11) adde
Gravity and Matter in Causal Set Theory
The goal of this paper is to propose an approach to the formulation of
dynamics for causal sets and coupled matter fields. We start from the continuum
version of the action for a Klein-Gordon field coupled to gravity, and rewrite
it first using quantities that have a direct correspondent in the case of a
causal set, namely volumes, causal relations, and timelike lengths, as
variables to describe the geometry. In this step, the local Lagrangian density
for a set of fields is recast into a quasilocal expression
that depends on pairs of causally related points and
is a function of the values of in the Alexandrov set defined by those
points, and whose limit as and approach a common point is .
We then describe how to discretize , and use it to define a
discrete action.Comment: 13 pages, no figures; In version 2, friendlier results than in
version 1 are obtained following much shorter derivation
A Lorentzian Gromov-Hausdoff notion of distance
This paper is the first of three in which I study the moduli space of
isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I
introduce a notion of Gromov-Hausdorff distance which makes this moduli space
into a metric space. Further properties of this metric space are studied in the
next papers. The importance of the work can be situated in fields such as
cosmology, quantum gravity and - for the mathematicians - global Lorentzian
geometry.Comment: 20 pages, 0 figures, submitted to Classical and quantum gravity,
seriously improved presentatio
Numerical determination of entanglement entropy for a sphere
We apply Srednicki's regularization to extract the logarithmic term in the
entanglement entropy produced by tracing out a real, massless, scalar field
inside a three dimensional sphere in 3+1 flat spacetime. We find numerically
that the coefficient of the logarithm is -1/90 to 0.2 percent accuracy, in
agreement with an existing analytical result
Discreteness without symmetry breaking: a theorem
This paper concerns sprinklings into Minkowski space (Poisson processes). It
proves that there exists no equivariant measurable map from sprinklings to
spacetime directions (even locally). Therefore, if a discrete structure is
associated to a sprinkling in an intrinsic manner, then the structure will not
pick out a preferred frame, locally or globally. This implies that the
discreteness of a sprinkled causal set will not give rise to ``Lorentz
breaking'' effects like modified dispersion relations. Another consequence is
that there is no way to associate a finite-valency graph to a sprinkling
consistently with Lorentz invariance.Comment: 7 pages, laTe
The moduli space of isometry classes of globally hyperbolic spacetimes
This is the last article in a series of three initiated by the second author.
We elaborate on the concepts and theorems constructed in the previous articles.
In particular, we prove that the GH and the GGH uniformities previously
introduced on the moduli space of isometry classes of globally hyperbolic
spacetimes are different, but the Cauchy sequences which give rise to
well-defined limit spaces coincide. We then examine properties of the strong
metric introduced earlier on each spacetime, and answer some questions
concerning causality of limit spaces. Progress is made towards a general
definition of causality, and it is proven that the GGH limit of a Cauchy
sequence of , path metric Lorentz spaces is again a
, path metric Lorentz space. Finally, we give a
necessary and sufficient condition, similar to the one of Gromov for the
Riemannian case, for a class of Lorentz spaces to be precompact.Comment: 29 pages, 9 figures, submitted to Class. Quant. Gra
Thermodynamics and area in Minkowski space: Heat capacity of entanglement
Tracing over the degrees of freedom inside (or outside) a sub-volume V of
Minkowski space in a given quantum state |psi>, results in a statistical
ensemble described by a density matrix rho. This enables one to relate quantum
fluctuations in V when in the state |psi>, to statistical fluctuations in the
ensemble described by rho. These fluctuations scale linearly with the surface
area of V. If V is half of space, then rho is the density matrix of a canonical
ensemble in Rindler space. This enables us to `derive' area scaling of
thermodynamic quantities in Rindler space from area scaling of quantum
fluctuations in half of Minkowski space. When considering shapes other than
half of Minkowski space, even though area scaling persists, rho does not have
an interpretation as a density matrix of a canonical ensemble in a curved, or
geometrically non-trivial, background.Comment: 17 page
Implications of area scaling of quantum fluctuations
Quantum fluctuations of a certain class of bulk operators defined in spatial
sub-volumes of Minkowski space-time, have an unexpected area scaling property.
We wish to present evidence that such area scaling may be ascribed to a
boundary theory. We first highlight the implications of area scaling with two
examples in which the boundary area of the spatial regions is not monotonous
with their volume. Next, we prove that the covariance of two operators that are
restricted to two different regions in Minkowski space scales linearly with
their mutual boundary area. Finally, we present an example which demonstrates
why this implies an underlying boundary theory.Comment: 12 pages, 5 figure
Relationships between various characterisations of wave tails
One can define several properties of wave equations that correspond to the
absence of tails in their solutions, the most common one by far being Huygens'
principle. Not all of these definitions are equivalent, although they are
sometimes assumed to be. We analyse this issue in detail for linear scalar
waves, establishing some relationships between the various properties. Huygens'
principle is almost always equivalent to the characteristic propagation
property, and in two spacetime dimensions the latter is equivalent to the
zeroth order progressing wave propagation property. Higher order progressing
waves in general do have tails, and do not seem to admit a simple physical
characterisation, but they are nevertheless useful because of their close
association with exactly solvable two-dimensional equations.Comment: Plain TeX, 26 page
A numerical study of the correspondence between paths in a causal set and geodesics in the continuum
This paper presents the results of a computational study related to the
path-geodesic correspondence in causal sets. For intervals in flat spacetimes,
and in selected curved spacetimes, we present evidence that the longest maximal
chains (the longest paths) in the corresponding causal set intervals
statistically approach the geodesic for that interval in the appropriate
continuum limit.Comment: To the celebration of the 60th birthday of Rafael D. Sorki
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