Formal series convergence in deformation quantization and $C^*$-condition

A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^*$-algebras instead of mere $*$-algebras (which $(\mathcal{C}^{\infty}(M)[[t]],\star)$ is) after deforming the initial commutative $C^*$-algebra of observables is evoked here: http://ncatlab.org/nlab/show/C-star+algebraic+deformation+quantization

However this is done very briefly and in the language of higher symplectic geometry which I'm not familiar with.

Can anyone give simpler explanations and arguments/theorems justifying the link between convergence and the $C^*$-condition? Or some reference for an introduction?

• Not sure if I understand the question. Are you maybe asking why people generally demand algebras of quantum observables to be C* ? In any case, to get some feeling for the definition, check out the list of examples of strict deformation quantizations starting on p. 4 of the Marc Rieffel, "Deformation quantization and operator algebras" math.berkeley.edu/~rieffel/papers/deformation.pdf – Urs Schreiber Sep 2 '13 at 0:17
• @Urs: Thanks for the reference. What I was meaning is the following: strict deformation is defined as the restriction to the cases where the deformed algebra is a $C^*$-algebra, and many comments say that this corresponds to asking formal series of the deformation parameter (coefficients of the series belong to the non-deformed algebra) to converge. Is there a simple explanation/argument/proof of this statement? – Issam Ibnouhsein Sep 2 '13 at 0:51
• I see, I guess you are asking in which way the algebra defined by a strict C*-algebraic deformation can really be understood as being generated from converging formal power series. Hm, I can't seem to be able to point to a lot of explicit discussion of this, admittedly. The following article has something in this direction: M. Bordemann, M. Brischle, C. Emmrich, Stefan Waldmann, "Subalgebras with Converging Star Products in Deformation Quantization" arxiv.org/abs/q-alg/9512019 . Also the book by Waldmann "Poisson-Geometrie und Deformationsquantisierung" should have more on this. – Urs Schreiber Sep 2 '13 at 8:16
• Ok, thanks for the references I'll have a look at them! – Issam Ibnouhsein Sep 2 '13 at 18:55

OK, let me give a try on this question. There are several problems hidden underneath which one has to address.

First, for physical reasons a formal deformation is not sufficient. $\hbar$ is a constant of nature but not a formal parameter... More severely, the formal star product algebras do not allow for a reasonable notion of spectra and thus can not produce physically reliable prediction concerning possible values of measurements. For this (and many related reasons) formal deformation quantization is not the final answer.

Second, the $C^*$-world provides all we need for a good quantum mechanical interpretation: good notion of spectra compatible with a spectral calculus etc. So in some sense, this is the situation we all want to reach: finding a $C^*$-algebra containing particular elements which have a physical interpretation. Note that just saying "this $C^*$-algebra is the algebra of observables of my system" is physically still meaningless. You have to specify an interpretation of which algebra element corresponds to which (physical) observable (ie. which, at least in principle, realisable measurement apparatus).

Third, and this is the bad side of the story: except for very simple situations (CCRs) it is very very hard to write down a $C^*$-algebra corresponding to a certain quantum system of which one only knows its classical counter part. So this is the problem of quantization: given a classical physical system, one wants to guess its quantum description. Of course, there are physically relevant situations where this is known and well-understood, but I'm taking here of more general classical systems: eg. for systems with gauge degrees of freedom, the physically relevant degrees of freedom form the "reduced phase space" which can carry an complicated geometry such that there are simply no physical observables which allow for a simply CCR quantization.

Together, this indicates that the formal DQ might be not a solution but a first step: since for formal DQ one has very well-understood and powerful existence and classification results, one tries to use them and, as a second step, guess/construct the desired $C^*$-algebraic framework. But this is not at all easy.

So one way to go is to take the formal power series in the star product and ask for their convergence (in a mathematically meaningful way). This is tricky and has been achieved ony in (few) examples. In fact, I have some current projects in this directions. Ideally, one obtains a topological non-commutative algebra afterwards, which might not yet be $C^*$. But, and this is the first non-trivial step, it will be an algbebra over $\mathbb{C}$ and not just over $\mathbb{C}[[\hbar]]$. Of this algebra, one can then study HIlbert space representations by, in general, still unbounded operators. Arriving at this stage there are several standard techniques to build nice $C^*$-algebras out of it.

Why can one hope that this works? In those examples where one has a $C^*$-algebraic deformation in one of the many senses of "strictness", it always involves some reasonable behaviour for $\hbar \to 0$. If one requires not just continuity, but some weak sort of smoothness, then one can "differentiate" the continuous field of $C^*$-algebras (from the right) at $hbar = 0$. This will give then a formal star product. As usual, one has only smoothness but not analyticity and hence the formal star product will not directly sum up to the continuous field of $C^*$-algebras. So in my opinion, this is the relation one can hopefully expect in quite some generality.

• Can you point to one explicit example that discusses C*-algebras which are explicitly the completion of algebras of converging formal power series inside a formal deformation quantization? I thought that's what Issam is asking for. – Urs Schreiber Sep 6 '13 at 8:01
• @Urs: haha, you got me. There are not so many examples around. The trivial one is to consider as subalgebra of smooth functions on $\mathbb{R}^{2n}$ the span of exponentials. They form a sub-algebra for the (formal) Weyl star product and give the $C^*$-algebraic Weyl algebra as completion. But beyond...? – Stefan Waldmann Sep 6 '13 at 8:41
• Thanks, that's already good to have this confirmed! I wasn't sure. For what it's worth, here is my feeling about the issue: in simple cases where everything is under control, such as the quantization of the 2-sphere, the strict C*-deformation produces the algebras of endomorphisms of the spaces of states that are produced by the geometric quantization process. But in geometric quantization we know fully well and in generality how to quantize the observables: find a maximal subgroup of the classical group of observables which preserves the polarization, then prequantize that. – Urs Schreiber Sep 6 '13 at 9:27
• Ok, thanks for this answer. It gives some perspective on the comments that seemed to suggest that the link between convergence and $C^*$-condition was trivial. Do you have a reference for the details of the first non-trivial step (getting an algebra over $\mathbb{C}$ by imposing convergence)? – Issam Ibnouhsein Sep 6 '13 at 9:47
• This seems to be related to issue of divergence of perturbation theory in QFT. For a model like phi^4 the gradings by the coupling or hbar are essentially the same. Typically these series diverge but in some examples one can prove the next best thing which is Borel summability. I am not too familiar with the DQ framework, but I suppose you could apply it to the quartic oscillator in 1d. That may be a good test example for this problem of summing the series in hbar. – Abdelmalek Abdesselam Sep 10 '13 at 16:53