Reeh–Schlieder theorem
Encyclopedia
The Reeh–Schlieder theorem is a result of relativistic local quantum field theory
, stating that the vacuum
is a cyclic vector for the field algebra of any open set in Minkowski space
. It was published by Helmut Reeh and Siegfried Schlieder (1918-2003) in 1961.
One may remark the states created by applying elements of the local algebra
to the vacuum state
are, therefore,
not strictly localized in its region
, but can in effect approximate any state. In a quantitative sense, the localization remains true. The long range effects of the operators of the
local algebra will diminish rapidly with distance, as seen by the cluster properties of the Wightman functions. And with increasing distance, creating a unit vector localized outside requires operators of ever increasing
operator norm
.
This theorem is also cited in connection with quantum entanglement
. But it is subject to some doubt whether
the Reeh–Schlieder theorem can usefully be seen as the quantum field theory
analog to quantum entanglement
, since the
exponentially-increasing
energy needed for long range actions will prohibit any macroscopic effects.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, stating that the vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
is a cyclic vector for the field algebra of any open set in Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
. It was published by Helmut Reeh and Siegfried Schlieder (1918-2003) in 1961.
One may remark the states created by applying elements of the local algebra
to the vacuum state
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
are, therefore,
not strictly localized in its region
, but can in effect approximate any state. In a quantitative sense, the localization remains true. The long range effects of the operators of thelocal algebra will diminish rapidly with distance, as seen by the cluster properties of the Wightman functions. And with increasing distance, creating a unit vector localized outside requires operators of ever increasing
operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
.
This theorem is also cited in connection with quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
. But it is subject to some doubt whether
the Reeh–Schlieder theorem can usefully be seen as the quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
analog to quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
, since the
exponentially-increasing
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
energy needed for long range actions will prohibit any macroscopic effects.
External links
- Siegfried Schlieder, Some remarks about the localization of states in a quantum field theory, Comm. Math. Phys. 1, no. 4 (1965), 265–280 online at Project Euclid
- hep-th/0001154 Christian Jaekel, "The Reeh–Schlieder property for ground states"