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I'm pretty sure that the Wigner-Weyl Transform is a different thing from the Wigner Function. The Wigner function is the Weyl-Wigner Transform of the Density Matrix of Hilbert Space, but the Weyl-Wigner Transform in general is given by
Please correct me if I'm wrong, because it will mean that my thesis has to be re-written, and I've only got four months left.
As soon as I've finished, I'll write a Weyl-Wigner Transform article for Wikipedia. By Special:Contributions/130.102.2.60 .
QMPS, which also includes the original seminal papers, adduced last in the references to this article. Conventionally, the Weyl transform maps phase-space (kernel) functions (sometimes called "symbols", a bit awkwardly) to hermitean operators; the reverse transform is the Wigner transform you write, which maps hermitean operators to phase-space kernel functions---which may or may not contain hbar, depending on whether these operators are Weyl-ordered or not (If not, the transform implicitly Weyl-orders them and generates hbar-dependence, in general, whence calling such kernels "classical" may be confusing).
I suspect people dubbed it "Wigner transform" since the Wigner function is the most celebrated example of it; and as you correctly point out, it is the Wigner transform of the Density Matrix (cf. QMPS).
This article has the Wigner transform in property 7, and the article on Weyl quantization details the Weyl transfrom, its inverse. Cuzkatzimhut 16:09, 18 January 2007 (UTC)
Some additional comments should be added along the following lines.
First, there is a simpler characterization of the Wigner function in relation to Weyl quantization that gets across what it is really about: the Wigner function is just the expectation of the delta function, understood in the sense of a weak operator limit. Weyl quantization may therefore be thought of as an operator whose kernel is just the Weyl quantization of the delta function. In comparison, if you take the expectation of the delta function on a classical state, you get its probability distribution.
Second, there is a close link between Wigner functions and probability distributions, which in turn provides a link to coherent state quantization. Namely, the Gaussian convolution of a Wigner function with a spread in each (p,q) sector equal to Planck's constant or larger is a probability distribution. This convolution is related to the transition probability taken with a coherent state. Conversely, given the transition probability taken with a family of coherent state |p,q> as (p,q) range over phase space, the Wigner function can be reproduced. Therefore, though Wigner functions may not be probability distributions in themselves, they are characterized as the "inverse Gaussian convolutions" of probability distributions.
Comment in response by
Cuzkatzimhut 00:53, 12 June 2007 (UTC): I would reluctantly concur with the first point, provided no confusing statements were made, and the comments were properly relegated to the Weyl Quantization pages, instead, where they would appear more apposite. What is refered to as a "weak operator limit delta function", centering an operator around its classical limit in Weyl ordering, is really the quantizer: the Fourier transform of a generic element of the Heisenberg group. Its
expectation, in Moyal's original 1946 language, is the Fourier transform of the characteristic (moment-generating) function.
I suspect that few readers would learn much from operator-valued distribution functions, as suggested, but I may be convinced otherwise.
I suspect property 7 covers the basics, and A Royer's classic 1977 paper (Phys Rev A15 , pp 449-450), interpreting the Wigner function as the expectation value of the (parity) reflection operator in phase space should suffice, instead of formal excursions on operator-valued delta functions.
I fear the second point needs to be finessed with far too much work or detail to be made sound. Indeed, the convolution mentioned, as per de Bruijn's (1967) and Cartwright's (1976) theorems, is positive semidefinite; but it cannot serve as a plain probability measure in phase space, as physicists have long known, but electrical engineers often miss, possibly due to lack of attention in the marginals. (This problem may be remedied by proper account of the *-product within integrals, not ignorable as in the case of the Wigner function, but at the cost of further complication.) Conversely, the inverse Husimi transform of an arbitrary positive semidefinite function viewed as a "probability" in phase space rarely satisfies the highly constrained ancillary conditions for a Wigner function: the outcome rarely fits into the form of the first formula of the article for some psi, as readily demonstrated in standard texts on the subject. Thus, connecting to the Husimi function as blithely as suggested merely adds delicate elements of potential confusion to the non-expert, and may well be deprecated.
Wigner-Ville? Who is Ville? I'm presuming someone just hasn't connected the spelling and pronunciation of Weyl? Cesiumfrog ( talk) 06:50, 1 April 2010 (UTC)
I think if that minimum info goes away, the essence of P(x,p) will be impenetrable. Much better to merge it with the next section, which it explains, and refer to the Main article of the W-W transform. In that case, the title of the subsection can cover both. Cuzkatzimhut ( talk) 13:15, 9 June 2012 (UTC)
In the first formula, the meaning of "y" is not listed. For those which are not familiar with the subject it would be good to make sure that every variable is defined.
