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51233f33eb7bc8737d5c8536eb1c9520e9ce91c6 | subsection | 12 | 18 | Datasets and Metrics | We used two publicly available datasets: VRD (Visual Relationships Dataset) and VG (Visual Genome V1.2 dataset ).VRD dataset. It contains 5,000 images with 100 object categories and 70 relationships. In total, VRD contains 37,993 relationship triplet annotations with 6,672 unique triplets and 24.25 relationship per obj... | {
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9904e776fe92fca450ad9a54c55fa32a3a82b9c9 | subsection | 13 | 18 | Settings | In our experiments, we only focused on the relationship prediction task, i.e., classifying any two object regions into relationship classes. The reasons are two-fold. First, relationship prediction plays the core role in relationship detection, a more comprehensive task that also needs to detect the two objects. Second... | {
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84a8d29b1163f6413c1c4fe0455c7f97fa19ee48 | subsection | 14 | 18 | Settings | And the other settings are the same with STA.We also compared with state-of-the-art visual relationship prediction methods such as VTransE , Lu's-V , Lu's-VLK , and Peyre's-A . Note that except for Lu's-VLK which is a multimodal model, all the methods compared here are visual models.
[Figure: Performances (R@100%) of r... | {
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0e8510bc1cbbfd5415c47bb71f91c45d8455282b | subsection | 15 | 18 | Results and Analysis | Table REF , REF show the performance of compared methods on two datasets of different experiment settings. As we can see, the proposed STA has the best performances compared with the other baselines and state-of-the-art on both datasets. For example, compared to the Base+OA, the proposed STA can boost the Recall@100 of... | {
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73fb95fe709cdaa398794e279f2d45e8bdbcbcca | subsection | 16 | 18 | Results and Analysis | REF shows the accuracy of each relationship, listed in an ascending, left-right order according to their biases to specific subject-object configuration by N_R(r)/N_C(r), where N_C(r) is the number of configurations and N_R(r) is the number of training samples of the r-th relationship. Notice that smaller bias indicate... | {
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0d808cf0d6eec5b663455f3abcc51e67c8f548f7 | subsection | 17 | 18 | Conclusions | We proposed a novel Shuffle-Then-Assemble visual relationship feature learning strategy for improving visual relationship models. The key idea is to discard the original one-to-one paired object alignments, and then try to recover them in an unsupervised pair discovery fashion by using a cycle-consistent adversarial do... | {
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47654ff3f4f273ca2576942004bbf5ca952ab1aa | abstract | 0 | 4 | Abstract | It is common to model inductive datatypes as least fixed points of functors.
We show that within the Cedille type theory we can relax functoriality
constraints and generically derive an induction principle for Mendler-style
lambda-encoded inductive datatypes, which arise as least fixed points of
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4bc6d04cb1fbbf578802e2b64ffc58feb2c9efda | subsection | 1 | 4 | Introduction | It is widely known that inductive datatypes may be defined in pure
impredicative type theory. For example, Church encodings identify
each natural number n with its iterator λ s. λ z. sⁿ z.
The Church natural numbers can be typed in System F by means of
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aebbbf2ddb2047987ab6b7b47f9718093f91981b | subsection | 2 | 4 | Introduction | We start by giving a detailed description of lambda-encoded
naturals with an induction principle and a constant-time predecessor
function that only requires linear space to encode a numeral. We
also give examples of infinitary datatypes. Finally, we present an
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a51de083c5bf4df0288c48f5ce86176f61b2f04e | subsection | 3 | 4 | Background | In this section, we briefly summarize the main features of Cedille's
type theory. For full details on CDLE, including semantics and
soundness results, please see the previous
papers , . The main metatheoretic property
proved in the previous work is logical consistency: there are types
which are not inhabited. CDLE is a... | {
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49df3706ab39fc5ac593caa7c476850f6c698a05 | abstract | 0 | 22 | Abstract | In this paper, we present preliminary results illustrating the effect of
cosmic rays on solid-phase chemistry in models of both TMC-1 and several
sources with physical conditions identical to TMC-1 except for hypothetically
enhanced ionization rates. Using a recent theory for the addition of cosmic
ray-induced reaction... | {
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} | 10.3847/1538-4357/aac5ee | 1805.05764 | On Cosmic Ray-Driven Grain Chemistry in Cold Core Models | [
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5227e92c966a0b6b9d0e66e7d0ebdb450b0676c5 | subsection | 1 | 22 | Introduction | Cosmic rays are a form of high-energy (MeV - TeV) ionizing radiation composed mostly of protons thought to form both in supernovae and galactic
nuclei , , .
It has long been speculated that these
energetic particles can have significant physicochemical effects on the
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dfc93c4767548d1be2146cf37a5add021f4c9d14 | subsection | 2 | 22 | Introduction | Inelastic events, in turn, are typically approximated as consisting of collisions
that cause either the ionization or electronic excitation of target species.
The ionization of species in a material results in the formation of so-called “secondary
electrons” . Around 10^4 secondary electrons can be
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c9e4c3bea9be0726c0e52fc03440e970b69e0779 | subsection | 3 | 22 | Introduction | As described in detail in , the G values
for processes (REF )-(REF ) can be calculated using the following
expressions:G_\mathrm {R1} = P_\mathrm {e}\left(\frac{100\,\mathrm {eV}}{W}\right)G_\mathrm {R2} = (1-P_\mathrm {e})\left(\frac{100\,\mathrm {eV}}{W}\right)G_\mathrm {R3} = P_\mathrm {dis}\left(\frac{100\,\mathrm ... | {
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af0517f5fad21e7dc4f9b35241a73d5c52dd5e0f | subsection | 4 | 22 | Introduction | Thus, the
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and \xi , the average number of excitations that can result from this amount of energy, is a function of the
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cff2c65172fc7a6c93ba5f9d643770b8f29db535 | subsection | 5 | 22 | Introduction | The organization of the rest of this paper is as follows: in § we give
details concerning the code and physical conditions used here, while §
contains a description of the reactions and processes added to the network for this work. § concerns the description and discussion of our major findings,
while in §, we summariz... | {
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7720398b4b91ad8455d0378b739ce7c64a310200 | subsection | 6 | 22 | Model | In this work, we focus on the chemistry of cold cores, such as TMC-1. Despite
the low temperatures of these regions, their chemical complexity has been
highlighted by recent detections of species such as HC_5O
, HC_7O
, , and the aromatic molecule
benzonitrile . The effects of radiation
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3f2b50ae583270b5975898773fa9077da8f7fe93 | subsection | 7 | 22 | Network | lccrrr
Parameters used in calculating G values and rate coefficients.
