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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 Wigner's theorem ： ウィキペディア英語版 Wigner's theorem Wigner's theorem, proved by Eugene Wigner in 1931,〔 (in German), (English translation).〕 is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states. According to the theorem, any symmetry transformation of ray space is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space. The representation of a symmetry group on Hilbert space is either an ordinary representation or a projective representation. ==Rays and ray space== It is a postulate of quantum mechanics that vectors in Hilbert space that are scalar nonzero multiples of each other represent the same pure state. A ray is a set :$\backslash underline\; =\; \backslash \backslash \},\; \backslash Psi\; \backslash in\; \backslash mathcal,$ and a ray whose vectors have unit norm is called a unit ray. If , then is a representative of . There is a one-to-one correspondence between physical pure states and unit rays.〔Here the possibility of superselection rules is ignored. It may be the case that a system cannot be prepared in specific states. For instance, superposition of states with different spin is generally believed impossible. Likewise, states being superpositions of states with different charge are considered impossible. Minor complications due to those issues are treated in 〕 The space of all rays is called ray space. Formally, if is a complex Hilbert space, then let be the subset :$B\; =\; \backslash$ of vectors with unit norm. If is finite-dimensional with complex dimension , then has real dimension . Define a relation ≅ on by :$\backslash Psi\; \backslash cong\; \backslash Phi\; \backslash Leftrightarrow\; \backslash Psi\; =\; e^\backslash Phi,\; \backslash quad\; \backslash alpha\; \backslash in\; \backslash mathbb\; R.$ The relation ≅ is an equivalence relation on the set . Unit ray space, , is defined as the set of equivalence classes :$S\; =\; B/\backslash cong.$ If is finite, has real dimension hence complex dimension . Equivalently, one may define ≈ on by :$\backslash Psi\; \backslash approx\; \backslash Phi\; \backslash Leftrightarrow\; \backslash Psi\; =\; z\backslash Phi,\; \backslash quad\; z\; \backslash in\; \backslash mathbb\; C^$ *, and set :$S\; =\; H/\backslash approx.$ This makes it clear that unit ray space is a projective space. It is also possible to skip the normalization and take ray space as〔This approach is used in , which serves as a basis reference for the proof outline to be given below.〕 :$R\; =\; H/\backslash cong,$ where ≅ is now defined on all of by the same formula. The real dimension of is if is finite. This approach is used in the sequel. The difference between and is rather trivial, and passage between the two is effected by multiplication of the rays by a nonzero ''real'' number, defined as the ray generated by any representative of the ray multiplied by the real number. Ray space is sometimes awkward to work with. It is, for instance, not a vector space with well-defined linear combinations of rays. But a transformation of a physical system is a transformation of states, hence mathematically a transformation of ray space. In quantum mechanics, a transformation of a physical system gives rise to a bijective unit ray transformation of unit ray space, :$T:\; S\; \backslash ni\; \backslash underline\; \backslash subset\; \backslash mathcal\; \backslash mapsto\; S\text{'}\; \backslash ni\; \backslash underline\; =\; T\backslash underline\; \backslash subset\; \backslash mathcal.$ The set of all unit ray transformations is thus the permutation group on . Not all of these transformations are permissible as symmetry transformations to be described next. A unit ray transformation may be extended to by means of the multiplication with reals described above according to〔 defines general ray transformations on to begin with, which means that it is not necessarily bijective on (i.e. not necessarily norm preserving). This is not important since only symmetry transformations are of interest anyway.〕 :$T:R\; \backslash rightarrow\; R\text{'};\; T(\backslash lambda\; \backslash underline\; \backslash Psi)\; \backslash equiv\; \backslash lambda\; T\backslash underline\; \backslash Psi,\; \backslash quad\; \backslash underline\; \backslash Psi\; \backslash in\; S,\; \backslash lambda\; \backslash in\; \backslash mathbb\; R.$ To keep the notation uniform, call this a ray transformation. This terminological distinction is not made in the literature, but is necessary here since both possibilities are covered while in the literature one possibility is chosen. 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「Wigner's theorem」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.