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92ee3cc12a6a2324bd2b8c5d89088034db0e8343 | subsection | 68 | 197 | 1. | This term arises from the piece of \phi _0 and \phi _\omega that goes like \frac{-z^\Delta }{2\Delta -d}. In particular, it has no dependence on t_B anywhere. On the surface \mathcal {E},
since \partial _\tau \phi =0, the integrand in (REF )
only depends on \nabla ^2\phi ^2. Working to leading
order in R means only kee... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03911387547850609,
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0.040242746472358704,
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0.005716819316148758,
-0.04783974587917328,
0.009549642913043499,
0.010388669557869434,
... | |
da1654a1b0be07b0c646d6215700fc801754888d | subsection | 69 | 197 | 1. | Again picking
out the \langle \mathcal {O} \rangle _g\,\delta \langle \mathcal {O} \rangle term in the integrand (REF ), we find\delta S^{(2)}_{\mathcal {T},1} &= -2\pi \langle \mathcal {O} \rangle _g\,\delta \langle \mathcal {O} \rangle \frac{\Omega _{d-2} z_0^{-d+1}}{(2\Delta -d)^2}
\int _0^Rdt\int _0^{R-t}dr\,r^{d-2... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06444930285215378,
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0.013014975935220718,
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0.03018009662628174,
0.0... | |
17155054d9c3e5f6a6c37a32c6a3b53cde5c4a12 | subsection | 70 | 197 | 1. | The divergent integral
evaluates to\delta S^{(2)}_{\mathcal {E},\text{div.}}
= -2\pi g\lambda _\omega \frac{\Omega _{d-2} R^d}{d^2-1}
\log \left( \frac{R}{z_0} \right)\left(1+\frac{d}{2}\gamma _E+\frac{d}{2}\log \frac{\omega R z_0}{4L}\right),and the remaining finite piece with z_0\rightarrow 0 is\delta S^{(2)}_{\mathc... | {
"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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0.0019018062157556415,
0... | |
d6a7790db5f9ea355ec032d78ef1af70f12939ee | subsection | 71 | 197 | 2. | On the surface \mathcal {E}, this term comes from the part of one field going like z^\Delta , and
the other going like z^{d-\Delta }. Hence, when we evaluate this term in \nabla ^2\phi ^2 for the
bulk integral, we will be acting on a term proportional to z^d, which is annihilated by the
Laplacian. So the bulk will only... | {
"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... | |
d9887d64ed76ed7f274d9845af63f45f2bde0b5b | subsection | 72 | 197 | 2. | This term receives no contribution from the region t\sim z, so we can
evaluate it in the region t\gg z, using the asymptotic form for F(t/z). Evaluating
the derivatives in this expression (and recalling that only the z-derivatives in the Laplacian
will produce a nonzero contribution at z\rightarrow 0), this leads to& \... | {
"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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0.035801492631435394,
0.024264439940452576,
0.... | |
7261ce3aa8b594da251c448ecf6020067e61f32c | subsection | 73 | 197 | 2. | The boundary
term at t=0 is2\pi g \lambda _\omega \frac{\Omega _{d-2} R^d}{d^2-1} \frac{d}{2}\log \left( \frac{2L}{z_0} \right) \left(\gamma _E+
\log {\frac{\omega z_0}{2}}
\right).At the other boundary t=c\gg z_0, the asymptotic formulas (REF ) and
(REF ) produce the term-2\pi g \lambda _\omega \frac{\Omega _{d-2} R^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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... | |
1fe57bb6c8410a62efdfd7999b19b7bbdce4688d | subsection | 74 | 197 | 2. | For the region (z+v,c), we can take \delta /z\rightarrow 0
and L/z, a/z\rightarrow \infty , which produces the integral2 \lambda _\omega \int _{z_0+v}^c dt \left(\frac{1}{\sqrt{t^2-z_0^2}} - \frac{t}{t^2-z_0^2}\right)
\rightarrow \lambda _\omega \log \frac{8v}{z_0},where we have taken the limits c/z_0\gg 1, v/z_0\ll 1.... | {
"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.... | |
762aa71015c127638fc702670b5e781fce071bd9 | subsection | 75 | 197 | 3. | The final type of term arises when both fields behave as z^{d-\Delta } F(t/z).
The \mathcal {E} surface term will go like R^{2(d-\Delta )}, and hence will be subleading
compared to the R^d terms. In fact, this calculation is essentially the same as the
change in vacuum entanglement when deforming by a constant source, ... | {
"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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... | |
461186dbefe45f407370e0e5130cfdecff5bef43 | subsection | 76 | 197 | 3. | We start with the first bracketed term in
equation (REF ),\delta S^{(2)}_{\mathcal {T},1} &=
2\pi g\lambda _\omega \Omega _{d-2} \int _c^R dt\int _0^{R-t} dr\, r^{d-2}
\left[\frac{R^2-r^2-t^2}{2R}\right] \frac{d}{2t} \left(\gamma _E+ \log \frac{t^2\omega }{L}\right) \\
&= 2\pi g \lambda _\omega \frac{\Omega _{d-2} R^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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0.010... | |
ff62c062f2e7d73ebe8a440b08d65c2ccf930e75 | subsection | 77 | 197 | 3. | Only the z-derivatives
in the Laplacian term \nabla ^2 \phi ^2
contribute in the limit z\rightarrow 0. Since \phi ^2 scales as z^d,
the z-derivatives in the Laplacian annihilate it, and hence this piece is zero.
The integral then becomes\delta S^{(2)}_{\mathcal {T},2}&=2\pi g\lambda _\omega \Omega _{d-2}\left(\frac{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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0.... | |
83309132def6a59101974426610400e54fa0d936 | subsection | 78 | 197 | 4. | The final divergence in \delta comes from the expectation value of the CFT stress tensor,
in \delta S^{(1)}. At order g \lambda _\omega , this is given by\delta \left\langle T^0_{00}(0) \right\rangle = -\int d^d x_a d^d x_b g\lambda _\omega (x_b)
\left\langle T^0_{\tau \tau }(0) \mathcal {O}(x_a)\mathcal {O}(x_b) \righ... | {
"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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0.03097973... | |
83a6504b40c33c61d6e79b0e939dcd86d241a6d5 | subsection | 79 | 197 | 4. | For \tau _a<0,
the contribution from the operator insertion is&\hphantom{=}
-g\lambda _\omega \frac{1}{V} \int d\vec{x} d\vec{x_a} d\vec{x_b}\int _\delta ^\mu d\tau _b
\int _{-\mu }^{-\delta } d\tau _a \partial _{\tau _a}\left\langle \mathcal {O}(x_a)\mathcal {O}(x_b) \right\rangle \delta (\vec{x}-\vec{x_a}) \\
&= -g\l... | {
"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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ada1667aea92b6efd7c16f600a665c98df4839f4 | subsection | 80 | 197 | Entanglement equilibrium for higher order gravity | This chapter is based on my paper “Entanglement equilibrium for higher order gravity,"
published in Physical Review D in 2017, in collaboration with Pablo Bueno, Vincent Min, and
Manus Visser . | {
"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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... | |
fc30727d927d34840d3b8e7d701a428d592e29f9 | subsection | 81 | 197 | Summary of results and outline | This chapter explores an extension of the entanglement equilibrium argument described in
section to higher curvature theories. For general relativity, the equilibrium
condition applied to the entanglement was\delta S_\text{EE}\big |_V = \frac{\delta A\big |_V}{4G} + \delta S_\text{mat} = 0 \, .It is not a priori clear ... | {
"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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-0.021885206922888756,
... | |
40a905b3e2f4a2558e24fbb2a52667484f09d87d | subsection | 82 | 197 | Summary of results and outline | Using a modified generalized
volume defined byW^{\prime } = W + W_\text{JKM} \, ,the identity (REF ) continues to hold with \delta (S_\text{Wald}+S_\text{JKM})\big |_{W^{\prime }}
replacing \delta S_\text{Wald}\big |_W . As discussed in section REF , the subleading divergences for the
entanglement entropy involve a par... | {
"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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... | |
eeffe8a4fb14205f26559b33e391f6cadc0eab25 | subsection | 83 | 197 | First law of causal diamond mechanics | Jacobson's
entanglement equilibrium argument
compares the surface area of a
small spatial ball \Sigma in a curved spacetime to the one that would be obtained in a MSS.
