Talk:Wigner quasiprobability distribution

First post , 4 July 2006

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

$A(q,p)=\int _{-\infty }^{\infty }ds\langle q-s/2|{\hat {A}}|q+s/2\rangle e^{ips/\hbar }$

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 .

A: Indeed, you are right, and can check the facts in the Book

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)

Wigner Function and Probability Distributions

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.

Ville

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)

See the 6th reference. A French information scientist and most definitely not Weyl. He is not known in physics, but the signal processing community pays homage to his independent introduction of that function on the time-frequency plane.Cuzkatzimhut (talk) 11:19, 1 April 2010 (UTC)

Split proposal for The Wigner–Weyl transformation section

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)

Definition of variables

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.

Recent additions in the Evolution equation for Wigner function section

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 would not mess with Marinov, introduced, as you can see, in the Sep 23,2012 edit by User:Taulalai. Some of the discussion of contrasting his path integral to Sharan's for the propagator, with the obvious collapse of the two *s in the propagation of P into the Moyal phase space propagator is found in User talk:Taulalai. You must convince him, which I doubt. I had to struggle to keep Sharan in. Personally, I am baffled by your wild claim that Leaf reproduces Marinov's construction. Please be specific with formula numbers and specific statements involving them--not a mere "behold"! It seems to me Leaf is just formally expanding on Moyal's (10.2), a Dyson expansion in Moyal brackets, without telling you how to calculate the expansion--surely you are not chalking up a phase-space path integral to Moyal? I am also confused about what you mean by "the main equation", as it was now, soundly, but with regrettable inflammatory rhetoric, reverted by Pendleton. If it is the p-fourier transform of Moyal's eqn for conventional hamiltonians, I am not sure what the reference you adduce or its referee (!? invisible statements by anonymous referees? some discussion here!) claims. I am familiar with it from the middle eqns of eqn (18) in the Overview of ref [6] in this article, and it is quite standard. Have you tried it at the Method of quantum characteristics? It wouldn't be a bad idea to move all this stuff, including the characteristics and phase-space path integrals to that article--and be as scholarly and expansive as you want, since it is an "advanced topic". Cuzkatzimhut (talk)

The reviewer of PRL has recommended the paper for publication, so he was impressed by some of the equations, hopefully not by those deleted by Wikipedia readers. Sect. 4 of Leaf’s paper is devoted to the path integral in phase space. For example, Eq. 4.5 looks quite modern. Sharan and Marinov reformulated the Leaf equations. As far as I know, these integrals have never been calculated. Their existence is interesting from a fundamental point of view. Trompedo (talk) 14:13, 14 November 2012 (UTC)

I'm not sure what a particular paper and its referee have to do with this discussion? With due respect to Leaf's paper, which is normally acknowledged in these discussions, his (4.5) is the trivial symbolic iteration of Moyal's group property (9.8); am I missing something? Moyal goes on to recast the infinite iteration thereof to (10.2). Are you arguing that this abstract formalism excursion belongs in an introductory article? Why might you be loath improving the real McCoy, Method of quantum characteristics? Cuzkatzimhut (talk) 14:41, 14 November 2012 (UTC)

The last equation, removed by Ms. Pendleton, was accompanied by a reference to a paper. Apparently, this equation was in the paper, and it was approved. We usually respect the reviewers, although from my point of view this equation is not new, and from the point of view of Ms. Pendleton this equation is neither new nor interesting. No problem. I do not have access to the second paper of ref. 6, but I see that it was written 40 years after Leaf. Is not there a logical circle? The work done 40 years ago must seem elementary. Unfortunately I do not have time enough to write for Wikipedia. Trompedo (talk) 15:32, 14 November 2012 (UTC)

I believe the book referred to in [6] is available from google books on the web for free. Moyal's paper, replicated there, written essentially in 1946 and published in 1949, cf eqn (7.6) contains the essentials. I would take no issue with your appreciation of Leaf's pedagogical treatment in his paper, as long as you don't make wild claims of originality on its part. You are aware of Moyal's further work on these things, no? In any case, I have an anxious sense we are discussing at the wrong page for the wrong article here.Cuzkatzimhut (talk) 16:25, 14 November 2012 (UTC)