Species E_\mathrm {ion}a W_\mathrm {exc}b W_\mathrm {s}(eV) (eV) (eV)
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505b9f44b9d160ce989c09c8a2ea5554c14a8fe8 | subsection | 8 | 22 | Network | For reactions of the formA + B^* \rightarrow \text{products}.we use the following formula for calculating the rate coefficients, k_\mathrm {ST}(cm^3s^{-1}):k_\mathrm {ST} =
f_\mathrm {br}
\left[
\frac{
\nu _0^B +
\nu _0^A
}{
{N_\mathrm {site}n_\mathrm {dust}}
}
\right]where f_\mathrm {br} is the product branching fract... | {
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"Christopher N. Shingledecker",
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86e75eaa4cd16dbe98ae657dd915b9a31e3a18bb | subsection | 9 | 22 | Network | (REF ). We emphasize that in our actual network, most suprathermal species are produced from the radiolysis of more than one species, and all have more than one destructive reaction.The suprathermal reactions we have added to our network can be grouped into two
classes. Class 1 refers to those that are similar to react... | {
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7428b1ebb18130ddfba38cdc6b7c68d32ace098e | subsection | 10 | 22 | Network | Many of these new reactions were
drawn from combustion chemistry. Since cosmic rays, such as other forms of
ionizing radiation, produce highly non-thermal species, some of the endothermic
reactions previously considered in the context of high-temperature systems
become relevant when considering irradiated low-temperatu... | {
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289a5edc7a856d39591d6879fe7a5c45e9d563f8 | subsection | 11 | 22 | Results & Discussion | Given the relative novelty of the radiation chemistry we have added to our chemical
network, it is natural to question what effect these new reactions will have
on the abundances of important cold core species. To that end, in Fig.
REF we show the calculated abundances of the cyanopolyynes in our TMC-1
models, both wit... | {
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0c05ac4877ff72edfc0294a714d93eab39ab3afd | subsection | 12 | 22 | HOCO | As shown in Fig. REF , abundances of HOCO are increased in simulations
including radiation chemistry in the three phases of the model: gas, ice
surface, and ice bulk. This increase is due primarily to the following surface
reaction\mathrm {OH^*}(s)+ \mathrm {CO}(s) \rightarrow \mathrm {HOCO}(g)where the HOCO product un... | {
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2c813645b4a42678897b14cfcc7cd32b471e06e4 | subsection | 13 | 22 | NO | NO_2 is another species the abundance of which is enhanced in simulations that
include radiation chemistry. As shown in Fig. REF , as for the case of HOCO, NO_2
abundances are increased in all three model phases, although the connection
between these enhancements and radiation chemistry is slightly more complex
than in... | {
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1a20dfe8db1e886db79321e335ad2b12fab5f823 | subsection | 14 | 22 | HC | The ketenyl radical, HC_2O, was first observed in the cold (T_\mathrm {kin}
\approx 15 K) starless cores Lupus-1A and L483 by
, who derived a column density of \sim 5\times 10^{11} cm^{-2} for both sources. Chemical simulations
were run assuming HC_2O formation via the reaction of OH and C_2H.
It was noted that such si... | {
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74e0e7ac3fd0e088e558a97f82f342210c7d5c85 | subsection | 15 | 22 | HCOOCH | As with the other species highlighted thus far, the abundance of methyl formate
(HCOOCH_3) is enhanced in all three phases of the model. In simulations
including radiation chemistry, the main production pathways for gas-phase
methyl formate are\mathrm {HCO^*}(s) + \mathrm {CH_3O}(s) \rightarrow \mathrm {HCOOCH_3}(g)and... | {
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90cffad7165e418ad8dc445aafed6f03e03488aa | subsection | 16 | 22 | CH | Unlike the other species highlighted thus far, surface and bulk abundances of
the COM, ethanol, were not significantly enhanced in our simulations including
radiation chemistry. However, as shown in Fig. 7, the gas-phase abundance
is enhanced by ca. an order of magnitude by the Class 2 insertion reaction\mathrm {CH_2^*... | {
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e9a84ad6c27dfde5b9125c1cd82e2a1fb9063a4e | subsection | 17 | 22 | Results Using Enhanced Ionization Rates | Additional simulations were run in order to examine the effect of the new radiation chemistry at high \zeta . As mentioned in § - and shown in Table - we assume that the simulated hypothetical sources are physically identical to TMC-1 except for having higher ionization rates. The results from these model runs for HOCO... | {
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"Christopher N. Shingledecker",
"Jessica D. Tennis",
"Romane Le Gal",
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fbc996892c4335d937c4f463badbcae991631cf8 | subsection | 18 | 22 | Conclusions | We have utilized the theory described in in an initial attempt to incorporate radiation chemistry into an existing chemical network. Simulations of the cold core TMC-1 were run, both with and without the new cosmic ray-induced reactions. We also modeled several hypothetical sources which were physically identical to TM... | {
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913f4989037e3f8adab982492415381447b5859a | subsection | 19 | 22 | Radiolysis Reactions | llccccc
New solid-phase radiolysis processes
Number Process f_\mathrm {br} G-value Type
7cH_2O1 \mathrm {H_2O}\leadsto \mathrm {O^*}+\mathrm {H_2^*} 0.500 3.704 \textup {I}2 \mathrm {H_2O}\leadsto \mathrm {OH^*}+\mathrm {H^*} 0.500 3.704 \textup {I}3 \mathrm {H_2O}\leadsto \mathrm {OH}+\mathrm {H} 1.000 1.747 \textup... | {
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"Christopher N. Shingledecker",
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"Romane Le Gal",
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6d50626abbf2655cf542e59378aa9da27e8a8ae2 | subsection | 20 | 22 | Class 2 Reactions | llccccr[ht]
New Class 2 reactions involving suprathermal species.