The comparison is made using balls of equal volume V, a choice justified
by an Iyer-Wald variational identity
for the conformal Killing vector \zeta ^... | {
"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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8389c41bf5780aa27929c803d3dcdc03bad35c8e | subsection | 84 | 197 | Iyer-Wald formalism | We begin by recalling the Iyer-Wald formalism , . A general diffeomorphism invariant theory may
be defined by its Lagrangian L[\phi ], a spacetime d-form locally constructed
from the dynamical fields \phi , which include the metric and matter fields. A variation of this
Lagrangian takes the form\delta L = E\cdot \delta... | {
"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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0.0029... | |
f29c1a2b6af22f51375432de9d565cfac7c10a47 | subsection | 85 | 197 | Iyer-Wald formalism | By combining equations (REF ), (REF ) and (REF ), one finds that-\int _{\partial \Sigma } \delta Q_\zeta +\delta H_\zeta = \int _\Sigma \delta C_\zeta \, .When the linearized constraints hold, \delta C_\zeta = 0, the variation of the Hamiltonian
is a boundary integral of \delta Q_\zeta . This on-shell identity forms 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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0.01... | |
9999fcda7b2998f3842ef370b976eb25028cd9e6 | subsection | 86 | 197 | Geometric setup | Thus far, the only restriction that has been placed on the vector field \zeta ^a is that it
vanishes on \partial \Sigma .
As such, the quantities \delta H_\zeta and \delta Q_\zeta appearing
in the identities depend rather explicitly on the fixed vector \zeta ^a, and therefore these
quantities are not written in terms 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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... | |
380e33522e263198e7b3f21cb5ddc48fbdac7302 | subsection | 87 | 197 | Geometric setup | On the other hand, the covariant derivative of \alpha is nonzero, so\nabla _d (£_\zeta g_{ab}) \big |_\Sigma = \frac{2}{N} u_d g_{ab} \, .The fact that the covariant derivative is nonzero on \Sigma
is responsible for making \delta H_\zeta nonvanishing.A conformal Killing vector with a horizon has a well-defined surfac... | {
"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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... | |
d6f7beca452eead0cb836a079bfeb3fea167c076 | subsection | 88 | 197 | Local geometric expressions | In this subsection we evaluate the Iyer-Wald identity (REF ) for an arbitrary higher derivative theory of gravity and for the geometric setup described above. The final on-shell
result is given in (REF ), which is the first law of causal diamond mechanics for higher derivative gravity.Throughout the computation we assu... | {
"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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0.00815779622644186,
0.02031436190009117,
0.011027142405509949,
-0.03075389936566353,
0.05360183119773865,
-0.036691002547740936,
0.031... | |
747f5922b595a82fdffc55102e46b114279ade14 | subsection | 89 | 197 | Wald entropy. | By virtue of equation (REF ) and the fact that \zeta ^a vanishes on \partial \Sigma , one can show that the integrated Noether charge
is simply related to the Wald entropy ,-\int _{\partial \Sigma }
Q_\zeta &= \int _{\partial \Sigma } \, E^{abcd} \, \epsilon _{ab} \nabla _c \zeta _d \\
&=\frac{\kappa }{2\pi }S_\text{Wa... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.0019662294071167707,
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0.018430301919579506,
-0.012991226278245449,
-0.011... | |
75a0bf6b1444167c50f402a3787a9f4b569b4881 | subsection | 90 | 197 | Generalized volume. | The gravitational part of \delta H_\zeta is related to the symplectic current \omega [\delta g, £_\zeta g] via (REF ). The symplectic
form has been computed
on an arbitrary background
for any higher curvature gravitational theory whose Lagrangian is a function
of the Riemann tensor, but not its covariant derivatives . ... | {
"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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0.06511228531599045,
-0.011327523738145828,
0.009... | |
e709c5afd73faf7c60c01a176a478c425b4d6fd9 | subsection | 91 | 197 | Generalized volume. | Since the tensors E^{abcd}, S^{ab}, and T_i^{abcda_1\ldots a_i} are
all constructed from the metric and
curvature, they will also have vanishing Lie derivative along
\zeta ^a when evaluated on \Sigma .Replacing \delta _2 g_{ab} in equation (REF ) with £_\zeta g_{ab} and using (REF ),
we obtain&\omega ^g[\delta g, £_\ze... | {
"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... | |
73aae14b085018ab9cd4d7c36e16d85a5d6c358e | subsection | 92 | 197 | Generalized volume. | This leads us to define a generalized volume functionalW = \frac{1}{(d-2)E_0} \int _\Sigma {\eta }(E^{abcd}u_a u_d h_{bc} - E_0) \,,and the variation of this quantity is related to the variation of the gravitational Hamiltonian by\delta H_\zeta ^g = -4E_0 \kappa k\, \delta W \,,where we have expressed N in terms of \ka... | {
"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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0.01... | |
635d64b078e4b801e98952acc0d18fbef9c34aa0 | subsection | 93 | 197 | Variation at fixed | We now show that the first two terms in (REF ) can be written in
terms of the variation of the Wald entropy at fixed W, defined as\delta S_\text{Wald}\big |_{W} = \delta S_\text{Wald} - \frac{\partial S_\text{Wald}}{\partial W} \delta W \, .Here we must specify what is meant by \frac{\partial S_\text{Wald}}{\partial W}... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03588881716132164,
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0.008... | |
a0b18867db3d28a42f5484791e7b0c35c9d7a523 | subsection | 94 | 197 | Variation at fixed | This has the effect of changing the Noether current and Noether charge byJ_\zeta &\rightarrow J_\zeta + \mathrm {d}Y[£_\zeta \phi ] \, ,\\
Q_\zeta &\rightarrow Q_\zeta + Y[£_\zeta \phi ] \,.This modifies both the entropy and the generalized volume by surface terms on \partial \Sigma
given byS_{\text{JKM}} &= - \frac{2... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.030195051804184914,
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1ebacc5a14a7d8ba982075f67f36e097c9089465 | subsection | 95 | 197 | Entanglement Equilibrium | The original entanglement equilibrium argument for Einstein gravity stated that
the total variation away from the vacuum of the entanglement of a region at fixed volume is zero.
