Reeh-Schlieder theorem

The Reeh-Schlieder theorem of quantum field theory states that - in any good approximation - all states of a relativistic particle can be generated from the vacuum by the field operators located in any open space-time domain .

The exact formulation says that the vacuum state is cyclic and separating for any algebra of local observables .

• Cyclic means here that the closed envelope of the set of all states that result from applying the local operators to the vacuum is already the entire state space.
• separating here means that no local operator applied to the vacuum state can result in 0. In particular, the expectation value of all self-adjoint local operators B of the form A ^* A in the vacuum state is greater than zero.

The Reeh-Schlieder theorem can be derived from the properties of concrete quantum field theories as well as from the various axiom systems of QFT .

The Reeh-Schlieder theorem does not imply a violation of micro-causality, but that the field operators localized to a domain do not only generate states that are restricted to this domain. The obvious interpretation is that a quantum box, a particle at the time at the site created or destroyed, is incorrect. ${\displaystyle \Phi (t,{\vec {x}})}$ ${\displaystyle t}$ ${\displaystyle {\vec {x}}}$

The theorem was first given in 1961 by Helmut Reeh and Siegfried Schlieder .

literature

• Rudolf Haag : Local quantum Physics. Fields, Particles, Algebras . 2nd revised and enlarged edition. Springer, Berlin et al. 1996, ISBN 3-540-61049-9 ( Texts and monographs in physics ).

Individual evidence

1. ↑ H. Reeh, S. Schlieder: Comments on the unitary equivalence of Lorentz invariant fields . In: Il Nuovo Cimento . 22 (1961) 1059-1068