Feynman proposed path integral for the amplitudes in 1951. Your remarks can be interpreted to mean that Moyal discovered path integral before Feynman. How it can be?? Sect. 4 of Leaf’s paper provides what we call path integral for the Wigner function. Just one fair reference for the same result prior to Leaf (but after Feynman, please) could disprove the priority of Leaf. If such a reference does not exist, the citation Leaf’s paper would grandiosely decorate the article. Trompedo (talk) 16:16, 15 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)

Eqn (7.6) was adduced as an overlap to the deleted equation. Eqn (9.8) is the group recursion easily iterated, which, in his own way (eqn (10.2)) Moyal, referencing Feynman in his (Moyal's) last ref, certainly addresses. Moyal's paper, together with Groenewold's are the two pillars of phase-space quantization: The guys invented the whole thing, when Terletsky or Husimi or Bass or Yvon didn't! Could I also prevail on you to read Groenewold's [paper] which demolished the Dirac unstated picture that, somehow, phase-space had something to do with classical mechanics and Hilbert space with quantum mechanics, when the two (PS & HS) are basically as equivalent as different coordinate systems? Cuzkatzimhut (talk) 20:55, 25 November 2012 (UTC)

Obviously, something is wrong with the numbering of equations. The Moyal paper of 1949 has only eight sections. I read the Groenewold paper before; there one may find the Moyal evolution equation and the so-called Stratonovich basis. However, I do not remember the path integral there. I also think that the importance of the papers of Groenewold and Moyal is still underestimated. Trompedo (talk) 23:11, 27 November 2012 (UTC)

Let's see... Moyal's paper has 26 pages [99-124], 17 sections, and 5 appendices, plus an un-numbered summary, no? I never suggested Groenewold's comes close to path integrals--but I am intrigued anyone would call it the "Stratonovich basis" ... it is the invertible Weyl correspondence. But, then, I never understood why some call it "Mackey's parameterization of the Heisenberg group". I would agree with you that, ever since G & M developed the complete theory of QM in phase space in the 1940s, several people went on to re-invent the same constructs paying scant attention to the masters. Cuzkatzimhut (talk) 01:32, 28 November 2012 (UTC)

Sorry, it was the paper: J. E. Moyal, STOCHASTIC PROCESSES AND STATISTICAL PHYSICS, Journal of the Royal Statistical Society. Series B (Methodological), Vol. 11, No. 2 (1949), pp. 150-210. Now I have downloaded the right text. Eq. 4.5 is of no interest, 7.6 is the equation deleted by Ms. Pendleton, 9.8 is the composition for developing the path integral, 10.2 is the Taylor expansion of the propagator. Sure, everything is ready for the iteration, and even the term "Markoff process" is present. However, the path integral is missing. The evolution from Moyal 1949 to Leaf 1968 looks like the evolution from Dirac 1933 to Feynman 1948. I would congratulate Leaf. Trompedo (talk) 08:31, 28 November 2012 (UTC)

Ineed: continued on User talk:Trompedo ... ... Cuzkatzimhut (talk) 13:17, 28 November 2012 (UTC)

Wrong sign in 4th formula

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

Good point! Now fixed, but you must flip the sing of the dummy variable y in the definition to see it. Interestingly, the error is replicated in the Wigner-Weyl transform article, as well! Cuzkatzimhut (talk) 02:20, 14 November 2012 (UTC)

Proposal to create a new article "Moyal equation of motion"

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

${\partial P(x,p,t) \over \partial t}=-\{\{P(x,p,t)~,~H(x,p)\}\}$

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)

I sense some resistance by just about everybody in discussing Moyal's equation of motion anywhere but in the Method of quantum characteristics article, which is both extensive and poorly written and in need of help. Odd priority disputes, philosophical divergences, etc.. emerge all over the place, except there, the designated arena for this evolution. I think if you had the gumption to rename the "Method of quantum characteristics" page into something like "Time-evolution of the Wigner function" (no superfluous "equation" please! of course evolution is controlled by Moyal's equation), reorganize it, rewrite it, compare and contrast methods, path integral techniques, etc, without multiplexing into mini-articles and stubs summarizing particular papers, that would be greatly appreciated by the interested reader. But, the issue is technical, not encyclopedic, and after 60+ years of work has produced little explicit progress, beyond endless hyper-formal reformulations. The interested reader has aleady hit the books and abundant papers and reviews. Both ref [6] here (scribd), W Schleich's excellent book, and Wong's numerous papers so studiously effaced from this article could have a dimension to contribute in it. It is the study of phase space propagators, initiated by Moyal in 1949. Incidentally, virtually all the refs in all the articles you mentioned are available for free in google scholar and google books, so having people rediscover the wheel because they have no access to the original papers that did it all is a bit rich.Cuzkatzimhut (talk) 01:26, 16 November 2012 (UTC)