This statement is encapsulated in equation (REF ), which
shows both an area variation due to the change in geometry, and a matter piece
from v... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03350892290472984,
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... | |
5e92b6d11ff31a1d785a62c834f17c7061eb8506 | subsection | 96 | 197 | Subleading entanglement entropy divergences | As discussed in section , the subleading divergences in the entanglement
entropy are given by a local integral over the entangling surface.
When the entangling surface is the bifurcation surface
of a stationary horizon, this local integral is simply the Wald entropy
, . On nonstationary entangling surfaces,
the computa... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.043786194175481796,
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-0.00054542039288207... | |
85cd32274ea7513159c2519443819594e4b45bd4 | subsection | 97 | 197 | Subleading entanglement entropy divergences | In induced gravity scenarios, the divergences are determined by
the matter content of the theory, and the matching
to gravitational couplings
has
been borne
out in explicit examples , , . | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.014914817176759243,
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-0.0095439571887254... | |
fafda70fe119498a9e892e0a1d572366bcada3e3 | subsection | 98 | 197 | Equilibrium condition as gravitational constraints | We can now relate the variational identity (REF ) to entanglement entropy.
The reduced density matrix for the ball in vacuum takes the form\rho _\Sigma = e^{- H_\text{mod}}/Z \,,where H_\text{mod} is the modular Hamiltonian
and Z is the partition function, ensuring
that \rho _\Sigma is normalized.
Since the matter is c... | {
"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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... | |
7c0760e8b6067a536611233729e74fd819b7a868 | subsection | 99 | 197 | Field equations from the equilibrium condition | The entanglement equilibrium hypothesis provides a clear connection between the
linearized gravitational constraints and the maximality of entanglement entropy at
fixed W^{\prime } in the vacuum for conformally invariant matter. In this section,
we will
consider whether information about the fully nonlinear
field equat... | {
"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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0.014236188493669033,
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0.0136... | |
18bfef9495f0103841a68f15fbe6407c20a9ab78 | subsection | 100 | 197 | Field equations from the equilibrium condition | Analyzing how these states can be incorporated into the entanglement equilibrium
story deserves further attention.u^a \zeta ^b(G_{ab}(0) - 8\pi G \delta T_{ab}) = 0\,.The procedure outlined above applies at all points and all frames, allowing us to obtain the full tensorial Einstein equation.Since we have only been dea... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.039733871817588806,
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0.011558528989553452,
-0.0059661841951310635... | |
610c41edab5bbd4e6310ea3b5a3ec0bde073c5ec | subsection | 101 | 197 | Field equations from the equilibrium condition | We conclude that the linearized equations cannot reproduce the full nonlinear
field equations for higher curvature gravity, and it is only the linearity of the Einstein equation
in the curvature that allows the nonlinear equations to be obtained for general relativity.When linearizing around flat space, the higher curv... | {
"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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0.014436788856983185,
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0.02... | |
78ddfa7e4e4369edb888160e0be8159fe660dcbf | subsection | 102 | 197 | Comparison to other “geometry from entanglement” approaches | Several proposals have been put forward to understand gravitational dynamics
in terms of thermodynamics and entanglement. Here we will compare the
entanglement equilibrium program considered in this chapter
to two other approaches: the equation of state for
local causal horizons, and gravitational dynamics from hologra... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.013461959548294544,
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... | |
f35660756b6c5eb91d95498fb6654d107df9ea2a | subsection | 103 | 197 | Causal horizon equation of state | By assigning an entropy proportional to the area of local causal horizons, Jacobson
showed that the Einstein equation arises as an equation of state
. This approach employs a physical process first law for the local
causal horizon, defining a heat \delta Q as the flux of local boost energy across the horizon.
By assign... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.029675835743546486,
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0.019392281770706177,
-0.0004369843227323144... | |
c47fd39f85fea08f378b7acf696f4712eb6f29a1 | subsection | 104 | 197 | Holographic entanglement entropy | A different approach comes from holography and the
Ryu-Takayanagi formula . By demanding that areas of
minimal surfaces
in the bulk match the entanglement entropies of spherical regions in the boundary CFT, one
can show that the linearized gravitational equations must hold , , . The argument employs an equilibrium stat... | {
"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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... | |
8757350b209b97488654524a7b2cdc205953e80a | subsection | 105 | 197 | Thermodynamic interpretation of the first law of causal diamond mechanics | Apart from the entanglement equilibrium interpretation, the first law of causal diamond mechanics could also directly be interpreted as a thermodynamic relation. Note that the identity (REF ) for Einstein gravity bears a striking resemblance to the fundamental relation in thermodynamicsdU = T dS - p dV,where U(S,V) is ... | {
"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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... | |
2e1f744d58cbafdaf3b0282fb11453ab94e52a23 | subsection | 106 | 197 | Generalized volume and holographic complexity | The emergence of a generalized notion of volume in this analysis is interesting in
its own right. We showed that when perturbing around a maximally symmetric background, the
variation of the generalized volume is proportional to the variation of the gravitational part of the Hamiltonian. The fact that the Hamiltonian c... | {
"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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0.0090734101831913,
0.... | |
2d8097bd2b94f159196d3f30e420e05e39ee1158 | subsection | 107 | 197 | Higher order perturbations | In this chapter we restricted attention only to first order perturbations of the entanglement
entropy and the geometry. Working to higher order in perturbation theory could yield several
interesting results. One such possibility would be proving that the vacuum entanglement
entropy is maximal, as opposed to merely extr... | {
"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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bcc3fca60519addd2c4e3a45125c4b69700f4d77 | subsection | 108 | 197 | Nonminimal couplings and gauge fields | We restricted attention to minimally coupled matter throughout this chapter. Allowing for
nonminimal coupling can lead to new, state-dependent divergences in the entanglement
entropy . As before, these divergences will be localized on
the entangling surface, taking the form of a Wald entropy. It therefore
seems plausib... | {
"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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-0.0218362528830766... | |
994d573ffd1a0cd5e38361290e3e5813c11a8005 | subsection | 109 | 197 | Nonspherical subregions | The entanglement equilibrium condition was shown to hold for spherical subregions and
conformally invariant matter. One question that arises is whether an analogous
equilibrium statement holds for linear perturbations to the vacuum in an arbitrarily
shaped region. Nonspherical regions present a challenge because
there ... | {
"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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0.0045005870051681995,
... | |
15954dc48a1898682dea28bdb18d99c45c05fe8a | subsection | 110 | 197 | Physical process | As emphasized above, the first law of causal diamond mechanics is an equilibrium state construction
since it compares the entropy of \partial \Sigma on two infinitesimally related geometries
. One could
ask whether there exists a physical process version of this story, which
deals with entropy changes and energy fluxes... | {
"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... | |
0ff072d6aaaf2339c144f7e203562dfb20fe8b49 | subsection | 111 | 197 | Conformal Killing vector in flat space | Here we make explicit the geometric quantities introduced in section REF in the case of a Minkowski background, whose metric we write in spherical coordinates, i.e., ds^2 = - dt^2 + dr^2 + r^2 d \Omega _{d-2}^2. Let \Sigma be a spatial ball of radius \ell in the time slice t=0 and with center at r=0. The conformal Kill... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.030986418947577477,
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... | |
ddf3d543e401518318461572faf4620db5acc8e8 | subsection | 112 | 197 | Conformal Killing vector in flat space | More explicitly, if
£_{\zeta }g_{ab}=2\alpha g_{ab}, then £_{\zeta }\bar{g}_{ab}=0
as long as g_{ab} and \bar{g}_{ab} are related through \bar{g}_{ab}=\Phi \, g_{ab}, where \Phi satisfies£_{\zeta } \Phi +2\alpha \Phi =0\, .For the vector (REF ), this equation has the general solution\Phi (r,t)=\frac{\psi (s)}{r^2}\, \q... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03467702493071556,
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0.01459884364157915... | |
a79b833ec70d2ad273ae05800bba1e873d55f81f | subsection | 113 | 197 | Generalized volume in higher order gravity | The generalized volume W is defined in (REF ).
