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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