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Area scaling in Minkowski space

Area scaling in Minkowski space. Rindler space thermodynamics. Area scaling in Minkowski space. Bulk - boundary correspondence. t. Acceleration = a/ z. z =const. Proper time = .  = const. t= z /a sinh(a h ). x. x= z /a cosh(a  ). ds 2 = -a 2 z 2 d h 2 +d z 2 + S dx i 2.

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Area scaling in Minkowski space

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  1. Area scaling in Minkowski space • Rindler space thermodynamics. • Area scaling in Minkowski space. • Bulk - boundary correspondence.

  2. 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

  3. 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

  4. f(x,0)=y’(x) S=Tr(r ln r)  A f(x,b)=y’’(x) -bF=ln(Tr(r))  A Thermodynamics in curved space ’,’’= ’| e-bH|’’  e-SE+… Df

  5. f(x,0+)=y’(x) f(x,0)=y(x) f(x,0-)=y’’(x) t y’(x) y’’(x) x A different method to obtain rKabat and Strassler, 1994)

  6.  y’ y’’ Exp[-SE] Df in out f(x,0+)=y’(x) f(x,0-)=y’’(x) Trout(y’ y’’ Exp[-SE] DfDout in y’in y’’in Exp[-SE] Df f(x,0+) = y’in(x)yout(x) f(x,0-) = y’’in(x)yout(x) f(x,0+) = y’in(x) f(x,0-) = y’’in(x) A different method to obtain r rin(y’in,y’’in) =

  7. 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) A different method to obtain r  ’| e-bHR|’’

  8. t 0 ’,’’= ’| e-bHR|’’ x 0 ’,’’= ’| e-bHR|’’ Intermediate summary I

  9. V V Tr(rinOV) More relations in Minkowski space OV= V O ddx

  10. V V Isothermal compressibility: k (N-N)2 Schematic picture Statistical mechanics In Rindler space (if V is half of space) Q.M. in V of Minkowski space Statistical Mechanics in Minkowski space with d.o.f restricted to V Heat capacity in Rindler space: C  A (HR-HR)2 = Tr(rin(HR-HR)2) = Tr(e-bHR(HR-HR)2])

  11. Assumptions: V  S(V) Area scaling of fluctuations(R. Brustein and A.Y. , 2003) U.V. cutoff 0(OV)20 0OiV OjV 0  S(V)

  12. F(x)=2f(x) Since F(x) =  eiqxcosqF(q) ddq D(x)=V V d(xxy) ddx ddy and F (q) ~ qa = GVVxd-1 – GSS(V)xd+O(xd+1)  ∂ x(D(x)/xd-1)   S Area scaling of correlation functions OiV OjV  = V V Oi(x)Oj(y) ddx ddy =V V Fij(|x-y|) ddx ddy = D(x) Fij(x) dx OiV OjV  = - ∂ x(D(x)/xd-1)xd-1 ∂xf(x) dx Introduce U.V. cutoff short~ 1/L distances

  13. V V Unruh radiation and Area dependent thermodynamics Statistical ensemble due to restriction of d.o.f Intermediate summary II Area scaling of Fluctuations due to entanglement

  14. V2 OV1OV2 V1 OV1OV2  S(B(V1)B(V2)) OV1OV2 Evidence for bulk-boundary correspondence OV1OV2- OV1OV2  V1 V2 Pos. of V2 Pos. of V2

  15. Large N limit A working example

  16. 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 V Summary

  17. On the other hand: Hence: Proof that 0|OV |0=Tr(rinOV) Start with the vacuum state: Find rin:

  18. Proof that Sin=Sout

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