We restate the expression hereW= \frac{1}{(d-2)E_0} \int _\Sigma \eta \left( E^{abcd} u_a u_d h_{bc} - E_0 \right) \, ,where E_0 is a theory-dependent constant defined by the tensor E^{abcd}
in a maximally symmetric solution to the field equations through
E^{abcd}\overset{... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03934180736541748,
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0.... | |
df5d02a5bc1a756672678818fdae83739a8f137c | subsection | 114 | 197 | Quadratic gravity. | A general quadratic theory of gravity is given by the LagrangianL_\text{quad} = & \, \epsilon \bigg [ \frac{1}{16 \pi G} \big ( R -2 \Lambda \big ) + \alpha _1 R^2 + \alpha _2 R_{ab} R^{ab} \\
& + \alpha _3 R_{abcd}R^{abcd} \bigg ] \, .Taking the derivative of the Lagrangian with respect to the Riemann tensor leaves us... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.030279092490673065,
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0.04383142665028572,
-0.013933876529335976,... | |
b3bd7ce73bcac8e822391bd942bf5bacc73111d5 | subsection | 115 | 197 | Linearized equations of motion for higher curvature gravity using RNC | The variational identity (REF ) states that the vanishing of the linearized constraint equations \delta C_\zeta is equivalent to a relation between the variation of the Wald entropy, generalized volume, and matter energy density.
In , Jacobson used this relation to extract the Einstein equations,
making use of Riemann ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04976819455623627,
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0.0... | |
91f2af1e1ea9b24570a81b8cd64b2f042e8fc227 | subsection | 116 | 197 | Linearized equations of motion for higher curvature gravity using RNC | Taking the stress tensor T^{ab} to be constant for small enough balls, the variation of (REF ) reduces to\delta H^{m}_{\zeta } = \frac{\Omega _{d-2}\ell ^{d}}{d^2-1} \kappa u_a u_b \delta T^{ab} + \mathcal {O}\left(\ell ^{d+2}\right)\, ,where \Omega _{d-2} denotes the area of the (d-2)-sphere, \ell is the radius of our... | {
"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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-0.010... | |
ba14c7ca08a4d4ba06e828296b412a4269e82b63 | subsection | 117 | 197 | Linearized equations of motion for higher curvature gravity using RNC | To evaluate the spherical integral, it is useful to note that spherical integrals over odd powers of n^i vanish and furthermore\int d\Omega \, n^i n^j &= \frac{\Omega _{d-2}}{d-1}\delta ^{ij} \, , \\
\int d\Omega \, n^i n^j n^k n^l &= \frac{\Omega _{d-2}}{d^2-1}\left(\delta ^{ij}\delta ^{kl}+\delta ^{ik}\delta ^{jl}+\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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-0.029030553996562958,
0... | |
56728295d3e4a568ad0be67de175ef5e4bb8d235 | subsection | 118 | 197 | Local phase space and edge modes for diffeomorphism-invariant theories | This chapter is based on my paper “Local phase space and edge modes for
diffeomorphism-invariant theories," published in the Journal of High Energy Physics in
2018 . | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
0.002495343564078212,
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0.008340705186128616,
0.01... | |
08bbe69c600585e35a007aa1654c5afdf943c9c2 | subsection | 119 | 197 | Covariant canonical formalism | The covariant canonical formalism , , ,
provides a Hamiltonian description of a field theory's degrees of freedom
while maintaining spacetime covariance. This is achieved by working with the space \mathcal {S} of
solutions to the field equations.
As long as the field equations admit a well-posed initial
value formulat... | {
"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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0.0183738861232996,
0.042333... | |
d6a4df0df027c6ea4cb00d2fa73b76da811ef2d9 | subsection | 120 | 197 | Covariant canonical formalism | Despite being infinite-dimensional, many concepts from finite-dimensional differential
geometry, such as vector fields, one-forms, and Lie derivatives,
extend straightforwardly to \mathcal {S}, assuming it satisfies some technical
requirements such as being a Banach manifold , .
One begins by understanding the function... | {
"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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0.01254302728921175,
-0.00845357310026884,
0.... | |
9f5efc2941a9c5c368ced57ab8744bec841c0b50 | subsection | 121 | 197 | Covariant canonical formalism | I_V can be completely characterized by its action on one-forms I_V \delta \phi ^x=\Phi _V^x,
along with the antiderivation property, linearity, nilpotency I_\Phi ^2
=0, and requiring that it annihilate scalars.
Just as in finite dimensions, the action of the \mathcal {S} Lie derivative, denoted L_V,
is related to \delt... | {
"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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... | |
b1485a4d57649ba2561b7caf05b70e5c4071008f | subsection | 122 | 197 | Covariant canonical formalism | Then Y^*_\lambda \alpha _\lambda and Y^*_0\alpha _\lambda are related
to each other at all values of \lambda by a diffeomorphism, Y_\lambda ^* (Y_0^{-1})^*.
The first order change in these quantities at \lambda =0 is given by
L_V, and since the two quantities differ at first order by an infinitesimal diffeomorphism, we... | {
"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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0.03330303728580475,
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0.... | |
f8b0341f74561ea419038556ba1766801a9ccf2a | subsection | 123 | 197 | Covariant canonical formalism | Using that Y^* and (Y^{-1})^* are inverses of each other, we find
\begin{equation}\delta \alpha = \delta Y^* (Y^{-1})^*\alpha = Y^*[\delta (Y^{-1})^*\alpha + £_{{{_Y}} (Y^{-1})^* \alpha ]
= \delta \alpha +£_{\delta _{Y^{-1}}}\alpha + £_{Y^*{{_Y}} \alpha ,
}where the last equality involves the identity \ref {id:liexiYi}... | {
"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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0.014992107637226582,
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0.... | |
d29d08bc4f72f5387cf3fd397ddfaeda8ca6fb77 | subsection | 124 | 197 | Covariant canonical formalism | First, a diffeomorphism moves points on the mainfold
around, and hence changes the shape and coordinate location of the
surface.
