Null energy conditions in quantum field theory

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 67, Issue 4, 2003, Pages

  Fewster, Christopher J ^a   Roman, Thomas A ^a  

^a UNIVERSITY OF YORK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; ENERGY; MATHEMATICAL ANALYSIS; QUANTUM MECHANICS; QUANTUM THEORY; STRESS;

EID: 0037441213     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.67.044003     Document Type: Article

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35. In more detail, we consider theories described by von Neumann algebras (Formula presented) indexed by open sets O of Minkowski space and consisting of bounded operators on a Hilbert space H. The assignment (Formula presented) is assumed to be isotonous [i.e., (Formula presented) implies that (Formula presented) is a subalgebra of (Formula presented) and local [i.e., if (Formula presented) and (Formula presented) are spacelike separated then any element of (Formula presented) commutes with every element of (Formula presented). In addition, the group of translations is implemented on H by unitary operators (Formula presented) such that (Formula presented) and which obey the spectral condition: (Formula presented) where the spectral measure (Formula presented) is supported in the forward light cone (Formula presented) It is also assumed that there is a unique translationally invariant vacuum state (Formula presented) and that Ω is cyclic [i.e., (Formula presented) is dense in H for each O].
36. Under the assumptions in the previous reference, let K be a compact subset of Minkowski space such that (Formula presented) for all (Formula presented) (Formula presented) Suppose that there exist vectors (Formula presented) and (Formula presented) such that (Formula presented) (Formula presented) and with the property that, for all (Formula presented) (Formula presented) (Formula presented) (Formula presented) Then a theorem of Woronowicz 46 entails that the algebra (Formula presented) of observables localized on K consists only of scalar multiples of the identity. In the case of a bounded null line segment (Formula presented) where (Formula presented) is null, it is readily verified that (Formula presented) (Formula presented) satisfy these conditions for any spacelike vector (Formula presented) with (Formula presented) and (Formula presented) Such vectors (Formula presented) exist in dimensions (Formula presented) thus justifying the claim made in the text.
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