Number Reaction f_\mathrm {br} Source
7cC^*50 \mathrm {C^*} + \mathrm {H_2O} \rightarrow \mathrm {CH} + \mathrm {OH} 1.0 51 \mathrm {C^*} + \mathrm {CO} \rightarrow \mathrm {CCO} 1.0 52 \mathrm {C^*} + \mathrm {CH_3OH} \rightarrow \mathrm {CH_3CHO} 0.... | {
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92230e89cb666b748f2eb36ce1bf5caf9b60e333 | subsection | 21 | 22 | New HOCO Reactions | llcccr
New gas-phase HOCO destruction reactions
Number Reaction \alpha \beta \gamma Source6cNeutral-Neutralas^{-1} K60 \mathrm {HOCO}+\mathrm {Cl}\rightarrow \mathrm {HCl}+\mathrm {CO_2} 4.800\times 10^{-11} 0.000 0.000 61 \mathrm {HOCO}+\mathrm {O_2}\rightarrow \mathrm {O_2H}+\mathrm {CO_2} 1.900\times 10^{-12} 0.00... | {
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3b43bbbc8b9150e5fcc2bd7da315f24a27e3b40e | abstract | 0 | 197 | Abstract | Entanglement entropy first arose from attempts to understand the entropy of
black holes, and is believed to play a crucial role in a complete description
of quantum gravity. This thesis explores some proposed connections between
entanglement entropy and the geometry of spacetime. One such connection is the
ability to d... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
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eae6e3bf0eb0f9326437c0c6e1c41ae2bd1b5cea | subsection | 1 | 197 | Body | Chapter 1:1emABSTRACT[Table: NO_CAPTION]Entanglement entropy first arose from attempts to understand
the entropy of black holes, and is believed to play a crucial role in
a complete description of quantum gravity.
This thesis explores
some proposed connections between entanglement entropy and the geometry of spacetime.... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05090285837650299,
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... | |
9b9d101571d1a34723ebe83bc715d1a2b9e5cca8 | subsection | 2 | 197 | Body | The solution
in the bulk on the Euclidean section then takes the simple form\phi _0 = g z^{d-\Delta }.Given these two solutions, the bulk contribution to \delta S^{(2)} may be computed using
equation (REF ). Note that \partial _\tau \phi =0 on the \tau =0 surface, so we
only need the \nabla ^2 \phi ^2 term in the integ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.055042464286088943,
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... | |
473e147db3187bb202c4af9aec8f42378a664d7f | subsection | 3 | 197 | Body | For a given term of the form A z^\eta , the contribution to \delta S^{(2)} is\delta S^{(2)}_\eta &
= -\frac{\pi }{2} \Omega _{d-2} \int _{z_0}^R \frac{dz}{z^{d+1}} \int _0^{\sqrt{R^2-z^2}}
dr\, r^{d-2} \left[\frac{R^2-r^2-z^2}{2R}\right] A z^\eta \\
&=-A\frac{\pi \Omega _{d-2}}{4(d^2-1)} \left[R^\eta \,\frac{\Gamma (\... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
a3b3a81fc51b9eaca725a972e23a794a2c71167c | subsection | 4 | 197 | Body | Note that since \beta _\omega \propto \omega ^{2\Delta -d}, which is generically
a non-integer power, the integral over \omega will not vanish, so all the \beta _\omega
terms survive.The resulting bulk contribution to the entanglement entropy at order \lambda g is\delta S^{(2)}_{\mathcal {E},\lambda g} = -\frac{g \pi ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.01250... | |
fbcc313133ef13d19cb81b0d963fbbfd3c013208 | subsection | 5 | 197 | Body | The precise value of this term is\delta S^{(2)}_{\lambda ^2} = -\frac{\pi \Omega _{d-2}}{d^2-1} R^{2\Delta }\big (\delta \langle \mathcal {O} \rangle \big )^2
\,\frac{\Delta \Gamma (\frac{d}{2}+\frac{3}{2}) \Gamma (\Delta -\frac{d}{2}+1)}{(2\Delta -d)^2\Gamma (\Delta +\frac{3}{2})},which is quite similar to the R^{2\De... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.07044490426778793,
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0.003460438223555684,
0.026... | |
9677aa814a4789631d0348bbafaa627138254c3c | subsection | 6 | 197 | Body | The second type of term is finite, and takes the form\delta S^{(2)}_{\mathcal {T},\text{finite}} = -2\pi \Delta \int _{\Sigma } \zeta ^t g \beta .The relation between \beta and \delta \langle \mathcal {O} \rangle identified in (REF ) implies
from equation (REF ),\delta \langle \mathcal {O} \rangle = \lambda _\omega \fr... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04596099257469177,
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0.03204451873898506,
0.005142381880432367,
0.00... | |
bd967bf000b12dae74ffd063862cb1b3840711c9 | subsection | 7 | 197 | Body | However, as noted in equation (REF ), we
do find such a term at second order in \lambda .Extending the above calculation to \Delta <\frac{d}{2} requires the introduction of one novel
element: a modification of the coupling g(x) to include an IR cutoff. It is straightforward to
see why this regulator is needed. The pert... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.0491488091647625,
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0.00004987385545973666... | |
f4ad537cabe38b3ef33c6537540a3a75db949bc7 | subsection | 8 | 197 | Body | The important features are that \phi _0 on the t=0 surface takes the form\phi _0 = -\frac{\langle \mathcal {O} \rangle _g}{2\Delta -d} z^\Delta + g z^{d-\Delta },and the vev \langle \mathcal {O} \rangle _g is determined in terms of the IR cutoff L by\langle \mathcal {O} \rangle _g = 2gL^{d-2\Delta }\frac{\Gamma (\Delta... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.043744225054979324,
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0.05... | |
2aa5347f57fafdcd093556dafe22c19e780fb8c3 | subsection | 9 | 197 | Body | It
would be interesting to explore this relation further.At first order in g and \lambda , three types of terms will appear, proportional to each of
\langle \mathcal {O} \rangle _g\,\delta \langle \mathcal {O} \rangle , (g\delta \langle \mathcal {O} \rangle +\lambda (0)\langle \mathcal {O} \rangle _g), or