Second, since solutions related to each other by a diffeomorphism represent
the same physical
configuration, the true phase space \mathcal {P} is obtained by projecting all solutions in
a gau... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.010920069180428982,
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0.03375987708568573,
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0.016544172540307045,
-0.018070386722683907,
... | |
2620ef1e8218650f322f976be021cb6ee202a5fd | subsection | 125 | 197 | Covariant canonical formalism | \end{equation}The fields X can be defined through a \text{Diff($M$)}-valued function {X}:\mathcal {S}\rightarrow \text{Diff}(M). In a given solution s, X is identified with
the diffeomorphism in the image of the map,
X={X}(s). One way to interpret X is as defining
a map from (an open subset of) \mathbb {R}^d into the s... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03170720860362053,
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0.0038260743021965027,
... | |
6514219b3116cc58c56cf26d747668d44ce74e21 | subsection | 126 | 197 | Covariant canonical formalism | \end{equation}In particular,
the \mathcal {S} Lie derivative L_{\hat{\xi }} must annihilate X^* \phi for any \xi , so from
(\ref {eqn:LphX*a}),
\begin{equation}
0=L_{\hat{\xi }} X^*\phi = X^*(L_{\hat{\xi }} \phi +£_{I_{\hat{\xi }}{_X}\phi ) = X^*(£_\xi \phi +
£_{I_{\hat{\xi }} {_X} \phi ),
}and hence
\begin{equation}
I... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.058834291994571686,
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... | |
227381c5fc1d495f130ac0064aae2a87aa86bfbb | subsection | 127 | 197 | Covariant canonical formalism | Finally, we note that when
no confusion will arise, we will simply denote {_X^a by {^a to avoid excessive
clutter. When referring to other diffeomorphisms besides X, we will explicitly include the
subscript, as in {_Y^a.
}}\section {Extended phase space}
We now turn to the problem of defining a gauge-invariant symplec... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.025288043543696404,
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0.... | |
702f31433b58aafbaba36d307b5e8bd6d8c6cd55 | subsection | 128 | 197 | Covariant canonical formalism | If the boundary were at asymptotic infinity, such diffeomorphisms could be disallowed by
imposing boundary conditions on the fields,
or could otherwise be regarded as true
time evolution with respect to the fixed asymptotic structure,
in which case degeneracy would
not be expected \cite {Ashtekar1991}. For a local subr... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04852129518985748,
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... | |
e3c033b2f93a0d6a61b8ef31084476bbd9c0686a | subsection | 129 | 197 | Covariant canonical formalism | An equivalent
expression for \theta ^{\prime } can be obtained by introducing the Noether current
for a vector field \xi ^a,
\begin{equation}J_\xi = I_{\hat{\xi }} \theta -i_\xi L,
\end{equation}where i_\xi denotes contraction with the spacetime vector \xi ^a.
Due to diffeomorphism invariance,
J_\xi is an exact form wh... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.022609539330005646,
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0.02790340594947338,
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0.009031610563397408,
-0.011503098532557487,
0.0... | |
d20bdb6555c31d6d2fade61febb91271e244440e | subsection | 130 | 197 | Covariant canonical formalism | This gives
\begin{align}
\Theta &= \int _\sigma \theta [X^*\phi ;\delta X^*\phi ] \\
&= \int _\Sigma (\theta +i_{{{}}L) + \int _{\partial \Sigma } Q_{{{}}.
}
The second line uses the alternative expression (\ref {eqn:th^{\prime }2}) for
\theta ^{\prime }, and is written as an integral
of fields defined on the
original ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.029937420040369034,
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0.002044655615463853,
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0.02000405453145504,
0.0223081074655056,
0.05737242475152016,
0.032745007425546646,
-0.003288233419880271,
0.02651948854327202,
0.011077760718762875,
-0.0001... | |
fede682a06e0c7825b4cb685f19811d6cf1dbc4b | subsection | 131 | 197 | Covariant canonical formalism | The remaining three terms in the bulk \Sigma integral simplify to an exact form on-shell
d(i_{{}\theta +\frac{1}{2} i_{{}i_{{}L) (see identity \ref {id:liedxth}), so the final expression
is
\begin{equation}
\Omega = \int _\Sigma \omega + \int _{\partial \Sigma } \left[\delta Q_{{{}}+£_{{}Q_{{}+ i_{{}\theta +\frac{1}{2}... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03439781069755554,
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0.009133580140769482,
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0.012056020088493824,
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0.006172987632453442,
0.005898293107748032,
-0.0... | |
20d5c336de9d9bc519018f22d74e986d2226252d | subsection | 132 | 197 | Covariant canonical formalism | To see how these derivatives appear, we decompose \delta Q_{{} as
\begin{equation}
\delta Q_{{}= Q_{\delta ({{})} + \text{}_{{},
}where \text{}_\xi =\text{}_\xi [\phi ;\delta \phi ]\footnote {\text{} is the archaic Greek letter ``qoppa.^{\prime \prime }
\footnotesize }
is a variational one-form depending on a vector \... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.045827507972717285,
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0.012471367605030537,
0.0001027362304739654,
-0... | |
49697dd946dcd9149ca561d0e4107e5608ab4254 | subsection | 133 | 197 | Covariant canonical formalism | One way to see its multivaluedness is to note that i_{{}L is a top rank
form on \Sigma , so, by the Poincaré
lemma applied to \Sigma , it can be expressed as the exterior derivative of a (d-2)-form,
\begin{equation}
i_{{}L\big |_\Sigma = d h_X i_{{}L.
}Here, h_X is the homotopy operator that inverts the exterior deriva... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05121269449591637,
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0.014069756492972374,
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0.012101211585104465,
0.009964806027710438,
0.0093009... | |
73e622cfef3a0a08f0235397f8e9c563841b674b | subsection | 134 | 197 | Covariant canonical formalism | Normally it is required that
the ambiguous terms be locally constructed from the dynamical fields in a spacetime-covariant
manner. In the extended phase space, however, there is additional freedom provided by the X
fields as well as the surfaces \Sigma and \partial \Sigma to construct forms that would
otherwise fail to... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.021433008834719658,
0.0036687818355858326,
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0.02292797900736332,
0.010762268677353859,
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-0.0014711357653141022,
0.01334033161401748... | |
438a5c86a4a2ea22eb0e6e42ed505620705bf562 | subsection | 135 | 197 | Covariant canonical formalism | Since \Theta changes by an \mathcal {S}-exact form, the symplectic form \Omega =\delta \Theta
receives no change from this type of ambiguity, which can also be checked by tracking
the changes of all quantities in (\ref {eqn:OmS}).
Given that only \Omega , and not \Theta , is needed in the construction of the phase spa... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.051935575902462006,
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-0.030285580083727837,
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0.05056242644786835,
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0.006572047248482704,
0.007739224471151829... | |
219d2bb9da933f1a6f3a8bff5ea97cdd57f364b6 | subsection | 136 | 197 | Covariant canonical formalism | This affects which parts of {{}^a correspond to degenerate directions, and will lead to
different numbers of boundary degrees of freedom in the reduced phase space. As discussed
in section \ref {sec:ssa}, this ambiguity can also be used to reduce the surface symmetry
algebra to a subalgebra.
