subspaces

If the state space of an elementary system had relativistically invariant subspaces then it would be appropriate to associate these subspaces with elementary systems.

Archive 2009-02-01

Put more technically, the state space of an elementary system must not contain any relativistically invariant subspaces, i.e., it must be the state space of an irreducible representation of the relevant invariance group.

Archive 2009-02-01

(An irreducible representation is one in which there are no subspaces invariant under the group action, apart from the null vector and the entire Hilbert space).

Archive 2009-02-01

Such operators are in one-to-one correspondence with the closed subspaces of H.

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V over an involutive division ring is called a generalized Hilbert space if its lattice of closed subspaces

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In view of the above-mentioned one-one correspondence between closed subspaces and projections, we may impose upon the set

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Since a typical closed subspace has infinitely many complementary closed subspaces, this lattice is not distributive; however, it is orthocomplemented by the mapping

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Ordered by set-inclusion, the closed subspaces of H form a complete lattice, in which the meet (greatest lower bound) of a set of subspaces is their intersection, while their join (least upper bound) is the closed span of their union.

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Axiom VII: The partially ordered set of all questions in quantum mechanics is isomorphic to the partially ordered set of all closed subspaces of a separable, infinite dimensional Hilbert space.

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The Virasoro algebra acts on a fermionic Fock space and preserves certain subspaces generated by singular vectors; this means that the factor modules are well defined.

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