The recent hyper-formal additions of the last couple of days, involving a mere Fourier transformation of the momentum variable in the Bopp-shifted form of the Moyal bracket, are only relevant or helpful for a generic Hamiltonian of the form p2+V(x). Fourier transformation of translated arguments is a standard formal technique utilized for decades in this area, but the mere rewriting of Moyal's equation for the Wigner function does not help the reader seeking a concise of the object of this introductory article. I propose that these hyper-technical discussions are removed. Sadly, this subsection is the most poorly written of the article, and it is already badly stretched with discursive tangents to quantum characteristics and path integrals and such. The formal rewritings introduced may well be moved to the Method of quantum characteristics article, which is, indeed, addressing time evolution in excessive detail, for more formally and technically minded readers; or perhaps an independent article, or to the main article dealing with the formulation in phase space. But, in this section the remarks are visibly detrimental and should be removed. Cuzkatzimhut ( talk) 17:32, 13 November 2012 (UTC)
If the proposed equation is the basic result of the cited PRL paper, the reviewer of PRL obviously disagrees with you. But your remarks seem reasonable. You could improve this section yourself e.g. by replacing citation of Marinov with B. Leaf, J. Math. Phys. 9, 769 (1968). Leaf happily built phase space path integral 23 years earlier! Trompedo ( talk) 21:08, 13 November 2012 (UTC)
I insist this is not the right forum to discuss the history of science, which would be most appropriate for a physics forum. Nevertheless, your aspersions could be hardly left uncountenanced! Feynman, of course, wrote his widely popular PhD dissertation on path integrals in coordinate space, in 1942 RPF thesis, and the formula you are referring to, in coordinate space, is already in Dirac's "The Lagrangian in quantum mechanics" , Phys. Z. der Sowjetunion, 3, 64-71 (1933), specifically eqn (11). Feynman of course had already summarized it all in his review, Rev Mod Phys 20 (1948), 367-87, which Moyal, of course, references in his paper, ref (26), I hope you noticed. I would never dream of ascribing any path integral priorities to Moyal---that was the whole point of my rhetorical question above! I suspect a careful rereading of Moyal's paper, doi: 10.1017/S0305004100000487, will dispel any misconstruing. I merely pointed out infinitely recursive formulas in phase space start with Moyal's paper, and make an arc all the way to Marinov's. You may inquire User:Taulalai for further subtlety, if you are so inclined, and if you believe Leaf has anticipated Marinov, which I do not. However, I really believe Path integral formulation is the place to fuss these matters, anyplace but here! We may continue the evidently endless conversation on your talk page, User talk:Trompedo, to spare the fellow editors. Cuzkatzimhut ( talk) 19:58, 15 November 2012 (UTC)
After all, you motivated me to download the Moyal paper. Alas, Sect. 7.6, which you pointed out earlier, contains no hint of the path integral, while Sect. 9.8 is absent. Trompedo ( talk) 19:40, 25 November 2012 (UTC)
In the 4th Formula (preceded by "In the general case, which includes mixed states", the Sign in the Exponential is Not Consistent With the Sign Used in the Definition.
Indeed
ψ^*(x+y) ψ(x-y) = <x-y|ρ|x+y> if ρ=|ψ><ψ| (pure Case)
89.92.57.32 ( talk) 01:07, 14 November 2012 (UTC)Jean Hare
There have been suggestions made to make sure that mathematical details regarding the time evolution of the Wigner function should not clutter this article, so that this article remains readable for non-specialized readers. Currently, there exist sections Wigner quasiprobability distribution#Evolution equation for Wigner function and Density matrix#"Quantum Liouville", Moyal's equation and Phase space formulation#Time evolution, plus the very mathematically detailed Method of quantum characteristics. In order to create a place for more detailed information on the time evolution aspect, I suggest to create a dedicated new article "Moyal equation of motion" (as it is called for ex. here), or alternatively shorter "Moyal equation", for an article on the Wigner function's time evolution equation
Sources for such a new article could include on the one hand the original articles for the historical overview, plus on the other hand also more recent but free-access articles such as hep-th/9409120 or others. Then this article's section "Evolution equation for Wigner function" could be brief and concise, referring to that new article as main article. And the new article could contain greater detail, without being quite as technical and detailed as the article on the method of quantum characteristics. Any comments, objections or support at this point? -- Chris Howard ( talk) 22:52, 15 November 2012 (UTC)
User:Jheald proposed merger of the Signal processing/ transform article Wigner distribution function into this one. The relative ratios of the Traffic stats in the last 30 days of the respective articles are WDF/WQD ~ 0.42, at this point in time. The two articles, are, of course, related in mathematical detail, but there are conceptual and cultural chasms all but impossible to bridge. I can imagine an extra section with WDF here, but it looks like a major task, if it were to be done right.