}Give that \beta contribute... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04737512022256851,
0.005895180627703667,
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0.030952559784054756,
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0.0003479396691545844,
0.0009176640305668116,
-0.0... | |
e93aae423e38a7e047eb85a1b7695440bb09ea15 | subsection | 137 | 197 | Covariant canonical formalism | Analogous to
vector fields defined on M, w^a defines
a vector \hat{w} on \mathcal {S}, whose action on the pulled back fields X^*\phi is given by the
Lie derivative,
\begin{equation}L_{\hat{w}} X^*\phi = £_w X^*\phi = X^*£_{(X^{-1})^*w}\phi ,
\end{equation}while its action on \phi is trivial, L_{\hat{w}}\phi = 0. On th... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.048483721911907196,
0.01885647512972355,
-0.012082790024578571,
-0.026255659759044647,
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0.028803419321775436,
-0.008428966626524925,
-0.003648102516308427,
0.029489941895008087,
0... | |
83f7d4d44c3e79e1b3bd2116a45c5768f40ff110 | subsection | 138 | 197 | Covariant canonical formalism | This occurs if W^a is tangent to \partial \Sigma or vanishing at
\partial \Sigma , and hence defines a mapping of the surface into itself. If W^a is tangential,
it generates a diffeomorphism \partial \Sigma , while vector fields that vanish on \partial \Sigma
generate transformations of the normal bundle to the surfac... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.014776187017560005,
0.03280206769704819,
-0.02186295948922634,
-0.023541204631328583,
0.0192693080753088,
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0.013654814101755619,
0.02200026996433735,
0.06044734641909599,
0.03908785805106163,
-0.008063205517828465,
0.04326821491122246,
0.0022198609076440334,
0.01696... | |
690e81ee8d8d3e1cd17a2ad2c66dfd004c77cf75 | subsection | 139 | 197 | Covariant canonical formalism | The transformations they induce on \mathcal {S} are pure gauge, and they drop out after passing
to the reduced phase space.
\end{equation}To identify the surface symmetry algebra, it is useful to first describe the larger algebra of
surface-preserving diffeomorphisms, which contains the surface symmetries as a subalgeb... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.053749363869428635,
0.048804789781570435,
-0.03885459527373314,
-0.017016053199768066,
0.012552200816571712,
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0.034154195338487625,
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0.039556603878736496,
0.040105998516082764,
-0.017244968563318253,
0.021014444530010223,
0.016191957518458366,
0.0... | |
489eb89406d84ce82e87e8949b01a09c3fb36473 | subsection | 140 | 197 | Covariant canonical formalism | This
establishes that the group of surface-preserving diffeomorphisms is \text{Diff}(\partial \Sigma )
\ltimes {\text{Dir}_{\partial \Sigma } } .
}The surface symmetry algebra is represented as a subalgebra of \text{Diff}(\partial \Sigma )
\ltimes {\text{Dir}_{\partial \Sigma } } . The Hamiltonian for a surface-preserv... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.07173418253660202,
-0.0028865376953035593,
-0.01176745817065239,
-0.0056814998388290405,
0.0009448498021811247,
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0.002657598815858364,
0.009104136377573013,
0.02496960014104843,
0.031746190041303635,
0.003285272978246212,
0.026999523863196373,
-0.009790952317416668,
... | |
cdc66afb1d7aae33c10e19996ecf34d659ae95ea | subsection | 141 | 197 | Covariant canonical formalism | The fact that only the traceless part of \nabla _aW^b
contributes to the Noether charge, which follows from the antisymmetry of E^{abcd} from
equation (\ref {eqn:Qxi}) in c and d, translates to the requirement that
W{_i^j} be traceless when W^a vanishes on \partial \Sigma . This means that
the 2\times 2 matrices W{_i^j... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.059469230473041534,
0.022854004055261612,
-0.024379638954997063,
-0.01157193724066019,
0.026546040549874306,
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-0.0000858169369166717,
0.028819235041737556,
0.011427002027630806,
0.033808059990406036,
-0.006403850857168436,
0.013410327024757862,
0.0020329079125076532... | |
27baddd3872f5f9d784859d34c703a21d3068f11 | subsection | 142 | 197 | Covariant canonical formalism | A particular class of theories
in which this occurs are f(R) theories (which include general relativity),
where the Lagrangian is a function of the Ricci scalar,
and E^{abcd} = \frac{1}{2}f^{\prime }(R)(g^{ac}g^{bd}-g^{ad}g^{bc}). In more general theories, however,
n_{ab}E{^a^b^c_d} will have a tangential component on ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06262271106243134,
0.01284040231257677,
-0.026687894016504288,
0.0038528835866600275,
0.025451919063925743,
-0.041565366089344025,
-0.0020923828706145287,
0.012230044230818748,
0.039001863449811935,
0.016006633639335632,
-0.036499395966529846,
0.012115602381527424,
0.01327528152614832,
... | |
d31901396f2ecf4e0053553ef54ba6abe24f5412 | subsection | 143 | 197 | Covariant canonical formalism | Determining how to fix
the ambiguity remains an important open problem for the extended phase space program.
\end{align}}\subsection {Surface translations}
While the surface-preserving transformations are present for generic surfaces, in situations
where the fields satisfy certain boundary conditions at \partial \Sigm... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.034360840916633606,
0.006038322579115629,
0.0011014312040060759,
-0.00032923734397627413,
0.019255496561527252,
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-0.004478199873119593,
0.008635983802378178,
0.04437003657221794,
0.04430900514125824,
0.010955187492072582,
0.02509928122162819,
0.01082549523562193,
0... | |
3a8422c84c2aa7171a0275dea30008f8093790bc | subsection | 144 | 197 | Covariant canonical formalism | \end{equation}With this, the Poisson bracket is given by
\begin{align}
\lbrace H_{\hat{w}}, H_{\hat{v}} \rbrace &= -I_{\hat{w}} \delta \int _{\partial \Sigma } \left(Q_V-i_V B\right) \\
&= \int _{\partial \Sigma } \left(-I_{\hat{w}}\delta Q_V - £_W Q_V+I_{\hat{w}} i_{\delta V} B + £_W
i_V B \right) \\
&=\int _{\partial... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.02007065713405609,
-0.014018933288753033,
-0.02736630104482174,
-0.017750702798366547,
-0.014904179610311985,
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0.03296776860952377,
0.014850758947432041,
0.05656414479017258,
0.023611638695001602,
-0.01458365935832262,
0.023489536717534065,
0.00447583245113492,
-0.0... | |
7e2247c31c5b79ae53670d3d1bb4f0bbc7c4e896 | subsection | 145 | 197 | Covariant canonical formalism | The symmetry generators are simply given by the integrated Noether charge, which is modified to
modified to Q_W\rightarrow Q_W-i_W B by the ambiguity. Hence, the generators H_{\hat{w}}
are the same as in (\ref {eqn:Hhw}), and their Poisson brackets still involve the central
charges K[\hat{w}, \hat{v}]. Finally, note th... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.010468513704836369,
-0.0020315172150731087,
-0.042881228029727936,
-0.009446077980101109,
0.005730214063078165,
-0.05328870192170143,
0.015008429996669292,
0.003105455543845892,
0.03302312269806862,
0.02632388472557068,
-0.0012389393523335457,
0.0056424676440656185,
0.018220705911517143,
... | |
c59a4a7a779a2fc786f810037f141e85bb9c1ae5 | subsection | 146 | 197 | Covariant canonical formalism | In practice, equation (\ref {eqn:iwthds}) may only be obeyed for some
specifically chosen normal vectors \cite {Brown1986a}. The resulting
algebra will then be a subalgebra of the generic case considered in this section.
