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It is a classical fact that a formal deformation quantization of a Lie-Poisson structure is provided by the universal enveloping algebra of the corresponding Lie algebra. Remarkably, this statement generalizes to some extent to more general (polynomial) Poisson algebras. In particular it holds for every such up to degree three in $\hbar$ ! This is due to Penkava-Vanhaecke 00.
I have added a quick summary of this theorem to deformation quantization in a new subsection: Existence – Deformation by universal enveloping algebras. I also gave this an entry on its own at polynomial Poisson algebra.
This is maybe remarkable, since there is possibly no physical measurement known which could detect contributions of higher than third order in $\hbar$. Though I’d need to check. This is subtle because order in $\hbar$ is different from the usual loop order that is commonly stated (which is order in the coupling constant) and the relation between the two is complicated.)
Also (and that’s how I came across this article) at least in special cases this gives a way to quantize just by universal constructions on Lie algebras, hence this might potentially tell us something about the quantization of Poisson bracket Lie n-algebras (for which no analog of the corresponding Poisson algebra, i.e. with an associative product around, is known).
I do not understand all this mistification. Penkava calls this “quantized enveloping algebra” to emphasize that it is deformed unless one deals with linear Poisson structure. One should not mislead somebody by just saying universal enveloping!
Also 3.15 is misleading talking “polynomial Lie-Poisson structure” as if it were something more general and it is only about linear Poisson structure and the result is just the same as for the linear Poisson structure in smooth case.
I also do not understand this optimism about measurement nondistinguishibility. Penkava says that there are examples when their quantized enveloping does not extend to order 4, that means in those cases you can not extend this solution to all orders at all and you have to take another solution which clearly is not quantized enveloping at order 3, thus the physics will DIFFER already at the order 3!
I do not understand all this mistification. Penkava calls this “quantized enveloping algebra” to emphasize that it is deformed unless one deals with linear Poisson structure. One should not mislead somebody by just saying universal enveloping!
It seems to me this paragraph is requesting that the word “quantized” be inserted into the entry at one point. I won’t mind if you insist, its just terminology, but I thought there are good reasons not to: a) In the case of a Lie-Poisson structure the Poisson UEA reduces to the usual one, which is not called “quatized” either, b) the construction of the Poisson UEA indeed has the universal property from an adjunction with Poisson algebras, and c) saying “quantized” somehow defeats the whole point of the theorem, which is to say that this UEA provides by universal construction a deformation up to order 3.
But I will not fight about such issues. If you insist, please feel free to change “universal enveloping” to “quantized universal enveloping” in the article.
Also 3.15 is misleading talking “polynomial Lie-Poisson structure” as if it were something more general
It’s the more precise statement. We may retain the polynomials or all smooth functions in the Poisson algebra. It does not make a big difference, but it serves to be precise nevertheless.
I also do not understand
It’s a simple point: The usual prescription of formal deformation quantization is one proposal for formalizing the process of quantization in a certain regime. As you know, there are other proposals, in other, sometimes overlapping regimes. One can come up with more quantization schemes. It is a natural question to ask to which degree these different quantization schemes have been tested by experiment.
Here an interesting question is: What if we declared that quantization of a given physically relevant (polynomial) Poisson algebra is not to be given by the usual precription of formal deformation quantization, but what if we declared that the real quantization is in fact that universal enveloping algebra.
The theorem under discussion says that for skew-symmetric $\pi_1$ these two formalizations of quantization would be instinguishable up to and including order $\hbar^3$. That order also seems to be the maximal order of $\hbar$ which is presently distinguishable by scattering experiment (see at loop order). Hence it would logically be possible that at order $\hbar^4$ ordinary deformation quantization disagrees while UEA-quantization coincides with nature.
That does not prove anything, but it seems a fun fact to notice.
The main problem with this speculation taken at face vaue is the skew-symmetry of $\pi_1$ that enters this theorem: For the formal algebras that appear in relativistic field theory (e.g. the Wick algebras of the free fields, but also the interacting algebras) $\pi_1$ is given by a Hadamard propagator (for the Wick algebras) or by the Feynman propagator (for the time-ordered products) and both of these have a non-vanishing symmetric component.
But this story changes as we de-transgress from phase space to the n-plectic structure on the jet bundle that induces it under transgression. There, first of all, one actually encounters the polynomiality that we would need for the theorem, and second the symmetric component disappears upon this de-transgression. That’s why I am thinking it might be interesting to consider the universal enveloping algebras (in the evident appropriate sense) of the Poisson bracket Lie $n$-algebra which is induced by the pre-symplectic current on the jet bundle.
If you don’t like this speculation, I kindly invite you to ignore it.
these two formalizations of quantization would be instinguishable up to and including order
If one of them does not exist at the order bigger than 3 then it is easy to distinguish from it!
