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

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

An ‘artificial’ horizon. V V in out

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

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

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

A different viewpoint in out = No accessRestricted measurements

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

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 

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 )

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

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

Energy fluctuations

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

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)

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)

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

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

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 ) =

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}

Evidence for bulk-boundary correspondence V1V1  O V1 O V2  A 1  A 2 OV1OV2OV1OV2 V2V2 OV1OV2OV1OV2  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)

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

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*.

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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