g \lambda (0)... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.00622728792950511,
0.031014334410429,
0.011... | |
a023511b89a8c5190e3e29277de5673658753151 | subsection | 10 | 197 | Body | The finite
term is given by\delta S^{(2)}_{\mathcal {T},2} = 2\pi \frac{\Omega _{d-2} R^d\Delta }{(d^2-1)(2\Delta -d)}(g\delta \langle \mathcal {O} \rangle
+\lambda (0)\langle \mathcal {O} \rangle _g),which is exactly analogous to the term (REF ) found for the case \Delta >\frac{d}{2}.Finally, the term with coefficien... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.07446359097957611,
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0.023666605353355408,
0.0006518425070680678,
... | |
c787e337d0fcd224e1b8980aeb8491d159cafcce | subsection | 11 | 197 | Body | The need
for this renormalization scale can be seen by examining the expression for \langle \mathcal {O} \rangle _g,
which depends on the \mathcal {O}\mathcal {O} two-point function with \Delta =\frac{d}{2}:\langle \mathcal {O} \rangle _g = -\int d^d x \frac{g c^{\prime }_\Delta }{x^d} =
-gc^{\prime }_\Delta \frac{\pi ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.051172081381082535,
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0.0357319638133049,
0.0... | |
1cedb594a99582673cc93d1af6f1c0c82333ec9c | subsection | 12 | 197 | Body | This is easily remedied by dividing by (2\Delta -d)^2 , ,
so that
the new constant appearing in the two point function isc^{\prime }_\Delta = \frac{\Gamma (\Delta )}{2\pi ^{\frac{d}{2}} \Gamma (\Delta -\frac{d}{2}+1)} \;\xrightarrow{} \;
\frac{\Gamma (\frac{d}{2})}{2\pi ^{\frac{d}{2}}} .This change affects the normaliz... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04079757630825043,
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0.038753122091293335,
0... | |
7547cebfc8548da5ad3e03e9cf3a0fc6654b2854 | subsection | 13 | 197 | Body | We also need \phi _0 in the region t_B\gg z, given by\phi _0 = g z^{\frac{d}{2}}\log \frac{L}{t_B} .For \phi _\omega , the solution on the \mathcal {E} surface is still given by a modified Bessel
function as in equation (REF ), but must be divided by (2\Delta -d) according
to our new normalization (REF ),\phi _\omega =... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.033532530069351196,
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3d0071af7aba39c0f8f7aad0b4a6db4947266cb4 | subsection | 14 | 197 | Body | \\
&\left.\left. + \frac{d}{4}H_{\frac{d+1}{2}}
\left(\gamma _E+\log \frac{R^2\omega }{4L}\right)
- \log \mu R -\frac{1}{8}
\left(H^{(2)}_{\frac{d+1}{2}} +H_{\frac{d+1}{2}} (H_{\frac{d+1}{2}}-2)\right) \right]
\right\rbrace .This is the answer for a single frequency \omega in the state deformation function
\lambda (x).... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.021903574466705322,
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... | |
7052acd869fe82b993f1e00675c3ced3f8a5b6d8 | subsection | 15 | 197 | Body | This cutoff is related to the renormalized vev \langle \mathcal {O} \rangle ^\text{ren.}_g
via (REF ), which does depend on the renormalization scheme.
Thus the dependence on L in
the above answer can be traded for \langle \mathcal {O} \rangle _g^\text{ren.}, at the cost of introducing
(spurious) \mu -dependence,\delta... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.08106797933578491,
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0.028034992516040802,
0.05... | |
fd279654292089d7f6d6c3f2bbf9e93b3e52bf9b | subsection | 16 | 197 | Body | However, one still has to be mindful of the branch prescription,
which is appropriately handled by adding the complex conjugate as directed in the expressions
above (denoted by “c.c.”). When t_B>z, the branch in the hypergeometric function
along the real axis is dealt with by replacing t_B\rightarrow t_B+i \delta , and... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.021933984011411667,
0.03... | |
492c914fc4ed24e28530c064bcc474560b62b0b3 | subsection | 17 | 197 | Body | In the first
case, \phi _0 reduces to\phi _0\big |_{t_B=0} = g z^{d-\Delta } - z^\Delta \, \frac{g L^{d-2\Delta } \Gamma (\Delta -\frac{d}{2}+\frac{1}{2})}{
\sqrt{\pi }\, \Gamma (\Delta -\frac{d}{2}+1)}
\; [_2]{F}{_1}\left(\Delta -\frac{d}{2}, \Delta -\frac{d}{2}+\frac{1}{2}; \Delta -\frac{d}{2}+1;
\frac{-z^2}{L^2}\rig... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.08173933625221252,
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0... | |
6255c0c17b30a243f1941fdf9f0b3075f085c1fd | subsection | 18 | 197 | Body | Thus the
bulk solution on the Euclidean section (REF ) is still valid. The real time
behavior of the solution is given by the following integral,\phi _\omega = \lambda _\omega z^{d-\Delta }\frac{\Gamma (\Delta -\frac{d}{2}+\frac{1}{2})}{\sqrt{\pi }\,
\Gamma (\Delta -\frac{d}{2})}\left[\int _0^{\infty } dy\,\cos (\omega... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.0597098171710968,
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0.006666222587227821,
0.0... | |
4a572f6bbef65b9fa7db1a6d9cd8240845096584 | subsection | 19 | 197 | Body | The final answer for the real time behavior of \phi _\omega near z=0
is\phi _\omega =-\frac{\delta \langle \mathcal {O} \rangle }{2\Delta -d}z^\Delta +\lambda _\omega z^{d-\Delta } F(t_B/z).where we have identified \delta \langle \mathcal {O} \rangle as\delta \langle \mathcal {O} \rangle =
\lambda _\omega \frac{2\,\Gam... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05439727380871773,
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0.018... | |
ca9753fb510022f333aab126325654fbcffb6bf4 | subsection | 20 | 197 | Body | Finally, it is also useful to have the form of the function ()
along \mathcal {T}, where t_B\gg z,\phi _0\rightarrow gz^{\frac{d}{2}} \log \frac{L}{t_B}.At t_B=0,
the modified Bessel function solution for \phi _\omega is still valid, and the appropriate
normalization is given in equation (REF ).