}}}}\section {Discussion}
}Building on the results of \cite {Donnelly2016F}, this chapter has desc... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03213836997747421,
0.05356395244598389,
-0.03332868218421936,
-0.014245569705963135,
0.016557518392801285,
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0.016038665547966957,
0.00016440681065432727,
0.01261270884424448,
0.0057798707857728004,
-0.008820503018796444,
0.03772367164492607,
-0.014566037803888321,
0... | |
c027e6973d26c80f8962c8028aacf9ad59d2ca5c | subsection | 147 | 197 | Covariant canonical formalism | The additional abelian factor \mathbb {R}^{2\cdot (d-2)}
arises generically; however,
it is not present in
f(R) theories, in which the tensor E^{abcd} is constructed solely
from the metric and scalars. We also noted that for any theory, there exists a choice
(\ref {eqn:bmod})
of ambiguity terms that can be added to \th... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.036227475851774216,
-0.004551324527710676,
-0.022172924131155014,
0.007187506649643183,
0.008278603665530682,
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0.015870502218604088,
0.028276963159441948,
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0.03418262302875519,
0.012208078987896442,
... | |
0f7658afd84c1a667061100672936c97b7adf2d7 | subsection | 148 | 197 | Covariant canonical formalism | In fact, the generators (\ref {eqn:Hhw}) are
invariant with respect to additional changes to the Lagrangian L\rightarrow L+d\alpha ,
since such a change shifts the Noether charge Q_W\rightarrow Q_W + i_W\alpha , but
also induces the change B\rightarrow B+\alpha . An ambiguity that does affect the phase
space is the shi... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04497489333152771,
-0.009557927958667278,
-0.01122083980590105,
0.023738445714116096,
-0.013844884000718594,
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0.016980005428195,
0.026255697011947632,
0.04241187497973442,
0.017605505883693695,
-0.011243723332881927,
0.016766421496868134,
0.020031219348311424,
0.001... | |
e0ae5ed9a158148762c767ca01f0d91e1323aea8 | subsection | 149 | 197 | Covariant canonical formalism | The dimension of the representation has some expression in terms of the Casimirs of the
group, and hence this term will take the form of an expectation value of local operators
at the entangling surface. It is conjectured that this term provides a statistical
interpretation for the Wald-like contributions in the genera... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04127134382724762,
-0.006185359787195921,
-0.04865867272019386,
-0.010913096368312836,
-0.0125157181173563,
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0.04609447345137596,
0.01459149643778801,
0.011950984597206116,
0.04859761893749237,
-0.006677593570202589,
0.009356263093650341,
-0.011859406717121601,
0.02... | |
52e270cd8bcd812d1139c8ace21749f9b8ddd4fd | subsection | 150 | 197 | Covariant canonical formalism | Another obstruction
to smoothness comes from issues related to ergodicity and chaos in totally constrained systems
\cite {Dittrich2015}. It would be interesting to understand if these issues are problematic
for the phase space construction
given here, and whether the X fields ameliorate any of these problems.
}}Another... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.015853215008974075,
0.023756934329867363,
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0.029173577204346657,
0.0036981934681534767,
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0.06408421695232391,
0.031401265412569046,
0.02766302041709423,
0.027251049876213074,
-0.035368382930755615,
-0.007289579603821039,
-0.016356732696294785,
... | |
6c837c6fb687c0328d7deb9e1a343af3d7a8efb6 | subsection | 151 | 197 | Covariant canonical formalism | I_U\alpha is a contraction of the vector U with the one-form \alpha , so
the Lie derivative first acts on U to give the vector field commutator L_V U = [V,U],
and then acts on \alpha , with the contraction I_U now being applied to
L_V\alpha . Hence,
on an arbitrary form, L_V I_U \alpha = I_{[V,U]}\alpha + I_U L_V\alpha... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03755791485309601,
-0.009320830926299095,
-0.04549053683876991,
-0.01646019145846367,
-0.0002841251844074577,
0.04347687214612961,
-0.002642936073243618,
0.05491815507411957,
0.041066575795412064,
0.003890035906806588,
-0.029930394142866135,
0.004942633677273989,
-0.0060600657016038895,
... | |
cb693159d7307c6418beeb851708708da4af455c | subsection | 152 | 197 | Covariant canonical formalism | On the other hand,
\begin{align}
L_VI_U Y^*\alpha = I_{[V,U]} Y^*\alpha + I_U L_V Y^*\alpha = I_{[V,U]} Y^*\alpha + I_U
Y^*\left(L_V\alpha + I_{\bar{{}(Y;V)}\alpha .
}
Since \right.U was arbitrary, equating these expressions shows that
\bar{{}^a(Y;V)=I_V\chi _Y^a, showing that the formula holds for
forms of degree n.
}... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.020773107185959816,
0.007238535210490227,
-0.012271549552679062,
-0.023108365014195442,
0.006353274453431368,
0.015469173900783062,
-0.010333132930099964,
0.03559359908103943,
0.07014930248260498,
0.009905765764415264,
-0.003052624175325036,
0.02475678361952305,
-0.01996416226029396,
0.... | |
dbb265de93f2f5afb7d128433ce7d6c3b0168974 | subsection | 153 | 197 | Covariant canonical formalism | \end{align}
}\item \frac{1}{2}[{{_Y},{{_Y}]^a = {_Y^b\nabla _b{_Y^a
\begin{}
This is a consequence of the formula for the commutator of two vectors, [\xi ,\zeta ] =
\xi ^b\nabla _b \zeta ^a - \zeta ^b\nabla _b \xi ^a, along with the fact that since {^a is an \mathcal {S}
one-form, it anticommutes with itself. Alternat... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.05606398358941078,
-0.003906472586095333,
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0.006046639289706945,
0.013138565234839916,
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0.05408022552728653,
0.032533589750528336,
-0.010979323647916317,
-0.027421604841947556,
-0.0043985964730381966,
-0.01942554488778114... | |
5d40c2b9086292286b9ba69da6fa0dccfc44e3e6 | subsection | 154 | 197 | Covariant canonical formalism | \end{}
}\item £_{{}i_{{}= \frac{1}{2}(i_{[{{},{{}]} + d i_{{}i_{{}- i_{{}i_{{}d)
\begin{}
The identity for ordinary spacetime vectors \xi ^a and \zeta ^b \cite {Edelen2005}
\begin{equation}£_\xi i_\zeta = i_{[\xi ,\zeta ]} + i_\zeta £_\xi \end{equation}along with the fact that {{}^a are anticommuting gives
\begin{alig... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.059737589210271835,
0.009615400806069374,
-0.05192316696047783,
-0.0011046264553442597,
-0.012873954139649868,
0.0048458571545779705,
0.018452413380146027,
-0.021047044545412064,
0.05210631713271141,
-0.025167930871248245,
-0.025259505957365036,
0.0014699735911563039,
-0.00528847053647041... | |
5d977e85ad08bb1951590b47e90a36312039ecf2 | subsection | 155 | 197 | Covariant canonical formalism | Then
\begin{align}
L_{[V,\hat{\xi }]}\phi = L_VL_{\hat{\xi }} \phi - L_{\hat{\xi }} L_V \phi = L_V £_\xi \phi - £_\xi I_V\delta \phi =£_{(I_V\delta \xi )\,\hat{}}\,\phi = L_{(I_V\delta \xi )\,\hat{}} \,\phi ,
\end{align}
hence, [V,\hat{\xi }] = (I_V\delta \xi ^a)\,\hat{}.
