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- Projective_Hilbert_space abstract "In mathematics and the foundations of quantum mechanics, the projective Hilbert space of a complex Hilbert space is the set of equivalence classes of vectors in , with , for the relation given by when for some non-zero complex number .The equivalence classes for the relation are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory, the wave functions and represent the same physical state, for any . It is conventional to choose a from the ray so that it has unit norm, , in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine within the ray, since could be multiplied by any with absolute value 1 (the U(1) action) and retain its normalization. Such a can be written as with called the global phase.This freedom means that projective representations of quantum states are important in quantum theory. For example, a density matrix of a pure quantum state is represented by a projective ray; it is impossible to recover the phase from a density matrix. The same is true for its generalisation, pure states in a representation of a C*-algebra.The same construction can be applied also to real Hilbert spaces.In the case is finite-dimensional, that is, , the set of projective rays may be treated just as any other projective space; it is a homogeneous space for a unitary group or orthogonal group , in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writesso that, for example, the projectivization of two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line . This is known as the Bloch sphere. See Hopf fibration for details of the projectivization construction in this case.Complex projective Hilbert space may be given a natural metric, the Fubini–Study metric, derived from the Hilbert space's norm.".
- Projective_Hilbert_space wikiPageID "796760".
- Projective_Hilbert_space wikiPageRevisionID "590226163".
- Projective_Hilbert_space date "January 2014".
- Projective_Hilbert_space hasPhotoCollection Projective_Hilbert_space.
- Projective_Hilbert_space reason "There must be some conditions on it…".
- Projective_Hilbert_space subject Category:Hilbert_space.
- Projective_Hilbert_space comment "In mathematics and the foundations of quantum mechanics, the projective Hilbert space of a complex Hilbert space is the set of equivalence classes of vectors in , with , for the relation given by when for some non-zero complex number .The equivalence classes for the relation are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space.".
- Projective_Hilbert_space label "Espace projectif de Hilbert".
- Projective_Hilbert_space label "Projective Hilbert space".
- Projective_Hilbert_space sameAs Espace_projectif_de_Hilbert.
- Projective_Hilbert_space sameAs m.03cs88.
- Projective_Hilbert_space sameAs Q3058230.
- Projective_Hilbert_space sameAs Q3058230.
- Projective_Hilbert_space wasDerivedFrom Projective_Hilbert_space?oldid=590226163.
- Projective_Hilbert_space isPrimaryTopicOf Projective_Hilbert_space.