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By M. Aizenman (Chief Editor)

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44(3), 313–338 (1999) 20. : Quantum flows as markovian limit of emission, absorption and scattering interactions. Commun. Math. Phys. 254, 489–512 (2005) 21. : The hamiltonian operator associated with some quantum stochastic evolutions. Commun. Math. Phys. 222, 181–200 (2001) 22. : Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21, 517–540 (1955) 23. : Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93(3), 301–323 (1984) 24. : Completely positive maps and entropy inequalities.

G(ξ ) (W M (φ)))ϒ M , for all φ ∈ S(M) and t ∈ R. (20) t M may be reducible also if is irreducible: in other words λ M may be a mixture also if λ 46 V. Moretti As a consequence of (20) we can conclude that the unique unitary representation U (ξ ) (ξ ) of {gt }t∈R on H M which leaves ϒ M invariant, is nothing but the restriction of U to (ξ˜ ) {gt }t∈R ⊂ G B M S and H M ⊂ H. This result allows us to compute explicitly the self(ξ ) adjoint generator of the unitary representation of {gt }t∈R .

In other words h (ξ ) has no zero modes. 3. Reformulation of the uniqueness theorem for λ. It is clear that there are asymptotically flat spacetimes which do not admit any isometry. 1 are meaningless. However those statements remain valid if referring to the asymptotic theory based on QFT on + and the universal state λ. Indeed λ is invariant under the whole G B M S group—which represents asymptotic symmetries of every asymptotically flat spacetime—and λ satisfies a positivity energy condition with respect to every smooth one-parameter subgroup of G B M S made of future-directed timelike or null 4-translations—which correspond to Killing-time evolutions whenever the spacetime admits a timelike Killing field, as established above.

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