 Entanglement & area thermodynamics of Rindler space  Entanglement & area  Entanglement & dimensional reduction (holography) Entanglement, thermodynamics & area אוניברסיטת בן - גוריון Ram Brustein sorry, not today! Series of papers with Amos Yarom, BGU (also David Oaknin, UBC) hep-th/ to appear

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

Rindler space

Lines of constant  - constant acceleration horizon Addition of velocities in SR proper acceleration

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

1. In general:

Result inout

H eff – generator of time translations Time slicing the interval [0,  0 ]: 2.

Guess: result

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

inout inout For half space H eff =H Rindler, H Rindler = boost

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

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

Optical metric In 4D Euclidean Rindler Compute:

הפוך

S MS S,T unitary

MM MM M SS M 1 o

 Entanglement & area thermodynamics of Rindler space  Entanglement & area  Entanglement & dimensional reduction (holography) Entanglement, thermodynamics & area אוניברסיטת בן - גוריון Ram Brustein sorry, not today! Series of papers with Amos Yarom, BGU (also David Oaknin, UBC) hep-th/ to appear

inout inout For half space H eff =H Rindler, H Rindler = boost

הפוך

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

EV=EV= For free fields

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

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

For half space

Rindler specific 

E + = …  contributions from the near horizon region

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

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

Proof:

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

Need to evaluate I    k  General cutoff Numerical factors depend on regularization

F(x) (  E V ) 2 for a d-dimensional sphere V D V (x)=

K 27 = KdKd

Fluctuations live on the boundary Covariance V1V1 V2V2 V3V3 V1V1 V1V1 V2V2

EE The “flower” Circles 5 < R < 75 R=40, dR=4, J R=20, dR=2, J R=10, dR=1, J Increasing m

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

 i +  j = 2  logarithmic  i +  j = d   -function

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

First, n-point functions of single fields

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

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