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Area scaling from entanglement in flat space quantum field theory

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 = ¼ A. S  A. T H. out. in. V. V. An ‘artificial’ horizon.

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Area scaling from entanglement in flat space quantum field theory

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

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

  3. out in V V An ‘artificial’ horizon.

  4. Entropy: Sin=Tr(rinlnrin) Srednicki (1993) Sin=Sout

  5. out Entanglement entropy of a sphere Srednicki (1993) in Entropy R2

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

  7. out in A different viewpoint Restricted measurements No access =

  8. Area scaling of fluctuationsR. Brustein and A.Y. , (2004) OaV12 OaV1ObV2 V2 V1 Assumptions:

  9. Area scaling of correlation functions OaV1ObV2 =  V1  V2 Oa(x) Ob(y) ddx ddy = V1  V2 Fab(|x-y|) ddx ddy = D(x) Fab(x) dx = D(x) 2g(x) dx = - ∂x(D(x)/xd-1) xd-1 ∂xg(x) dx Geometric term: Operator dependent term D(x)=V V d(xxy) ddx ddy

  10. V2 V1 Geometric term D(x)=V1 V2d(xxy) ddx ddy =  d(xr) ddr ddR ddR  Ax +O(x2) d(xr) ddr  xd-1 +O(xd) D(x)=C2 Axd + O(xd+1)

  11. Geometric term D(x)=  d(xr) ddr ddR V1=V2 ddR  V + Ax +O(x2) d(xr) ddr  xd-1 +O(xd) D(x)=C1Vxd-1 ± C2 Axd + O(xd+1)

  12. Area scaling of correlation functions OaV1ObV2 =  V1  V2 Oa(x)Ob(y) ddx ddy = V1  V2 Fab(|x-y|) ddx ddy = D(x) Fab(x) dx = D(x) 2g(x) dx = - ∂x(D(x)/xd-1) xd-1 ∂xg(x) dx UV cuttoff at x~1/L  ∂ x(D(x)/xd-1) 1/L   A D(x)=C1Vxd-1 + C2 Axd + O(xd+1)

  13. Energy fluctuations

  14. V V Intermediate summary Tr(rinOV) Tr(rinOV2)

  15. Trout(y’ y’’ rin(y’in,y’’in) =   Exp[-SE] DfDout f(x,0+)=y’(x) f(x,0)=y(x) f(x,0+)=y’(x) f(x,0-)=y’’(x) t f(x,0-)=y’’(x) in y’in y’’in Exp[-SE] Df f(x,0+) = y’in(x)yout(x) y’in(x) y’(x) y’’(x) f(x,0-) = y’’in(x)yout(x) x y’’in(x) f(x,0+) = y’in(x) f(x,0-) = y’’in(x) Finding rin

  16. in y’in y’’in Exp[-SE] Df f(x,0+) = y’in(x) f(x,0-) = y’’in(x) t y’in(x) x y’’in(x) Finding rho Kabbat & Strassler (1994)  ’| e-bK|’’

  17. t Acceleration = a/z z=const Proper time =  = const t=z/a sinh(ah) x x=z/a cosh(a) ds2 = -a2z2dh2+dz2+Sdxi2 Rindler space(Rindler 1966) ds2 = -dt2+dx2+Sdxi2 HR = Kx

  18. t ah≈ ah+i2p x Radiation at temperature b0 = 2p/a Unruh Radiation(Unruh, 1976) = 0 ds2 = -a2z2dh2+dz2+Sdxi2 Avoid a conical singularity Periodicity of Greens functions rR= e-bHR=  e-bK= rin

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

  20. t in y’in y’’in Exp[-SE] Df y’in(x) x y’’in(x) f(x,0+) = y’in(x) f(x,0-) = y’’in(x) Other shapesR. Brustein and A.Y., (2003) rb=y’in|e-bH0|y’’out d/dt H0 = 0 SE = 0bH0dt x(x,t), h(x,t), +B.C. H0=K, in={x|x>0}

  21. V2 OV1OV2 V1 OV1OV2  A1A2 OV1OV2 Evidence for bulk-boundary correspondence R. Brustein D. Oaknin, and A.Y., (2003) OV1OV2- OV1OV2  V1 V2 Pos. of V2 Pos. of V2

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

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

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

  25. Fin

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