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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Projective Hilbert space ： ウィキペディア英語版 Projective Hilbert space In mathematics and the foundations of quantum mechanics, the projective Hilbert space $P(H)$ of a complex Hilbert space $H$ is the set of equivalence classes of vectors $v$ in $H$, with $v\; \backslash ne\; 0$, for the relation $\backslash sim$ given by :$v\; \backslash sim\; w$ when $v\; =\; \backslash lambda\; w$ for some non-zero complex number $\backslash lambda$. The equivalence classes for the relation $\backslash sim$ 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 $\backslash psi$ and $\backslash lambda\; \backslash psi$ represent the same ''physical state'', for any $\backslash lambda\; \backslash ne\; 0$. It is conventional to choose a $\backslash psi$ from the ray so that it has unit norm, $\backslash langle\backslash psi|\backslash psi\backslash rangle\; =\; 1$, in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine $\backslash psi$ within the ray, since $\backslash psi$ could be multiplied by any $\backslash lambda$ with absolute value 1 (the U(1) action) and retain its normalization. Such a $\backslash lambda$ can be written as $\backslash lambda\; =\; e^$ with $\backslash phi$ called the global phase. Rays that differ by such a $\backslash lambda$ correspond to the same state (cf. quantum state (algebraic definition), given a C *-algebra of observables and a representation on $H$). No measurement can recover the phase of a ray, it is not observable. One says that $U(1)$ is a gauge group of the first kind. If $H$ is an irreducible representation of the algebra of observables then the rays induce pure states. Convex linear combinations of rays naturally give rise to density matrix which (still in case of an irreducible representation) correspond to mixed states. The same construction can be applied also to real Hilbert spaces. In the case $H$ is finite-dimensional, that is, $H=H\_n$, the set of projective rays may be treated just as any other projective space; it is a homogeneous space for a unitary group $\backslash mathrm(n)$ or orthogonal group $\backslash mathrm(n)$, in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writes :$P(H\_)=\backslash mathbbP^$ so that, for example, the projectivization of two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line $\backslash mathbbP^$. 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. ==Product== The Cartesian products of projective Hilbert spaces is not a projective space. Their categorical product is equivalent to the tensor product of respective (vector) Hilbert spaces and, in quantum physics, describes states of a composite quantum system. Segre mapping is an embedding of the Cartesian product of two projective spaces into their categorical product. It describes how to make states of the composite system from states of its constituents. It is only an embedding not a surjection; most of the categorical product space does not lie in its range and represents ''entangled states''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Projective Hilbert space」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.