1. Entropy, holography and the second law Daniel R. Terno PERIMETER INSTITUTE FOR THEORETICAL PHYSICS

2. Purpose: holographic principle generalized second law to increase our confusion about Non-covariance of entropy Number of degrees of freedom Entropy & temperature Subjects:entropy temperature Warning:

3. Entropy Meaning: minimum over all possible measurements von Neumann Shannon more entropies… (=POVM)

4. Relative entropy classical quantum measure of distinguishability Perfectly distinguishable states:  Microcanonical entropy Temperature data compression # degrees of freedom quantum data compression # of degrees of freedom

5. here there 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). here there Geometric/entanglement entropy trace out “there”

6. Entropy: non-covariance = ? no correlations no Bell-type violations not irreducible Transformations do not split into here and there spaces

7. Simple example No transformation law for reduced density matrices Noncovariance of spin entropy Peres and Terno, Rev. Mod. Phys. 76, 93 (2004)

8. Degrees of freedom: ambiguity Yurtsever, Phys. Rev. Lett. 91, 041302 (2003) Bekenstein, Lett. Nuovo Cim. 4, 737 (1972) …. Busso, Rev. Mod. Phys. 74,825 (2002) Lorentz boost: factors 1/γ # of degrees of freedom is frame-dependent

9. Entropy: renormalization Relative entropy Unruh effect bosons: General: cut-off Holzhey, Larsen and Wilczek, Nucl. Phys. B424, 443 (1994) D. Marolf, D. Minic, and S. F. Ross, hep-th/0310022. ?

10. Cosmic thermo Bekenstein Jacobson, Phys. Rev. Lett. 75, 1260 (1995) && Unruh effect & a bit more Temperature

11. Unruh + Audretsch and Müller, Phys. Rev. D 49, 4056 (1994) wavepacket basis Matter outside the horizon n particles in the mode (k,m) x t (k,m) (k,-m) (k’,m’) (-k’,m’)

12. Special case renormalized quantities temperature Of what? two subsystems General: is Temperature undefined ?

13. Questions Transplantability What to do without T? Corrections to Einstein equations?

14. Thanks to Charlie Bennett Florian Girelli Netanel Lindner Rob Myers David Poulin Terry Rudolph Lee Smolin Rafael Sorkin