Applied Mathematicsematics

Download e-book for iPad: Communications In Mathematical Physics - Volume 279 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Asymptotic flatness at future null infinity and + . Following [AH78,As80,Wa84]1 a smooth spacetime (M, g) is called asymptotically flat vacuum spacetime at future ˜ g) null infinity if there is a second smooth spacetime ( M, ˜ such that M can be viewed as an ˜ + is an embedded submanopen embedded submanifold of M˜ with boundary + ⊂ M. ˜ = ∅. ( M, ˜ g) ˜ is required to be strongly causal in a ifold of M˜ satisfying + ∩ J − (M; M) + 2 ˜ is strictly ∈ C ∞ ( M) neighbourhood of and it has to hold g˜ M = M g M , where positive on M.

Is the vacuum state for BMS-massless particles if one analyses the unitary representations of the BMS group within the Wigner-Mackey approach [DMP06]. λ is uniquely determined by a positive BMS-energy requirement in addition to the above-mentioned BMS invariance [Mo06] (actually the latter requirement can be weakened considerably). Finally, in the folium of λ there are no further pure BMS-invariant (not necessarily quasifree or positive energy) states. λ is universal: it does not depend on the particular asymptotically flat spacetime under consideration but it is defined in terms of the asymptotic extent which is the same for all asymptotically flat spacetimes.

D thesis [Ho00]. In that work, the definition of local vacuum states near a point x in spacetime is considered. The idea of the construction is to define the state by specifying the “initial data” of its 2-point function on the (say, past-) lightcone through x. The formula for this initial datum is the formula (46) when the lightcone in [Ho00] is replaced by − as in our work. In [Ho00] it is also established that the individuated state is of Hadamard form (in the sense of the microlocal condition (45)) employing a proof which is very analogous to our proof.

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