If the coefficient in front of fourth order thing is huge one could distinguish as well, but this is less important.
I still find “polynomial Lie-Poisson structure” very misleading as “linear Poisson structure” is a well established term and saying polynomial in that context instead strongly suggests that we go to more general case. If you want to be precise about the space of functions allowed then use the geometric language: linear Poisson structure on affine space versus linear Poisson structure on an adjoint orbit, or linear Poisson structure on a smooth manifold $\mathbb{R}^n$ etc.
What if we declared that quantization of a given physically relevant (polynomial) Poisson algebra is not to be given by the usual precription of formal deformation quantization, but what if we declared that the real quantization is in fact that universal enveloping algebra.
That means that you strongly broke the symmetry and took linear terms as linear, and quadratic as quadratic, and this concept depends on the local coordinates. In other coordinates linear will be something else and you will have different “universal enveloping”!! So it is not really universal. In other words, you choose a filtration on the algebra in order to define what is your “universal” enveloping. This can not easily go to more complicated geometries and Poisson algebras, so it is likely an artifact of the polynomial like algebras which is hard to globalize in choice independent manner.
I am not saying that I like “quantized” but that it is deformed. Quantized here means deformed and it is deformed in every case except of the usual linear Poisson structure.
In other coordinates linear will be something else
That’s why it is important to speak of polynomial Poisson algebras in this context.
one of them does not exist at the order bigger than 3
What you really mean is that it is not in general a deformation in the category of associative algebras. But that’s the point: There are other quatization prescriptions than Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer proposed. For instance geometric quantization is also not a deformation in the category of associative algebras. Here we have yet another kind of deformation, one that happens to agree with the BFFLS prescription at least up to third order in the deformation parameter, but possibly not beyond.
Since the physical validity of BFFLS quantization has arguably been verified in experiment only to just around order three, it is a legal question to wonder whether other quantization prescriptions are still experimentally viable, too.
Here we have yet another kind of deformation, one that happens to agree with the BFFLS prescription at least up to third order in the deformation parameter, but possibly not beyond.
There is no deformation in the cases when it does not extend beyond the third order. If there is an obstruction then there is no deformation period. Define me the “universal enveloping algebra deformation” in the case when it is not defined beyond the third order. If you take formally $h^4 = 0$ yes, but this has nothing to do with physics and measurement. There is no consistent quantum field theory as far as I know in which the physical constants in units of Joule seconds are nilpotent.
That’s why it is important to speak of polynomial Poisson algebras in this context.
I have nothing against talking polynomial Poisson algebras. It is not confusing. But I have against talking “polynomial Lie-Poisson structure”. The latter is confusing among the rest because there are Lie-Poisson groups and also their tangent Lie algebras. Lie-Poisson groups may be algebraic hence one works with polynomials there. When one talks nonlinear structures then Lie-Poisson inevitably reminds the practitioners of the Lie-Poisson groups. But those are not the same as affine space except in one case. In linear case it is standard to talk linear Poisson structure or to talk the corresponding Lie algebras or coadjoint orbits. If one mentions polynomial one would immediately thin you talk about Lie-Poisson groups and that is not the affine case. When I forget this discussion in 10 years and look at this page I will be mislead. Don’t you agree that if something is usually called linear it is less confusing to add in which context then to call it polynomial, especially when “Lie-Poisson” has another interpretation of polynomial when we talk about algebraic group case. Besides the geometric language is much clearer, fully precise and without any problem. So one can say linear Poisson structure on the affine space (as algebraic variety) for what you call polynomial (but it is polynomial affine, not algebraic), on $\mathbb{R}^n$ as a manifold, on the same as analytic manifold, and also on formal neighborhood of zero. The last would be an oxymoron in your approach as it would be then logically called formal Lie-Poisson structure but formal and Lie categories mean a different thing.
Historically it is highly inaccurate to attribute the geometric quantization of Kirrillov, Kostant, Sourieu to Flato et al. Flato introduced deformation quantization even when the unitary Hilbert space operators are not realized and are beyond the reach of the method. Deformation quantization in their form is not giving unitary operators, hence it is a mathematical precursor of a true physical problem. Unitary quantizations were studied before in various generalities and prescriptions, geometric Berezin, Toeplitz, Weyl bundle etc. It is now generally accepted that for true unitary quantization one needs more data in general.
Define me the “universal enveloping algebra deformation” in the case when it is not defined beyond the third order.
So one can say linear Poisson structure on the affine space (as algebraic variety) for what you call polynomial
If you insist, i have changed the name of example 3.1.
When I forget this discussion in 10 years and look at this page I will be mislead.
I took some time to write it out in full definitions.
I am surprised that such a simple summary of statements right from the literature causes such a headache over minutiea. But the entry is free for anyone to edit.
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