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In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture. Often the mapping from functions on phase space to operators is called the Weyl transform, whereas the inverse mapping, from operators to functions on phase space, is called the Wigner transform. This mapping was originally devised by Hermann Weyl in 1927 in an attempt to map symmetrized ''classical'' phase space functions to operators, a procedure known as ''Weyl quantization''.〔 〕 It is now understood that Weyl quantization does not satisfy all the properties one would want for quantization and therefore sometimes yields unphysical answers. Nevertheless, in a mathematical sense, the Weyl-Wigner transform is a well-defined integral transform between the phase-space and operator representations, and yields insight into the workings of quantum mechanics. Most importantly, the Wigner quasi-probability distribution is the Wigner transform of the quantum density matrix, and, conversely, the density matrix is the Weyl transform of the Wigner function. In contrast to Weyl's original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation within quantum mechanics; it need not connect "classical" with "quantum" quantities. For example, the phase-space function may depend explicitly on Planck's constant ħ, as it does in some familiar cases involving angular momentum. This invertible representation change then allows one to express quantum mechanics in phase space, as was appreciated in the 1940s by Groenewold〔 〕 and Moyal.〔 〕〔 〕 ==Example== The following illustrates the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be ''(q,p)'', and let ''f'' be a function defined everywhere on phase space. The Weyl transform of the function ''f'' is given by the following operator in Hilbert space, broadly analogous to a Dirac delta function, Here, the operators ''P'' and ''Q'' are taken to be the generators of a Lie algebra, the Heisenberg algebra: : where ħ is the reduced Planck constant. A general element of the Heisenberg algebra may thus be written as ''aQ+bP+c'' . The exponential map of this element of the Lie algebra is then an element of the corresponding Lie group, :: the Heisenberg group. Given a group representation ρ of the Heisenberg group on a Hilbert space, the operator :: denotes the element of the representation corresponding to the group element ''g''. By the Stone–von Neumann theorem, under suitable analytic hypotheses this representation is unique up to unitary equivalence, and the reader may imagine that it is the familiar one where P is the momentum operator and Q is the position operator. For this reason we will drop ρ in what follows. This Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator, : The inverse of the above Weyl map is the Wigner map, which takes the operator Φ back to the original phase-space kernel function ''f'', In general, the resulting function ''f'' depends on Planck's constant ''ħ'', and may well describe quantum-mechanical processes, provided it is properly composed through the star product, below.〔 〕 For example, the Wigner map of the quantum angular-momentum-squared operator L2 is not just the classical angular momentum squared, but it further contains an offset term − 3''ħ''2/2, which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit. 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–Weyl transform」の詳細全文を読む スポンサード リンク
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