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Related concepts
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
A one-parameter group (of unitary operators in a Hilbert space) is a homomorphism of groups
where is a Hilbert spaces, denotes its group of unitary operators and the additive group of real numbers.
More generally, one can define one-parameter semigroups of operators in a Banach space as homomomorphisms of monoids
where denotes the semigroup of bounded operators .
Typically, we also require a continuity condition such as continuity in the strong topology.
Strongly continuous one-parameter unitary groups of operators in a Hilbert space are in bijection with self-adjoint unbounded operators on :
This bijection sends
The operator is bounded if and only if is norm-continuous.
Strongly continuous one-parameter semigroups of bounded operators on a Banach space (alias -semigroups) satisfying are in bijection with closed operators with dense domain such that any belongs to the resolvent set of and for any we have
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See also
Last revised on June 21, 2022 at 07:35:23. See the history of this page for a list of all contributions to it.