We also need expression... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.059158604592084885,
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0... | |
40e9a7462382479cbe3465019778ac21cc3fe959 | subsection | 21 | 197 | Body | The calculation is divided into
four parts: the \mathcal {E} surface integral, the \mathcal {T} surface integral for
t_B\sim z_0, the \mathcal {T} surface integral for t_B\gg z_0, and the
\delta S^{(1)} divergence.A simple higher curvature gravity is obtained by replacing R in the Einstein-Hilbert action by a function ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.048033129423856735,
0.03103538230061531,
0.013076363131403923,
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0.047117628157138824,
-0.03083702363073826,
0... | |
5f86c31f9272903340957f2b3ba61532d4c6ad27 | subsection | 22 | 197 | Gravity as a regulator | The notion of quantizing the gravitational field is nearly as old as general relativity itself. In Einstein's
1916 paper on gravitational waves, he remarked that
electrons would be able to radiate gravitationally as well as electromagnetically, and
inferred from this that
the arguments for quantizing the electromagneti... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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f21a6d5a8de2dc80a260974cee887d0a3ffd3680 | subsection | 23 | 197 | Gravity as a regulator | The
Einstein equations then constrain the evolution of the energy and momentum associated
with the surface integral.
completely avoiding the difficulties encountered in classical
electromagnetism in which point particles require an infinite mass subtraction to compensate
the
divergent self-energy , .
Bergmann believed ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
28043b974fc94c8710df1b2d156f1c380d58417a | subsection | 24 | 197 | Gravity as a regulator | Gravity has naturally provided the necessary “counterterm.”
It is also worth noting the that since the R\rightarrow 0
mass is
proportional
to G^{-1/2}, an attempt to compute it perturbatively in integer powers of G would lead
to a divergent result at any finite order in perturbation theory.
The finiteness exhibited in... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
f31fcea0ee6661e8c61578f11ab5160f70a811f2 | subsection | 25 | 197 | Black hole entropy | Perhaps the most compelling evidence for gravity's UV finiteness comes from
the physics of black
holes.
Based on thought experiments in which the entropy of the universe is decreased by
sending packets of thermal matter
into a black hole, Bekenstein conjectured that black holes must possess an
intrinsic entropy in orde... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
64eaf18603ac99cab1b01fb154f4e8a5815aeb20 | subsection | 26 | 197 | Black hole entropy | One therefore might view the
Bekenstein-Hawking formula (REF ), as well as Wald's generalization ,
as the only entropies
consistent with Hawking's calculation that also incorporate diffeomorphism invariance
in the gravitational theory.Returning to Bekenstein's original motivation, the resolution of
the entropy loss con... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
88686287220a0fb95f7615db2c5b69439cdfd0c2 | subsection | 27 | 197 | Generalized entropy as entanglement entropy | Underlying both of these puzzles is a deeper question: why are these entropies
finite?
The hypothetical membrane theory on the black hole horizon must not be a continuum
field theory, with an infinitude of states, but rather should be discrete at Planckian length scales to
give the correct value for S_\text{BH}. The fi... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0... | |
5dba4fd2ce334390db8e6abbc148fd0123141d5e | subsection | 28 | 197 | Generalized entropy as entanglement entropy | More explicitly, if S_{\text{EE}} in (REF )
is split in a regulator-dependent way into an area term c_0 A/\epsilon ^{d-2} and a finite
piece S_\text{finite}^{(\epsilon )} (ignoring the subleading divergences for the moment),
and if the same regularization scheme is used in the matter field loops that change
the bare Ne... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.03557... | |
17a270d00c916821bca1f7abcabe8623fe1a0cc9 | subsection | 29 | 197 | Generalized entropy as entanglement entropy | Such higher curvature
corrections arise generically in quantum gravity theories
, in which case S_\text{BH} is replaced by the Wald entropy
, ,S_\text{gen} = S_\text{Wald} + S_\text{out}.The subleading divergences are then seen to simply correspond to the renormalization of the
higher curvature gravitational couplings ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0... | |
1230886b10ddc62601df0b8bec0199abd014cc3d | subsection | 30 | 197 | Examples where | The identification of black hole entropy with entanglement entropy may seem like a radical
proposal at first, but luckily it can be checked in situations
where a UV completion for gravity is known in some detail.
One such example comes from the AdS/CFT correspondence
, ,
in which a quantum gravity theory in anti-de Sit... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.0... | |
673d5c661058155550556307c2298a898816ecc6 | subsection | 31 | 197 | Gravitational dynamics from entanglement | When viewed as entanglement entropy,
it is clear that a generalized entropy can be assigned to surfaces other than black hole
horizon cross sections
, , , .
This is borne out explicitly in AdS/CFT, where the quantum-corrected RT formula
maps the generalized entropy of minimal-area surfaces to the entanglement entropy ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.... | |
cf59957642a4d616a511a2bc83a253283e929b23 | subsection | 32 | 197 | Gravitational dynamics from entanglement | Showing maximality would require consideration of second order perturbations.
the entanglement entropy of a fixed-volume subregion is equivalent to imposing the
linearized Einstein
equation.