"Wong's numerous papers" offer classical trajectories to calculate the quantum evolution of Wigner function. This is a mistake. Mi Tatara Buela (talk) 10:12, 29 March 2013 (UTC)

Proposed merger of Wigner distribution function

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

After 2 months of yawning, with concurrent negligent consensus on Talk:Wigner distribution function, the lapsed merge proposal template is taken down. Cuzkatzimhut (talk) 19:28, 20 September 2013 (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)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Negativity of the Wigner function

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)

Can you compose something pithy, concise, and informative, and propose it on this page?

Section 1, right at the start, does just that, gently, hinting at the numerous conceptual pitfalls novices regularly slip into. Hudson's point unfolds, in plainer language, in the 2nd paragraph.

I gather you have also consulted Phase space formulation#Phase space distribution, where the much more important violation of the 3rd axiom of probability is briefly mentioned, on account of the uncertainty principle. This is what is underlying the negative values, that are just readily apparent. All the points you raise are self-evident (I assume you have read up on Ole Steuernagel's work), but excessive reliance on them can lead the novice astray into goose-chase suppression schemes of the negative values instead of sound appreciation of the uncertainty principle.

I would strongly object to discussions of Bell-inequality material here, as opposed to a more specialized venue, such as the Bell's theorem wikis--it can only lead to grief and confusion here, in this article. Likewise for quantum computing. It may be more appropriate for these discussions, (possibly covering Revzen, M.; Mello, P.; Mann, A.; Johansen, L. (2005). "Bell's inequality violation with non-negative Wigner functions". Physical Review A. 71 (2). doi:10.1103/PhysRevA.71.022103.), in their respective wikis, to point to here, section 1, or elsewhere. Still, this here is an introductory venue.

But, of course, constructive proposals of informative paragraphs for Section 1 or others, or the phase space formulation wiki above are welcome, and you could propose some, here. Cuzkatzimhut (talk) 12:06, 27 November 2014 (UTC)

new comment

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)

It is correct, if you properly change variables. What, specifically, in formulas, is your problem? Reassure yourself of the sign change from bra-ket to the 1d wavefunction expression, and then from that to the 3d one. New comments go to the bottom of the talk page, not the top. Cuzkatzimhut (talk) 16:11, 18 December 2017 (UTC)

Constant in the momentum symmetrical Wigner function

If I'm not mistaken, the equation:

$W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\varphi ^{*}(p+q)\varphi (p-q)e^{-2ixq/\hbar }\,dq$

should be:

$W(x,p)=2\int _{-\infty }^{\infty }\varphi ^{*}(p+q)\varphi (p-q)e^{-2ixq/\hbar }\,dq$

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)

Removal of insistent insert on "truncations"

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)

Average value of operator in the Glauber-Sudarshan P representation

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

ρ = ∫dαdα*/(2πi) exp(-αα*) |α〉P(α,α*)〈α|.

The average value of the operator G is an integral over the phase space:

〈G〉= Tr[ρG] = ∫dαdα*/(2πi) exp(-αα*) P(α,α*)〈α|G|α〉.

The usual dot-product works inside the integral. --Edehdu (talk) 07:14, 20 May 2022 (UTC)

I fear you misread the statement. The star product of the Glauber ordering does not integrate itself out by parts! The expression you provide has a Gaussian optical phase-space measure . Converting to the phase space discussed, the Glauber-Sudarshan Cohen measure is distinctly nontrivial, $\exp(-\hbar (\tau ^{2}+\sigma ^{2})/4)$, as you may check in standard texts, specifies a nontrivial star product, and prevents its elimination inside the integral. Please review optical phase space. Cuzkatzimhut (talk) 13:33, 21 May 2022 (UTC)