In particular, unless the issue of negative values, ritually frowned upon by the signal-processing community, with little reason (leading to creative "remedies"), were dealt with responsibly, the merger could easily lead to a bonfire of confusions, eye-rolling, and pseudo-science. Enterprising wikipedians may give it a shot, but fretting about cross-terms without considering expectation values and the uncertainty principle as it subtly manifests itself in signal processing can only lead to grief and avoidable pointless mass debate, I suspect. Cuzkatzimhut ( talk) 14:30, 2 July 2013 (UTC)
I think that a section could be dedicated to the importance and the interpretation of the peculiar negative values of the Wigner function. In fact this is probably the most important difference between the Wigner function and a classical probability distribution and, indeed, such negative values are commonly interpreted as a signature of quantumness or non-classicality.
This naive intuition is also motivated by the following rigorous and fundamental results:
1) all pure states are described by a negative Wigner function with the only exception of Gaussian states. [R. L. Hudson, Rep. Math. Phys. 6, 249 (1974)],
2) it is impossible to violate Bell inequalities with homodyne measurments performed on quantum states having positive Wigner functions. Indeed, in this case the Wigner function constitutes a "hidden variable" model for the measurement statistics. [Konrad Banaszek and Krzysztof Wódkiewicz, Phys. Rev. Lett. 82, 2009 (1999)],
3) the Wigner function negativity is a necessary condition for an exponential speed up of a quantum computer with respect to a classical one. Indeed, every algorithm involving states and operations representable by positive Wigner functions, can be efficiently classically simulated. [A Mari, J Eisert, Phys. Rev. Lett. 109, 230503 (2012)].
Mine is just a suggestion and you can judge for yourself if this section would be appropriate or not. I am not an expert on editing Wikipedia. — Preceding unsigned comment added by 2001:760:2C00:8253:226:BBFF:FE67:5EBC ( talk) 10:17, 27 November 2014 (UTC)
I am pretty certain that the 3D and 1D definitions of the Wigner function are not consistent with each other. See the factors of tow in the exponent and in the arguments of the bras/kets. — Preceding unsigned comment added by 66.207.203.214 ( talk) 15:27, 18 December 2017 (UTC)
If I'm not mistaken, the equation:
should be:
Else, the units turn wrong. Can someone check me and fix it? Shohamjac ( talk) 08:53, 10 August 2019 (UTC)
‘’q’’ is momentum and the wave function is normalized in momentum space. Cuzkatzimhut ( talk) 10:08, 10 August 2019 (UTC)
I am not tempted by serial reversions of the sentence discussed here ( last one of section 5 : "The truncated Wigner approximation is a semiclassical approximation to the dynamics obtained by replacing Moyal's equation with the classical Liouville's equation.") which has reemerged. I strongly believe, nevertheless, that the sentence adds nothing to the small section inserted into.
Truncations are, of course, hinted at in the generic first sentence and its references, in that section, and the relation to Liouville dynamics in the very first section of the article. I can only see trouble ahead, if anyone took the bait and provided the redlinked wished-for article, which would then have to be linked. But as it stands, the sentence provides no meaningful information, beyond a misguided clarion call. Cuzkatzimhut ( talk) 17:56, 10 November 2019 (UTC)
The present article does not quite correctly state that "the Wigner distribution holds a privileged position among all these distributions, since it is the only one whose required star-product drops out". The Glauber-Sudarshan P representation mentioned in the article leads to an average value of the operator G, which coincides formally with that in classical mechanics. In this representation, the density matrices are diagonal in the basis of coherent states, so one can write
The average value of the operator G is an integral over the phase space:
The usual dot-product works inside the integral. -- Edehdu ( talk) 07:14, 20 May 2022 (UTC)
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I'm pretty sure that the Wigner-Weyl Transform is a different thing from the Wigner Function. The Wigner function is the Weyl-Wigner Transform of the Density Matrix of Hilbert Space, but the Weyl-Wigner Transform in general is given by
Please correct me if I'm wrong, because it will mean that my thesis has to be re-written, and I've only got four months left.