One can therefore derive the Einstein equation by assuming that entanglement
entropy is maximized at fixed volume. This is the o... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
8d0f50ed4c54eb3435d9ea0374dc70ee018abd58 | subsection | 33 | 197 | Gravitational dynamics from entanglement | In particular,
even after taking the small ball limit and employing Riemann normal coordinates,
one can only conclude the linearized higher curvature field equations hold from the
entanglement equilibrium requirement, see section . | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
7d3136e088bbc3094a59c2a7df3df8b1d7706fc0 | subsection | 34 | 197 | Edge modes | The discussion up to this point has been reticent about how gauge fields factor into
the identification of black hole entropy with entanglement entropy. This is a subtle point because
the definition of entanglement entropy of a subregion is ambiguous when gauge
constraints are present. The definition of entanglement en... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.025325056165456772,... | |
4e7b9cefd0c65536e927e9c5098882583f9b3d53 | subsection | 35 | 197 | Edge modes | The fact that
the density matrix arose from a global state satisfying the gauge constraint allows one to conclude
that each edge mode density matrix must be maximally mixed in its representation R_i,\rho _\text{edge}^i = \frac{\mathbb {1}}{\dim R_i }.The entropy simply follows from plugging the density matrix (REF ) in... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0... | |
c5c85fd1fdcf65bb2af99d4e2fee7f1062660785 | subsection | 36 | 197 | Edge modes | The algebra turns out to be universal for all diffeomorphism-invariant theories (for a given
choice of Noether charge ambiguities), and can include transformations that move the surface
if the fields satisfy appropriate boundary conditions. The identification
of this symmetry algebra and the symplectic structure for th... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.... | |
b6ebdb982d30e41ffa7861b51a9de1ed338c7db1 | subsection | 37 | 197 | Summary | A driving motivation behind this thesis is the idea that gravity tends to act as a universal
regulator, following from the underlying diffeomorphism symmetry. This statement
suggests that gravity renders finite the divergences appearing in entanglement entropy.
Applying this observation to black holes leads to the iden... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
1bbe41c715cd84b657b8928e8a08902c489dd957 | subsection | 38 | 197 | Excited state entanglement entropy in conformal perturbation theory | This chapter is based on my paper “Entanglement entropy of excited states in conformal
perturbation theory and the Einstein equation,” published in the Journal of High Energy
Physics in 2016 . | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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cd26ee330a677b224b987e327d3a8121f50db3e4 | subsection | 39 | 197 | Introduction | The entanglement equilibrium argument, outlined in section , proceeds by
replacing geometrical quantities that appear in the first law of causal diamond mechanics
(REF ) with an equivalent expression related to entanglement.
The discussion of section motivates interpreting the area term in
this equation with the leadin... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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7a688d6e6844070a8b53913776a4f25cd04cce9b | subsection | 40 | 197 | Introduction | The result for the change in entanglement entropy between the excited
state and vacuum is\delta S = \frac{2\pi \Omega _{d-2} }{d^2-1} \left[R^d\left(\delta \langle T^g_{00} \rangle
-\frac{1}{2\Delta -d} \delta \langle T^g \rangle \right) - R^{2\Delta } \langle \mathcal {O} \rangle _g\delta \langle \mathcal {O} \rangle... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
94b6b3cd1acfca99bc3269c3a62f6bc9f7986fb0 | subsection | 41 | 197 | Introduction | These results agree with the holographic calculations , and this
chapter therefore establishes that those results extend beyond holography.In both equations (REF ) and (REF ), the first terms scaling as R^d
take the form required for Jacobson's argument. However, when \Delta \le \frac{d}{2},
the terms scaling as R^{2\D... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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1c00e3163965f050e7eddd2e7f2d2e37e6706eb4 | subsection | 42 | 197 | Introduction | Thus, the issues of local observables
and entanglement in gravitational theories are intertwined.Despite these challenges, there are indications that a well-defined notion of local observables
and entanglement should exist in gravitational theories. Holography provides a compelling example,
where the entanglement of bu... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.00... | |
7743e0f991f2655d38a0e6462568a209488bd1db | subsection | 43 | 197 | Introduction | We find that for generic diffeomorphism-invariant theories, the transformations
that preserve the entangling surface generate the algebra
\text{Diff}(\partial \Sigma )\ltimes \left(SL(2,\mathbb {R})
\ltimes \mathbb {R}^{2\cdot (d-2)}\right)^{\partial \Sigma }. In certain cases, including general
relativity, the algebra... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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6e740a3535b51f5b800dfdef6d7bb144631847bd | subsection | 44 | 197 | Einstein equation from entanglement equilibrium | This section provides a brief overview of Jacobson's argument for
the equivalence of the Einstein equation and
the maximal vacuum entanglement hypothesis
. The hypothesis states that the entropy of a small geodesic
ball is maximal in
a vacuum configuration
of quantum fields coupled to gravity, i.e. the vacuum is an equ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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... | |
e803e6f2f99892e04e36a60a4c607addf0ed9ce4 | subsection | 45 | 197 | Einstein equation from entanglement equilibrium | With the standard Minkowski time t=x^0 and spatial
radial coordinate r, it is given by\zeta = \left(\frac{R^2-r^2-t^2}{2R}\right) \partial _t - \frac{rt}{R}\partial _r.If R is taken small enough such that \langle T_{00} \rangle is approximately constant throughout
the ball, equation (REF ) becomes\delta S_\text{IR} = 2... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06971346586942673,
0.05732131749391556,
-0.02475377544760704,
0.0032144044525921345,
0.0070774126797914505,
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-0.000817431602627039,
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0.021884657442569733,
-0.012331103906035423,
0... | |
198120ce5f1ceb51b067fffb8471a89278ddd027 | subsection | 46 | 197 | Einstein equation from entanglement equilibrium | In the simplest case, C would be given by the variation of some
scalar operator expectation value, C = \delta \langle X \rangle , with X independent of the quantum
state, since such an object has trivial transformation properties under Lorentz boosts.
