Entanglement

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Entanglement, thermodynamics area
Series of papers with Amos Yarom, BGU (also
David Oaknin, UBC) hep-th/0302186 to appear

• Entanglement
• area thermodynamics of Rindler space
• Entanglement area
• Entanglement dimensional reduction
  (holography)

sorry, not today!
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Thermodynamics, Area, Holography

• Black Holes
• Entropy Bounds
• BEB
• Holographic
• Causal
• Holographic principle
• Boundary theory with a limited DOF/planck area

thooft, Susskind
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Rindler space
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Lines of constant x -constant acceleration
Addition of velocities in SR
proper acceleration
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Minkowski vacuum is a Rindler thermal
state(Unruh effect)
TFD
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1.
In general
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Result
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2.
Heff generator of time translations
Time slicing the interval 0,b0
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Guess
result
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Results
If

1. The boundary conditions are the same
2. The actions are equal
3. The measures are equal

Then
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For half space HeffHRindler ,
HRindler boost
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Rindler area thermodynamics
Susskind Uglum Callan Wilczek Kabat Strassler De
Alwis Ohta Emparan
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Go to optical space
Compute using heat kernel method
High temperature approximation
Volume of optical space
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Compute
Euclidean Rindler
In 4D
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????
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S,T unitary
S
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Entanglement, thermodynamics area
Series of papers with Amos Yarom, BGU (also
David Oaknin, UBC) hep-th/0302186 to appear

• Entanglement
• area thermodynamics of Rindler space
• Entanglement area
• Entanglement dimensional reduction
  (holography)

sorry, not today!
22
For half space HeffHRindler ,
HRindler boost
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????
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(DEV)2

• System in an energy eigenstate ? energy
  does not fluctuate
• Energy of a sub-system fluctuates ?Entanglement
  energy fluctuations

Connect to Rindler thermodynamics
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For free fields
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For a massless field
Vanishes for the whole space!
Geometry
F(x)
Operator
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UV cutoff!! In this example Exp(-p/L)
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For half space
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Rindler specific heat
_at_ h0
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E ? contributions from the near horizon
region
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Other shapes
y
t
z
Heff complicated, time dependent, no simple
thermodynamics, area dependence o.k. For area
thermodynamics need Thermofield double
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Entanglement and area
Non-extensive!, depends on boundary (similar to
entanglement entropy)
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Proof
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is linear in boundary area
Show that
R is the radius of the smallest sphere containing
V
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Need to evaluate

• Ia ?? ka
• General cutoff

Numerical factors depend on regularization
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(DEV)2 for a d-dimensional sphere
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Fluctuations live on the boundary
V2
V1
V1
V1
V2
V3
Covariance
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The flower
Circles 5 lt R lt 75 R40, dR4, J R20, dR2,
J R10, dR1, J
Increasing m
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Boundary theory ?
This is possible iff
which is generally true for operators of interest
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didj 2 ?? logarithmic didj d ?? d-function
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Boundary correlation functions
Show
(massless free field, V half space, large of
fields N)
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First, n-point functions of single fields
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Then, show that in the large N limit equality
holds for all correlation functions
Only contribution in leading order in N comes
from
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Summary

• Entanglement
• area thermodynamics of Rindler space
• Entanglement area
• Entanglement dimensional reduction