Entanglement thermodynamics area Ram Brustein Series of papers

Entanglement, thermodynamics & area Ram Brustein גוריון - אוניברסיטת בן 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!

Thermodynamics, Area, Holography • Black Holes Bekenstein, Hawking • Entropy Bounds – BEB Bekenstein – Holographic Fichler & Susskind, Bousso – Causal Brustein & Veneziano • Holographic principle: ‘thooft, Susskind Boundary theory with a limited #DOF/planck area

Rindler space

Lines of constant x constant acceleration Addition of velocities in SR horizon proper acceleration

Minkowski vacuum is a TFD Rindler thermal state out = z < 0 (Unruh effect) Compare two expressions for rin (by writing them as a PI) 1. 2. in = z > 0

1. In general:

Result out in

2. Heff – generator of time translations Time slicing the interval [0, b 0]:

Guess: result

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

For half space Heff=HRindler , HRindler= boost out in

Rindler area thermodynamics Susskind Uglum Callan Wilczek Kabat Strassler De Alwis Ohta Emparan …

Go to “optical” space Compute using heat kernel method High temperature approximation In 4 D: Volume of optical space

Compute: Euclidean Rindler Optical metric In 4 D

S, T unitary S S M

1 M M o M M S S

Entanglement, thermodynamics & area Ram Brustein גוריון - אוניברסיטת בן 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!

For half space Heff=HRindler , HRindler= boost out in

V 2 (DE ) • System in an energy eigenstate energy does not fluctuate • Energy of a sub-system fluctuates “Entanglement energy” fluctuations Connect to Rindler thermodynamics

For free fields E V=

For a massless field X Vanishes for the whole space! Geometry F(x) Operator

F(x) = UV cutoff!! In this example Exp(-p/L) F(x)

For half space

Rindler specific heat @ h=0

E+ = … contributions from the near horizon region

Other shapes t y z Heff complicated, time dependent, no simple thermodynamics, area dependence o. k. For area thermodynamics need – Thermofield double

Entanglement and area |0> is not necessarily an eigenstate of |0> is an entnangled state w. r. t. V Show: Non-extensive!, depends on boundary (similar to entanglement entropy)

Proof:

Show that R is the radius of the smallest sphere containing V is linear in boundary area

Need to evaluate üIa ka üGeneral cutoff Numerical factors depend on regularization

(DEV)2 for a d-dimensional sphere V F(x) DV(x)=

Kd K 27 =

Fluctuations live on the boundary V 2 V 1 V 2 Covariance V 1 V 3 V 1

The “flower” Increasing m DE Circles 5 < R < 75 R=40, d. R=4, J R=20, d. R=2, J R=10, d. R=1, J

Boundary theory ? Express as a double derivative and convert to a boundary expression This is possible iff which is generally true for operators of interest

di+dj = 2 logarithmic di+dj = d d-function

Boundary* correlation functions Show (massless free field, V half space, large # of fields N)

First, n-point functions of single fields

Then, show that in the large N limit equality holds for all correlation functions Only contribution in leading order in N comes from

Summary ¯ Entanglement & area thermodynamics of Rindler space ¯ Entanglement & area ¯ Entanglement & dimensional reduction