As soon as I've finished, I'll write a Weyl-Wigner Transform article for Wikipedia. By Special:Contributions/130.102.2.60 .
QMPS, which also includes the original seminal papers, adduced last in the references to this article. Conventionally, the Weyl transform maps phase-space (kernel) functions (sometimes called "symbols", a bit awkwardly) to hermitean operators; the reverse transform is the Wigner transform you write, which maps hermitean operators to phase-space kernel functions---which may or may not contain hbar, depending on whether these operators are Weyl-ordered or not (If not, the transform implicitly Weyl-orders them and generates hbar-dependence, in general, whence calling such kernels "classical" may be confusing).
I suspect people dubbed it "Wigner transform" since the Wigner function is the most celebrated example of it; and as you correctly point out, it is the Wigner transform of the Density Matrix (cf. QMPS).
This article has the Wigner transform in property 7, and the article on Weyl quantization details the Weyl transfrom, its inverse. Cuzkatzimhut 16:09, 18 January 2007 (UTC)
Some additional comments should be added along the following lines.
First, there is a simpler characterization of the Wigner function in relation to Weyl quantization that gets across what it is really about: the Wigner function is just the expectation of the delta function, understood in the sense of a weak operator limit. Weyl quantization may therefore be thought of as an operator whose kernel is just the Weyl quantization of the delta function. In comparison, if you take the expectation of the delta function on a classical state, you get its probability distribution.
Second, there is a close link between Wigner functions and probability distributions, which in turn provides a link to coherent state quantization. Namely, the Gaussian convolution of a Wigner function with a spread in each (p,q) sector equal to Planck's constant or larger is a probability distribution. This convolution is related to the transition probability taken with a coherent state. Conversely, given the transition probability taken with a family of coherent state |p,q> as (p,q) range over phase space, the Wigner function can be reproduced. Therefore, though Wigner functions may not be probability distributions in themselves, they are characterized as the "inverse Gaussian convolutions" of probability distributions.
Comment in response by
Cuzkatzimhut 00:53, 12 June 2007 (UTC): I would reluctantly concur with the first point, provided no confusing statements were made, and the comments were properly relegated to the Weyl Quantization pages, instead, where they would appear more apposite. What is refered to as a "weak operator limit delta function", centering an operator around its classical limit in Weyl ordering, is really the quantizer: the Fourier transform of a generic element of the Heisenberg group. Its
expectation, in Moyal's original 1946 language, is the Fourier transform of the characteristic (moment-generating) function.
I suspect that few readers would learn much from operator-valued distribution functions, as suggested, but I may be convinced otherwise.
I suspect property 7 covers the basics, and A Royer's classic 1977 paper (Phys Rev A15 , pp 449-450), interpreting the Wigner function as the expectation value of the (parity) reflection operator in phase space should suffice, instead of formal excursions on operator-valued delta functions.
I fear the second point needs to be finessed with far too much work or detail to be made sound. Indeed, the convolution mentioned, as per de Bruijn's (1967) and Cartwright's (1976) theorems, is positive semidefinite; but it cannot serve as a plain probability measure in phase space, as physicists have long known, but electrical engineers often miss, possibly due to lack of attention in the marginals. (This problem may be remedied by proper account of the *-product within integrals, not ignorable as in the case of the Wigner function, but at the cost of further complication.) Conversely, the inverse Husimi transform of an arbitrary positive semidefinite function viewed as a "probability" in phase space rarely satisfies the highly constrained ancillary conditions for a Wigner function: the outcome rarely fits into the form of the first formula of the article for some psi, as readily demonstrated in standard texts on the subject. Thus, connecting to the Husimi function as blithely as suggested merely adds delicate elements of potential confusion to the non-expert, and may well be deprecated.
Wigner-Ville? Who is Ville? I'm presuming someone just hasn't connected the spelling and pronunciation of Weyl? Cesiumfrog ( talk) 06:50, 1 April 2010 (UTC)
I think if that minimum info goes away, the essence of P(x,p) will be impenetrable. Much better to merge it with the next section, which it explains, and refer to the Main article of the W-W transform. In that case, the title of the subsection can cover both. Cuzkatzimhut ( talk) 13:15, 9 June 2012 (UTC)
In the first formula, the meaning of "y" is not listed. For those which are not familiar with the subject it would be good to make sure that every variable is defined.
