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Area scaling from entanglement in flat space quantum field theory Introduction Area scaling of quantum fluctuations Unruh radiation and Holography

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Black hole thermodynamics J. Beckenstein (1973) S. Hawking (1975) S A THTH S = ¼ A

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An ‘artificial’ horizon. V V in out

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Entropy: S in =Tr( in ln in ) S in =S out Srednicki (1993)

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Entanglement entropy of a sphere out in Entropy R2R2 Srednicki (1993)

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Other Thermodynamic quantities Heat capacity: More generally: A ? ?

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A different viewpoint in out = No accessRestricted measurements

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Area scaling of fluctuations R. Brustein and A.Y., (2004) O a V1 O b V2 V1V1 Assumptions: V2V2 O a V1 2

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Area scaling of correlation functions O a V1 O b V2 = V1 V2 O a (x) O b (y) d d x d d y = V1 V2 F ab (|x-y|) d d x d d y = D( ) F ab ( ) d D( )= V V ( x y ) d d x d d y Geometric term: Operator dependent term = D( ) 2 g( ) d = - ∂ (D( )/ d-1 ) d-1 ∂ g( ) d

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Geometric term D( )= V1 V2 ( x y ) d d x d d y V1V1 V2V2 = ( r) d d r d d R d d R A 2 ) ( r) d d r d-1 +O( d ) D( )=C 2 A d + O( d+1 )

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Geometric term D( )= ( r) d d r d d R d d R V + A 2 ) ( r) d d r d-1 +O( d ) D( )=C 1 V d-1 ± C 2 A d + O( d+1 ) V 1 =V 2

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Area scaling of correlation functions O a V1 O b V2 = V1 V2 O a (x) O b (y) d d x d d y = V1 V2 F ab (|x-y|) d d x d d y = D( ) F ab ( ) d = D( ) 2 g( ) d ∂ (D( )/ d-1 ) = - ∂ (D( )/ d-1 ) d-1 ∂ g( ) d UV cuttoff at ~1/ D( )=C 1 V d-1 + C 2 A d + O( d+1 ) A

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Energy fluctuations

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Intermediate summary V V Tr( in O V ) Tr( in O V 2 )

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Finding in (x,0)= (x) x t ’(x) ’’(x) Tr out ( ’ ’’ in ( ’ in, ’’ in ) = in ’ in ’’ in Exp[-S E ] D (x,0 + ) = ’ in (x) (x,0 - ) = ’’ in (x) (x,0 + ) = ’ in (x) out (x) (x,0 - ) = ’’ in (x) out (x) Exp[-S E ] D D out ’’ in (x) ’ in (x) (x,0 + )= ’(x) (x,0 - )= ’’(x) (x,0 + )= ’(x) (x,0 - )= ’’(x)

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Finding rho x t ’ in (x) ’’ in (x) in ’ in ’’ in Exp[-S E ] D (x,0 + ) = ’ in (x) (x,0 - ) = ’’ in (x) ’| e - K | ’’ Kabbat & Strassler (1994)

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Rindler space (Rindler 1966) ds 2 = -dt 2 +dx 2 + dx i 2 ds 2 = -a 2 2 d 2 +d 2 + dx i 2 t= /a sinh(a ) x= /a cosh(a ) Acceleration = a/ Proper time = x t = const =const H R = K x

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Unruh Radiation (Unruh, 1976) x t ds 2 = -a 2 2 d 2 +d 2 + dx i 2 = 0 a ≈ a +i2 Avoid a conical singularity Periodicity of Greens functions Radiation at temperature 0 = 2 /a R = e - H R = e - K = in

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Schematic picture VEVs in V of Minkowski space VV Observer in Minkowski space with d.o.f restricted to V Canonical ensemble in Rindler space (if V is half of space) Tr( in O V ) = Tr( R O V ) =

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Other shapes R. Brustein and A.Y., (2003) in ’ in ’’ in Exp[-S E ] D (x,0 + ) = ’ in (x) (x,0 - ) = ’’ in (x) x t ’’ in( x) ’ in( x) = ’ in |e - H 0 | ’’ out d/dt H 0 = 0 S E = 0 H 0 dt (x,t), (x,t), +B.C. H 0 =K, in={x|x>0}

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Evidence for bulk-boundary correspondence V1V1 O V1 O V2 A 1 A 2 OV1OV2OV1OV2 V2V2 OV1OV2OV1OV2 V1 V2 V1 V2 O V1 O V2 - O V 1 O V2 Pos. of V 2 R. Brustein D. Oaknin, and A.Y., (2003)

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A working example Large N limit R. Brustein and A.Y., (2003)

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Summary V Area scaling of Fluctuations due to entanglement Unruh radiation and Area dependent thermodynamics A Boundary theory for fluctuations Statistical ensemble due to restriction of d.o.f V A Minkowski observer restricted to part of space will observe: Radiation. Area scaling of thermodynamic quantities. Bulk boundary correspondence*.

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Speculations Theory with horizon (AdS, dS, Schwarzschild) A Boundary theory for fluctuations V Area scaling of Fluctuations due to entanglement Statistical ensemble due to restriction of d.o.f V ??? Israel (1976) Maldacena (2001)

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Fin

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