Entropy, holography and the second law

Daniel R. Terno

PERIMETER INSTITUTE FOR THEORETICAL PHYSICS

Purpose: holographic principlegeneralized second law

to increase our confusion about

Non-covariance of entropy

Number of degrees of freedom

Entropy & temperature

Subjects: entropytemperature

Warning:

1Bc G k 1 2

( ) tr lnS

( ) lni ii

H p p p

tri ip E 0, 1i ii

E E

{ }( ) inf ( , )

EHS p E

Entropy

Meaning: minimum over all possible measurements

von Neumann

Shannon

more entropies…

(=POVM)

lnS N

1:

ST E

( || ) tr ln tr lnS

( || ) lni i ii

S p q p p qRelative entropy

classical

quantum

measure of distinguishability

Perfectly distinguishable states:

Microcanonical entropy

Temperature

NHn edata compression

# degrees of freedom

NSn equantum

data compression

" " SN e

# of degrees of freedom

herethere there

here( ) 0S

2( / )cS f A l Bombelli et al, Phys. Rev. D34, 373 (1986)Holzhey, Larsen and Wilczek, Nucl. Phys. B424, 443 (1994) Callan and Wilczek, Phys. Lett.B333, 55 (1994).

iijij

ic here there

Geometric/entanglement entropy

here trace out “there”

c Pl l

Entropy: non-covariancetrivial 1D representation

1 irrep by 1-particle states

1 2( ) ( ) ( )U U U =?

iijij

ic no correlationsno Bell-type violations

1 2( ) ( )U U not irreducible

Transformations do not split into here and there spaces

Simple example

( ) , [ ( , )] ,U p D W p p

( ) ( ) ,d p p p

*( ) ( ) ( )d p p p

No transformation law for reduced density matricesNoncovariance of spin entropy

Peres and Terno, Rev. Mod. Phys. 76, 93 (2004)

Degrees of freedom: ambiguity

Yurtsever, Phys. Rev. Lett. 91,041302 (2003)

24 P

AS

l

Bekenstein, Lett. Nuovo Cim. 4, 737 (1972) ….Busso, Rev. Mod. Phys. 74,825 (2002)

3 3PN L l4

max ( )E c G L2 2

PS L l

Lorentz boost: factors 1/γ# of degrees of freedom

" " SN e is frame-dependent

Pm

maxE

Entropy: renormalization

Relative entropy

Unruh effect

'', ' ', '

', ' '

1kr

k m k mrk m k

e r rZ

( )S

bosons: maxS

General: cut-offHolzhey, Larsen and Wilczek, Nucl. Phys. B424, 443 (1994)D. Marolf, D. Minic, and S. F. Ross,hep-th/0310022.

: lim ( , ) ( , )l

S S l S l

|| : lim , || ,l

S S l l ?

''

2

krk

r

Z e

a

Cosmic thermo

/S E T

0BH matS S

,E S

BHS

1 8T M

Bekenstein

/BHS E T

Jacobson, Phys. Rev. Lett. 75, 1260 (1995)

Q TdS

12 8R Rg g T

& 4S A & Unruh effect & a bit more

Temperature

Unruh +

Audretsch and Müller, Phys. Rev. D 49, 4056 (1994)

,k m wavepacket basis

' '( , ) (( )' ',', ) am amk k am m e ek

Matter outside the horizon

'', ' ', ' , ,

', ' ',

1 1 ( )!! !

k kqrkk m k m n

r qkm k

m kk

m

mk

n qe r r e r r

Z Z n q

n particles in the mode (k,m)

x

t

(k,m)

(k,-m)

(k’,m’)(-k’,m’)

''

2

krk

r

Z e

a

1ne Special case

renormalized quantitiesE ne

( ( 1) log( 1))S n n n n e

temperature

1 ln( 1)ndS dEdn dNT

Of what?

/ ( ln( 1)) 0S E T n n e

two subsystems

General:

is Temperature undefined?

/S E T

QuestionsTransplantability

What to do without T?

Corrections to Einstein equations?

Thanks to

Charlie BennettFlorian GirelliNetanel LindnerRob MyersDavid PoulinTerry RudolphLee SmolinRafael Sorkin