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 TH

out in V V An ‘artificial’ horizon.

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

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

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

out in A different viewpoint Restricted measurements No access =

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

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

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)

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)

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)

Energy fluctuations

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

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

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|’’

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

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

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) = =

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}

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

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

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*.

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)

Fin

Area scaling from entanglement in flat space quantum field theory