We find this to be the case for the first order state variations we ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04349430650472641,
0.027729526162147522,
-0.06800372898578644,
0.010713334195315838,
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-0.012414953671395779,
-0.006291413214057684,
-0.013315361924469471,
0.021945547312498093,
-0.... | |
f712128f36d0b49a90fd6af38032eb3dbb8d2b8c | subsection | 47 | 197 | Entanglement entropy of balls in conformal perturbation theory | Checking the conjecture (REF ) requires a method for calculating the
entanglement entropy of balls in a non-conformal theory. Faulkner has recently
shown how to perform this calculation in a CFT deformed by a relevant operator,
\int f(x)\mathcal {O}(x) . This deformation may be split into two parts,
f(x) = g(x) + \lamb... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.029087109491229057,
-0.01806880161166191,
-0.023806868121027946,
0.03787733614444733,
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0.010117919184267521,
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0.0365649089217186,
-0.015382899902760983,
0.0031055754516273737,
0.02922445721924305,
0... | |
1579f771fd9f144530a57aa6a18039c739722f70 | subsection | 48 | 197 | Entanglement entropy of balls in conformal perturbation theory | This leads to the operator
expression for \rho _0 given in equation (REF ).The path integral representation for \rho is given in a similar manner,\langle \phi _-| \rho |\phi _+ \rangle &= \frac{1}{N} \int _{{\phi (\Sigma _+) = \phi _+\\
\phi (\Sigma _-) = \phi _-}}\mathcal {D}\phi \, e^{-I_0 -\int f\mathcal {O}} \\
&=... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.005305818282067776,
0.019392365589737892,
-0.014593861065804958,
0.06243395432829857,
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0.029111433774232864,
0.004417065065354109,
-0.004539125598967075,
0.041500575840473175,
-0... | |
65586da1818012fbe862ece8bb8eba54b2e8b803 | subsection | 49 | 197 | Entanglement entropy of balls in conformal perturbation theory | Using the
cyclicity of the trace and
equation (REF ), the \beta integral is readily evaluated, and applying
(REF ) one finds\delta S^{(1)} = 2\pi \operatorname{Tr}(\delta \rho \, K) = 2\pi \delta \langle K \rangle .Note when \delta \rho is a first order variation, this is simply the first law of entanglement
entropy (s... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03723753243684769,
0.0031552703585475683,
-0.033727437257766724,
0.00366652337834239,
-0.013521497137844563,
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0.06776009500026703,
0.02950005978345871,
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0.016497448086738586,
0.04355570301413536,
0... | |
426b7a9b3ef6b7dcf866d6adf5b6e0ea1a64e8fa | subsection | 50 | 197 | Entanglement entropy of balls in conformal perturbation theory | The surface element on the horizon is d\Sigma ^a = \xi ^a d\chi dS, where \chi is a
parameter for \xi ^a satisfying \xi ^a\nabla _a\chi =1,
and dS is the area element in the transverse space. T^B_{ab}
is the stress tensor of a scalar field \phi satisfying the Klein-Gordon equation,\nabla _c\nabla ^c \phi -\Delta (\Delt... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.028206385672092438,
0.02103656344115734,
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0.03258455544710159,
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0.03801531344652176,
0.014103192836046219,
0.038442... | |
f84c41889808399f113d15e5324010ab10a932a3 | subsection | 51 | 197 | Entanglement entropy of balls in conformal perturbation theory | This can lead to a divergence in \delta when the contour is close to the branch point (which
can occur when t_B\sim \sqrt{z^2+(\vec{x}-\vec{x}_b)^2}), and this ultimately cancels
against a divergence in \langle T_{00} \mathcal {O}\mathcal {O} \rangle from \delta S^{(1)}. More details about
these divergences and the ori... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06488499045372009,
0.018235551193356514,
-0.04693937674164772,
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0.003805431304499507,
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0.012490207329392433,
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0.03262561187148094,
0.03500615432858467,
0.021... | |
55f22c170184ff978d3df6278eb3e65d64d43189 | subsection | 52 | 197 | Entanglement entropy of balls in conformal perturbation theory | When t_B\sim z, there are divergences associated with switching off the coupling in real
times, and these are regulated with the C(\delta ) contour prescription.Returning to the flux equation (REF ),
since \xi ^a is a Killing vector, this integral defines a conserved quantity, and
may be evaluated on any other surface ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03556413948535919,
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0.034526217728853226,
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0.031351394951343536,
0.049331892281770706,
0... | |
706d20147c371c44a47fed9c63c520c4afe473b6 | subsection | 53 | 197 | Entanglement entropy of balls in conformal perturbation theory | The
\mathcal {T} surface integral
is&2\pi \int _{\mathcal {T}} d\Sigma ^a \xi ^bT^B_{ab} \\
&= \frac{2\pi }{z_0^{d-1}} \int d\Omega _{d-2} \int _0^R dt\int _0^{R-t} dr\, r^{d-2}\left\lbrace
\left[\frac{R^2-r^2-t^2}{2R}\right]\partial _z\phi \partial _t\phi - \frac{z_0t}{R} \left[
(\partial _z\phi )^2-\frac{ \nabla ^2... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05873936414718628,
0.05239246040582657,
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0.012960802763700485,
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0.02592160552740097,
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0.048212047666311264,
0.035182591527700424,
-0... | |
01fe1a6e51c1710ccbc3b0f9410635f8295f5412 | subsection | 54 | 197 | Producing excited states | This section describes the class of states that are formed from the Euclidean path integral
prescription, and also discusses restrictions on the source function f(x). One requirement
is that the density matrix be Hermitian. For a density matrix constructed from a path integral
as in (REF ), this translates to the condi... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.021590467542409897,
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0.04281473904848099,
-... | |
73bce007636e4f2e2fdd41dd85b6db2f7818b91e | subsection | 55 | 197 | Producing excited states | The leading divergence is then\delta \langle \mathcal {O}(0) \rangle _\text{div} &= -\lambda (0)\int _{C(\delta )}d\tau \int d\Omega _{d-2}\int _0^\infty dr
\frac{r^{d-2}c_\Delta }{
(\tau ^2 +r^2)^\Delta } \\
&=
-\lambda (0)\frac{2\Gamma (\Delta -\frac{d}{2}+\frac{1}{2})}{\sqrt{\pi } \,\Gamma (\Delta -\frac{d}{2})}
\d... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06895402073860168,
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0.02411865070462227,
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-0.011929655447602272,
0.0021338369697332382,
0.0... | |
edc56bafbc7dd3d8f2865dbd019c22a76c35f710 | subsection | 56 | 197 | Producing excited states | The first order in \lambda piece, which gives \delta \langle T^0_{\tau \tau } \rangle , is\delta \langle T^0_{\tau \tau } \rangle _\text{div}
= -g \lambda (0) 2^{d-2\Delta }\frac{2\Gamma (\Delta -\frac{d}{2}+\frac{1}{2})}{\sqrt{\pi }\,
\Gamma (\Delta -\frac{d}{2})} \delta ^{d-2\Delta }.The divergence in the actual ener... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04080209136009216,
0.003309256862848997,
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0.03176886588335037,
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-0.02114872634410858,
-0.007896440103650093,
... | |
556caee58c3e778daf09b68a09e9cd16b64dc048 | subsection | 57 | 197 | Entanglement entropy calculation | Now we compute the change in entanglement entropy for the state formed by the path
integral with the deformed action I=I_0+\int f\mathcal {O}, with f(x) = g(x)+\lambda (x) being
a sum of the theory deformation g and the state deformation \lambda .