\end{}
}\item L_{\hat{\xi }} = £_\xi +I_{\delta... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.03417802229523659,
0.01469349768012762,
-0.046018265187740326,
0.009246375411748886,
-0.006999628152698278,
0.0016354717081412673,
0.028685124590992928,
0.0004503507516346872,
0.05862141028046608,
-0.016539720818400383,
-0.012519226409494877,
0.024061936885118484,
-0.014960513450205326,
... | |
ffd769c31a84abe3726746c73a17768871b4333d | subsection | 156 | 197 | Covariant canonical formalism | The left hand side is the graded
commutator of the derivation I_{\hat{{}} and the antiderivation \delta , which defines the
the antiderivation L_{\hat{{}} \cite {Kolar1993}.
}
}\item [V,{\hat{{}}] = (\delta I_V {{}^a)\,\hat{} - [I_V{{},{{}]\,\hat{}
\begin{}
This follows from the defining relation of the bracket \cite ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.068974070250988,
-0.018555855378508568,
-0.032137032598257065,
-0.0027028832118958235,
-0.01500033400952816,
0.017579231411218643,
0.026658782735466957,
0.047488339245319366,
0.06457926332950592,
0.015175821259617805,
-0.018967868760228157,
0.004444402176886797,
-0.03689807280898094,
0.... | |
c5a04ac55c124e12a387df7cd20d675a3b1a57f4 | subsection | 157 | 197 | Covariant canonical formalism | Similarly, we have
\begin{equation}
I_V I_{\delta {{}\;\hat{}} + I_{\delta {{}\;\hat{}} I_V = I_{(I_V\delta {{})\,\hat{}} = I_{[\nu ,{{}]\,\hat{}},
}where equation (\ref {eqn:ddx}) was used in the last equality.
}We then prove the identity through induction on the degree of the form on which it acts.
It is true for sca... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04450710862874985,
0.010607833042740822,
-0.022436710074543953,
-0.00958520732820034,
-0.03583768382668495,
0.01127177570015192,
0.07063747942447662,
0.009829415939748287,
0.04026397317647934,
-0.027336155995726585,
-0.020162513479590416,
0.023825649172067642,
-0.018269892781972885,
-0.... | |
e63257de49247a067ab3380164b399bf3826c2b6 | subsection | 158 | 197 | Covariant canonical formalism | The Lie derivative
term may be expressed
\begin{align}
£_{{{}}Q_{{{}} &= L_{{\hat{{}}} Q_{{{}} + I_{{\delta ({{})} \,\hat{}}\,Q_{{{}} \\
&= I_{\hat{{}}\delta Q_{{}-\delta I_{\hat{{}}Q_{{}-Q_{\delta ({{})} \\
&= I_{\hat{{}}\text{}_{{}+I_{\hat{{}}Q_{\delta ({{})} +\delta Q_{{}- Q_{\delta ({{})} \\
&= \text{}_{{}+ I_{{\ha... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.06482837349176407,
0.02624877728521824,
-0.03058287687599659,
-0.03354349359869957,
-0.04602692648768425,
-0.0036034840159118176,
0.03108648769557476,
0.04569118469953537,
0.019946018233895302,
-0.007874633185565472,
-0.010179031640291214,
0.015672961249947548,
-0.05979227274656296,
0.0... | |
90a226f1f0e97a51f32e5668f408a28665f7dbbd | subsection | 159 | 197 | Covariant canonical formalism | The relevant
term in \text{}[\phi ;£_{{}\phi ]_{{}
comes from the variation of the Christoffel symbol in (\ref {eqn:qoxi}),
which gives
\begin{align}
&\,-\epsilon _{ab}E^{abcd}\left(\nabla _c \nabla _{(d}{{}_{e)}+\nabla _e\nabla _{(d}{{}_{c)}
-\nabla _d\nabla _{(c}
{{}_{e)} {{}^e }}}}\right.\\
=&\,-\frac{1}{2}\epsilon ... | {
"cite_spans": []
} | 1808.03973 | Investigations on entanglement entropy in gravity | [
"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.04202236607670784,
0.02334575727581978,
-0.046416860073804855,
-0.015266294591128826,
-0.003610581625252962,
-0.03698700666427612,
0.06683295220136642,
0.025848180055618286,
0.03848235309123993,
0.01340473722666502,
-0.0062446086667478085,
0.0009283944382332265,
-0.031432852149009705,
-... | |
ccf6cce03ab67b8fc220dc38e839cb3352179154 | subsection | 160 | 197 | Covariant canonical formalism | This
shows that (\ref {eqn:dQlieQ2}) does not depend on second derivatives of {{}^d.
}}
}\clearpage {}
}}}
\small \normalsize }}}\right.\newpage =0mu plus 1mu\bibliographystyle {JHEPthesis}
\begin{}{100}
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"Antony J. Speranza"
] | [
"hep-th",
"gr-qc"
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0.010049101896584034,
0.06983248144388199,
0.0006409449852071702,
0.011025779880583286,
-0.017763333395123482,
-0.0024550482630729675,
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"gr-qc"
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0.03601637855172157,
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-0.0013429836835712194,
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-0.018038712441921234,
0.029637206345796585,
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0.020083710551261902,
0.003700835630297661,
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"hep-th",
"gr-qc"
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-0.043043170124292374,
0.01868256740272045,
-0.009249702095985413,
-0.03187026083469391,
-0.026604341343045235,
-0.0645952820777893,
0.04688958078622818,
-0.020514192059636116,
0.02616170048713684,
0.022605296224355698,
-0.007822562009096146,
0.03443453460931778,
-0.048019081354141235,
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0.020960891619324684,
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0.001155594945885241,
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0.012555177323520184,
0.021693149581551552,
-0.0194963738322258,
0.03000733070075512,
0.02503407746553421,
-0.02883266657590866,
0.04131156578660011,
0.017940325662493706,
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"gr-qc"
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-0.029286053031682968,
0.04080818593502045,
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-0.020388832315802574,
-0.01936633698642254,
-0.005356646608561277,
0.03644350916147232,
-0.029057137668132782,
0.025379210710525513,
0.01871011033654213,
0.02060248702764511,
0.019381599500775337,
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] | [
"hep-th",
"gr-qc"
] | 2,018 | en | Physics | [
-0.046627264469861984,
0.009406317956745625,
-0.012488355860114098,
-0.0345127172768116,
-0.01571534015238285,
0.00500831101089716,
0.035641781985759735,
-0.023084767162799835,
0.00579407811164856,
0.029889661818742752,
-0.029782859608530998,
0.02134539932012558,
-0.006003870628774166,
-0.... | |
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-0.0017194946995005012,
0.03811689466238022,
-0.03537028282880783,
0.026611635461449623,
0.019638288766145706,
-0.0054360064677894115,
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"Antony J. Speranza"
] | [
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"gr-qc"
] | 2,018 | en | Physics | [
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0.036435868591070175,
0.004028086084872484,
-0.00469943368807435,
0.006496051326394081,
0.007327607367187738,
0.02459573745727539,
-0.023542942479252815,
0.01349561382085085,
0.025221310555934906,
-0.005733156576752663,
0.007819674909114838,
0.007251317612826824,
-0... |
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