The recent hyper-formal additions of the last couple of days, involving a mere Fourier transformation of the momentum variable in the Bopp-shifted form of the Moyal bracket, are only relevant or helpful for a generic Hamiltonian of the form p2+V(x). Fourier transformation of translated arguments is a standard formal technique utilized for decades in this area, but the mere rewriting of Moyal's equation for the Wigner function does not help the reader seeking a concise of the object of this introductory article. I propose that these hyper-technical discussions are removed. Sadly, this subsection is the most poorly written of the article, and it is already badly stretched with discursive tangents to quantum characteristics and path integrals and such. The formal rewritings introduced may well be moved to the Method of quantum characteristics article, which is, indeed, addressing time evolution in excessive detail, for more formally and technically minded readers; or perhaps an independent article, or to the main article dealing with the formulation in phase space. But, in this section the remarks are visibly detrimental and should be removed. Cuzkatzimhut ( talk) 17:32, 13 November 2012 (UTC)
If the proposed equation is the basic result of the cited PRL paper, the reviewer of PRL obviously disagrees with you. But your remarks seem reasonable. You could improve this section yourself e.g. by replacing citation of Marinov with B. Leaf, J. Math. Phys. 9, 769 (1968). Leaf happily built phase space path integral 23 years earlier! Trompedo ( talk) 21:08, 13 November 2012 (UTC)
I insist this is not the right forum to discuss the history of science, which would be most appropriate for a physics forum. Nevertheless, your aspersions could be hardly left uncountenanced! Feynman, of course, wrote his widely popular PhD dissertation on path integrals in coordinate space, in 1942 RPF thesis, and the formula you are referring to, in coordinate space, is already in Dirac's "The Lagrangian in quantum mechanics" , Phys. Z. der Sowjetunion, 3, 64-71 (1933), specifically eqn (11). Feynman of course had already summarized it all in his review, Rev Mod Phys 20 (1948), 367-87, which Moyal, of course, references in his paper, ref (26), I hope you noticed. I would never dream of ascribing any path integral priorities to Moyal---that was the whole point of my rhetorical question above! I suspect a careful rereading of Moyal's paper, doi: 10.1017/S0305004100000487, will dispel any misconstruing. I merely pointed out infinitely recursive formulas in phase space start with Moyal's paper, and make an arc all the way to Marinov's. You may inquire User:Taulalai for further subtlety, if you are so inclined, and if you believe Leaf has anticipated Marinov, which I do not. However, I really believe Path integral formulation is the place to fuss these matters, anyplace but here! We may continue the evidently endless conversation on your talk page, User talk:Trompedo, to spare the fellow editors. Cuzkatzimhut ( talk) 19:58, 15 November 2012 (UTC)
After all, you motivated me to download the Moyal paper. Alas, Sect. 7.6, which you pointed out earlier, contains no hint of the path integral, while Sect. 9.8 is absent. Trompedo ( talk) 19:40, 25 November 2012 (UTC)
In the 4th Formula (preceded by "In the general case, which includes mixed states", the Sign in the Exponential is Not Consistent With the Sign Used in the Definition.
Indeed
ψ^*(x+y) ψ(x-y) = <x-y|ρ|x+y> if ρ=|ψ><ψ| (pure Case)
89.92.57.32 ( talk) 01:07, 14 November 2012 (UTC)Jean Hare
There have been suggestions made to make sure that mathematical details regarding the time evolution of the Wigner function should not clutter this article, so that this article remains readable for non-specialized readers. Currently, there exist sections Wigner quasiprobability distribution#Evolution equation for Wigner function and Density matrix#"Quantum Liouville", Moyal's equation and Phase space formulation#Time evolution, plus the very mathematically detailed Method of quantum characteristics. In order to create a place for more detailed information on the time evolution aspect, I suggest to create a dedicated new article "Moyal equation of motion" (as it is called for ex. here), or alternatively shorter "Moyal equation", for an article on the Wigner function's time evolution equation
Sources for such a new article could include on the one hand the original articles for the historical overview, plus on the other hand also more recent but free-access articles such as hep-th/9409120 or others. Then this article's section "Evolution equation for Wigner function" could be brief and concise, referring to that new article as main article. And the new article could contain greater detail, without being quite as technical and detailed as the article on the method of quantum characteristics. Any comments, objections or support at this point? -- Chris Howard ( talk) 22:52, 15 November 2012 (UTC)
User:Jheald proposed merger of the Signal processing/ transform article Wigner distribution function into this one. The relative ratios of the Traffic stats in the last 30 days of the respective articles are WDF/WQD ~ 0.42, at this point in time. The two articles, are, of course, related in mathematical detail, but there are conceptual and cultural chasms all but impossible to bridge. I can imagine an extra section with WDF here, but it looks like a major task, if it were to be done right.