The bulk term \delta S^{(2)} in
plays an important role in this case.A s... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.013444464653730392,
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-0.01745796576142311,
0.02965107187628746,
-0... | |
4025f2b72125b61940d262ea3129654922204811 | subsection | 58 | 197 | Entanglement entropy calculation | This leads to the equation for the z-dependence,\partial _z^2\phi -\frac{d-1}{z}\partial _z\phi -\left(\omega ^2+\frac{\Delta (\Delta -d)}{z^2}\right)\phi = 0.This has modified Bessel functions as solutions, and regularity as z\rightarrow \infty selects
the solution proportional to z^{\frac{d}{2}} K_\alpha (\omega z), ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.025840088725090027,
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... | |
326d89d054c534ef4036b085e1a4785b2ae55901 | subsection | 59 | 197 | Discussion | The equivalence between the Einstein equation and maximum vacuum entanglement
of small balls relies on
a conjecture about the behavior of the entanglement entropy of excited states, equation
(REF ). This chapter has sought to check the conjecture in CFTs deformed
by a relevant operator. In doing so, we have derived new... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.042579732835292816,
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0.004650958348065615,
0.02885959856212139,
0... | |
c967303c0193a87bb2af0c0292b82c353c9aa5cf | subsection | 60 | 197 | Discussion | Here, both
UV and IR divergences arise, and these are dealt with in the same manner as the
\Delta >\frac{d}{2} and \Delta <\frac{d}{2} cases. The answer before imposing that the state is nonsingular
is given in equation (REF ), and it depends logarithmically on an arbitrary
renormalization scale \mu . This scale \mu ar... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04021484777331352,
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0.004199217073619366,
0.0277201... | |
2971dfcb5e2cc12daf2be2fd1626b066361d08f5 | subsection | 61 | 197 | Discussion | With the additional assumption that matter is conformally invariant, we arrived at our main result (REF ), showing that the equilibrium condition \delta S_\text{EE}\big |_{W^{\prime }}=0
applied to small balls is
equivalent to imposing the linearized constraints \delta C_\zeta = 0.In section , we reviewed the argument ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.035377513617277145,
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0.026220262050628662,
0.007497499231249094,
0.01... | |
e4209b128f9718452862a7d0e09e217a7bbd8fd3 | subsection | 62 | 197 | Implications for the Einstein equation | We now ask whether the results (REF ) and (REF ) are consistent
with the conjectured form of the entanglement entropy variation (REF ). The
answer appears to be yes, with the following caveat: the scalar function C explicitly
depends on the ball size R. This comes about from the R^{2\Delta } in equation
(REF ), in whic... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06415964663028717,
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0.025868363678455353,
0.009736882522702217,
0.017856039106845856,
-0.009210358373820782,
0.00179418851621449,
0.027852367609739304,
0.00956137478351593,
0.020... | |
c9adf72dfa1883a02bd664ef501e2f7deed19e71 | subsection | 63 | 197 | Implications for the Einstein equation | This means \Lambda (R)
\ell _P^2 \ll 1, which then implies\frac{R}{\ell _P} \gg \Big (\ell _P^{2\Delta }\langle \mathcal {O} \rangle _g\delta \langle \mathcal {O} \rangle \Big )^{\frac{1}{d-2\Delta }}This now places a lower bound on the size of the ball for which the derivation is valid. However,
the R-dependence in \L... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.07189694792032242,
0.024275900796055794,
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0.03759636729955673,
0.02384866774082184,
0.001909190439619124,
-0.00016784091712906957,
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0.03301888704299927,
0.01163442712277174,
-... | |
c83b0d3f1bd8612f565e3eed93d51a9bdbc81ac5 | subsection | 64 | 197 | Future work | This work leads to several possibilities for future investigations.
First is the question
of how the entanglement entropy changes under a change of Lorentz frame. The
equivalence between vacuum equilibrium and the Einstein equation rests crucially on
the transformation properties of the quantity C appearing in equation... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
0.011563657782971859,
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0.0016561736119911075,
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-0.0026944999117404222,
-0.018093004822731018,
-0.011525518260896206,
0.023874834179878235,
... | |
34d8069c77b255c5da13bb2da6a06216ee9210aa | subsection | 65 | 197 | Future work | This
can be seen by examining the \mathcal {O}\mathcal {O}\mathcal {O} three point function,\iint d^d x_1 d^d x_2 \big \langle \mathcal {O}(0)\mathcal {O}(x_1)\mathcal {O}(x_2) \big \rangle = \iint d^d x_1 d^d x_2
\frac{c}{|x_1|^\Delta |x_2|^\Delta |x_1-x_2|^\Delta }.By writing this in spherical coordinates, performing... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.031532395631074905,
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0.022725913673639297,
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0.02989930473268032,
0.014789395965635777,
-0.03293655067682266,
-0.010531148873269558,
0.00609356677159667,
0.0... | |
1f85601e1e441e5bf9a9af8b0731b438e498dbf5 | subsection | 66 | 197 | Coefficients for the bulk expansion | This appendix lists the coefficients appearing in section REF for the expansion of
\phi _\omega and \nabla ^2 \phi _0\phi _\omega . Given its definition (REF ), the
coefficients appearing in the expansion (REF ) follow straightforwardly from
known expansions of the modified Bessel functions :a_n &= \frac{\Gamma (\frac{... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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-0... | |
1dd9ba57c5b376592920ebec9408606dd217c781 | subsection | 67 | 197 | Surface integrals | This appendix gives the details of the \mathcal {E} and \mathcal {T} surface integrals
for \Delta <\frac{d}{2} (section REF ) and for \Delta =\frac{d}{2} (section REF ). | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
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0.0... |
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