R.F. Werner: "Quantum harmonic analysis on phase space", J.Math.Phys. 25(1984) 1404--1411 Home/Search Document Not in Database Summary Related Articles Check |
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The Classical Limit of Quantum Theory - Werner (1995) Self-citation (Werner) (Correct)
....from the conceptual point of view. Technically it means that operator norms (see (d) above) cannot be estimated without artificial smoothness assumptions [Dau] It is well known that by averaging Wigner functions with a suitable Gaussian [Bop,Car] these difficulties disappear [Dav,Hol,We1] Moreover, the Gaussians can be chosen such that in the classical limit this smearing out becomes negligible anyhow. In their averaged form Wigner functions play an important role in our approach. For a discussion of states that have positive Wigner functions all the way to the classical limit ....
....a classical intermediate step. It is clear that something like this must be possible from the idea that the comparison described by the j hh 0 should be at least asymptotically transitive. Positive maps taking quantum observables to classical ones and conversely are wellknown [Bop,Sim,Dav,Tak,We1] These maps depend on the choice of a normal state, which is usually taken to be coherent, i.e. the ground state of some harmonic oscillator. Let h (x) h) Gammad=4 exp Gammax 2 2h (3:8) be the ground state vector of the standard oscillator Hamiltonian H osc h = 1 2 X i (P ....
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R.F. Werner: "Quantum harmonic analysis on phase space", J.Math.Phys. 25(1984) 1404--1411
Mixed States with Positive Wigner Functions - Bröcker, Werner (1994) Self-citation (Werner) (Correct)
....of the respective trace class operators. Then the positivity of the Wigner function becomes, via Bochner s Theorem [5] a positive definiteness condition, whereas the positivity of the operator itself becomes, via a quantum version of Bochner s theorem due to Kastler, Loupias, and Miracle Sole [6,7,8], a twisted positive definiteness condition. We can interpolate between the two conditions with a twisting parameter j, which has roughly the interpretation of h, such that j = 1 corresponds to the quantum case, and j = 0 to the classical Bochner s Theorem. The characterization of mixed states ....
....a trace, and the exponential by the Weyl operator. That this transform really deserves the name Fourier transform, and has many properties analogous to the classical Fourier transform (with the symplectic form used as a the scalar product between the two sets of variables) is shown in detail in [8]. Wigner s phase space distribution function [1] is defined by taking this analogy literally, i.e. by interpreting F ae as the Fourier transform of an ordinary function, then called the Wigner function W ae of ae. This way of computing the Wigner function is a bit cumbersome, however, and it is ....
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R. Werner:"Quantum harmonic analysis on phase space", J.Math.Phys. 25(1984) 1404-1411
Classical Mechanics as Quantum Mechanics with Infinitesimal h - Werner, Wolff (1995) Self-citation (Werner) (Correct)
.... any A 2 B(H) we define a function on phase space by (S A) hW ( Omega ; AW ( Omega i = h Omega ; ff Gamma (A) Omega i : 6) This is variously called the lower symbol [Sim] a smeared Wigner function [Car] the Husimi function [Tak] or the convolution with a coherent state [We1] of the operator A. In the other direction, we have the upper symbol [Sim] or P representation [KS] also going by many other names) which assigns an operator to each bounded measurable function f via S # f = Z dx dp (2 h) d f(x; p) ff (j Omega ih Omega j) 7) Unlike the ....
R.F. Werner:"Quantum harmonic analysis on phase space", J.Math.Phys. 25 (1984) 1404-1411
On the Concentration of Quantum States in Phase Space - Werner (1993) Self-citation (Werner) (Correct)
....estimates of states or observables are desired. There is a well known alternative to the Wigner function avoiding these difficulties, which is variously known as the Husimi function [14] a phase space observable [5,7] a Berezin upper lower symbol [13] or convolution with a coherent state [16]. One price to pay in all these approaches is that while the Wigner function has an intrinsic characterization in terms of the Weyl operators alone [10] these positive distribution functions depend on the choice of a reference state , usually a coherent state. In the latter case the frequency ....
....it has turned out that it fails in general: there are measures of concentration of quantum states in phase space which are not maximized by coherent states. We now fix the basic notations and conventions for phase space quantum mechanics. For a more complete exposition the reader is referred to [16,17]. By a phase space we will mean a 2d 1 dimensional real vector space X with a non degenerate antisymmetric bilinear form oe : X Theta X IR. The connection between quantum mechanics and the classical phase space structure is made by an irreducible Weyl system, i.e. an irreducible set of ....
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R.F. Werner:"Quantum harmonic analysis on phase space", J.Math.Phys. 25 (1984) 1404--1411
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