In particular, unless the issue of negative values, ritually frowned upon by the signal-processing community, with little reason (leading to creative "remedies"), were dealt with responsibly, the merger could easily lead to a bonfire of confusions, eye-rolling, and pseudo-science. Enterprising wikipedians may give it a shot, but fretting about cross-terms without considering expectation values and the uncertainty principle as it subtly manifests itself in signal processing can only lead to grief and avoidable pointless mass debate, I suspect. Cuzkatzimhut ( talk) 14:30, 2 July 2013 (UTC)
I think that a section could be dedicated to the importance and the interpretation of the peculiar negative values of the Wigner function. In fact this is probably the most important difference between the Wigner function and a classical probability distribution and, indeed, such negative values are commonly interpreted as a signature of quantumness or non-classicality.
This naive intuition is also motivated by the following rigorous and fundamental results:
1) all pure states are described by a negative Wigner function with the only exception of Gaussian states. [R. L. Hudson, Rep. Math. Phys. 6, 249 (1974)],
2) it is impossible to violate Bell inequalities with homodyne measurments performed on quantum states having positive Wigner functions. Indeed, in this case the Wigner function constitutes a "hidden variable" model for the measurement statistics. [Konrad Banaszek and Krzysztof Wódkiewicz, Phys. Rev. Lett. 82, 2009 (1999)],
3) the Wigner function negativity is a necessary condition for an exponential speed up of a quantum computer with respect to a classical one. Indeed, every algorithm involving states and operations representable by positive Wigner functions, can be efficiently classically simulated. [A Mari, J Eisert, Phys. Rev. Lett. 109, 230503 (2012)].
Mine is just a suggestion and you can judge for yourself if this section would be appropriate or not. I am not an expert on editing Wikipedia. — Preceding unsigned comment added by 2001:760:2C00:8253:226:BBFF:FE67:5EBC ( talk) 10:17, 27 November 2014 (UTC)
I am pretty certain that the 3D and 1D definitions of the Wigner function are not consistent with each other. See the factors of tow in the exponent and in the arguments of the bras/kets. — Preceding unsigned comment added by 66.207.203.214 ( talk) 15:27, 18 December 2017 (UTC)
If I'm not mistaken, the equation:
should be:
Else, the units turn wrong. Can someone check me and fix it? Shohamjac ( talk) 08:53, 10 August 2019 (UTC)
‘’q’’ is momentum and the wave function is normalized in momentum space. Cuzkatzimhut ( talk) 10:08, 10 August 2019 (UTC)
I am not tempted by serial reversions of the sentence discussed here ( last one of section 5 : "The truncated Wigner approximation is a semiclassical approximation to the dynamics obtained by replacing Moyal's equation with the classical Liouville's equation.") which has reemerged. I strongly believe, nevertheless, that the sentence adds nothing to the small section inserted into.
Truncations are, of course, hinted at in the generic first sentence and its references, in that section, and the relation to Liouville dynamics in the very first section of the article. I can only see trouble ahead, if anyone took the bait and provided the redlinked wished-for article, which would then have to be linked. But as it stands, the sentence provides no meaningful information, beyond a misguided clarion call. Cuzkatzimhut ( talk) 17:56, 10 November 2019 (UTC)
The present article does not quite correctly state that "the Wigner distribution holds a privileged position among all these distributions, since it is the only one whose required star-product drops out". The Glauber-Sudarshan P representation mentioned in the article leads to an average value of the operator G, which coincides formally with that in classical mechanics. In this representation, the density matrices are diagonal in the basis of coherent states, so one can write
The average value of the operator G is an integral over the phase space:
The usual dot-product works inside the integral. -- Edehdu ( talk) 07:14, 20 May 2022 (UTC)