Talk:Heisenberg group

message: Please put questions such as those at the end of the article into this talk page.--CSTAR 21:48, 1 May 2005 (UTC)

message: This article should address the following topics: 
* Angular position and angular momentum are conjugate in quantum mechanics. Is there an analog of the Heisenberg group for angular position and momentum? If not, why not?
* Derivations and Sheffer sequences are in a certain sense conjugate. Is there an analog of the Heisenberg group in this case? If not, why not?
* Cotangent bundles for general Riemannian manifolds posses natural canonical coordinates. Is there an analog of the Heisenberg group in this case? If not, why not?
:Reply to User:Linas: For the latter question, I think you may be interested in the theory of nilmanifolds. Also look at work of Stephen Semmes. But in any case please refrain from putting these questions in the body of the article.--CSTAR 04:15, 3 May 2005 (UTC)
replies: text: I put in the Mackey book that may be most relevant to the angular momentum question. For U(1), but for the compactness of the domain of the angle, the commutation relation is the same, [θ,L]=i, and the group essentially the same, with arguably different topology. For SU(2), the components of the angular momentum don't commute, and I am not even sure I've seen a workable "nice" version of the 15 CCRs, let alone the full group exponentiating them, but Mackey's book outlines its features.  As far as your second question is concerned, though, indeed: the fact the CCR of Δ with Sheffer bases identifies with the continuum Heisenberg commutation relation, see  here, makes the systematic Umbral calculus correspondence possible, by dint of identical combinatorics, for the same group−−−just a different representation. Cuzkatzimhut (talk) 16:42, 4 April 2013 (UTC)
replies: text: Thanks. Still pondering. 67.198.37.16 (talk) 19:35, 10 November 2020 (UTC)
thread_title:  Open questions 

message: This is a very, very limited approach to the concept ,Heisenberg group'. One actually should start with a (finite-dimensional) symplectic vector space as it was done in H. Tilgner, Ann.Inst.H.Poincare A13 (1970) 103-127 ,A Class of Solvable Lie Groups and Their Relation to the canonical Formalism'. This allows for a basisfree description of all canonical concepts in terms of the symplectic form. Then there is the Weyl algebra as a factorization of the universal enveloping algebra of the Heisenberg Liealgebra, in which this Liealgebra together with the symplectic Lie algebra is embedded as a Lie algebra. The simply connected Lie group, with the Heisenber algebra as its Lie algebra should be called Weyl group. Certainly this Weyl group is generated by the exponentials of this embedded Lie algebra. And note that there is a counterpart, if the symplectic form is subsituted by a pseudoorthogonal bilinear form, Jordan algebras playing the role of Lie algebras, the Clifford algebras playing the role of the Weyl algebras. There is no correspondance to the concept of (Lie) group here, but symmetric spaces in the sense of Otmar Loos may overtake this role. Hans Tilgner 13:50, October 19, 2005
:Hello Hans.  Thank you for making these suggestions for improvements to the article.  I agree with them.  I would like to see the article treat the general Heisenberg group on any symplectic vector space.  I'm reading a related paper by Babbit at the moment, and hope to update the article shortly.  I have also looked up your paper and hopefully will be able to learn some more things to help this article from that.  In the mean time, you are welcome to revise the article. -lethe talk [ +] 17:12, 3 July 2006 (UTC)
thread_title:  suggestions for improvement 

message: So I've added everything I wanted to add to this article.  It remains to make my additions coherent with the existing article.  For one thing, we have to choose a single notation.  My preferred way to accomplish this would involve among other things deleting a large amount of the existing material.  In particular, I think that the verification of the group law and inverse and the exponentiation in terms of matrices is tedious and not appropriate for this article.  We could just mention what the group and algebra representations are, and leave homework problems to the reader.  But before undertaking anything drastic, I would like to solicit your input, CSTAR.  I will also mention that I am going offline shortly for what may turn out to be an extended amount of time, so I may not attend to this matter further in a timely fashion.  -lethe talk [ +] 09:43, 11 July 2006 (UTC)
replies: text: Though the symplectic form of the Heisenbeg group is certainly the right one from a technical point of view, I'm not so sure that it's really appropriate for an encyclopedia article. My view of these things is "what could be simpler than matrices"? I Imagine some individual asking me what the Heisenberg group is. Knowing the technical advantages of the symplectic form, I probably will still respond with the matrix definition, even if it's non-canonical. --CSTAR 17:28, 11 July 2006 (UTC)
replies: text: I certainly can appreciate the view that the matrix representation is the easiest one to grasp.  It also seems to be the standard definition most often used in the literature, at least from what I've seen.  But the article already starts off with the 3x3 matrix case in the examples section, and then has a later section which sells itself as having the more "general" Heisenberg group, I think the more abstract formulation can fit nicely there.  I don't agree that the canonical formulation is inappropriate for the article, I think it might be relevant base material for some articles about geometric quantization (which has become my new favorite subject recently) that I envision in the distant future.  I think with some reorganization of the article we could arrive at a state which would satisfy both of us, and in fact some reorganization is necessary because my additions have left the article in a disorganized state with too much repetition.  The one complaint I had about the calculation of the matrix multiplication, inverse, and exponentiation being too remedial for this article would be mitigated if we lost the canonical formulation, or even moved it to a later section, but I might still like to get rid of it or at least tighten it up.  Seems like the sort of thing the reader ought to do for herself.  But I won't press the issue if you disagree. -lethe talk [ +] 21:44, 11 July 2006 (UTC)
replies: text:  BTW another book which I thoroughly recommend (and kick myself for not buying when it was on a yellow sale) is Varadarajan's book "Geometry of Quantum Theory" vols 1 and 2, republished as 1 volume by (surprise) Springer. It treats beautifully and carefully a vast number of subjects, including Mackey's theory of systems of imprimitivity, Gleason's theorem, Induced representations.--CSTAR 21:54, 11 July 2006 (UTC)
replies: text:  I think I actually picked up the copy you failed to!  I bought an old used copy of the pre-Springer two volume set a while ago.  I've only gotten around to reading the first couple of chapters from volume 1, which seemed to be about Stone duality of Boolean algebras and quantum logic.  I was looking more for stuff about polarizations and the Moyal product on manifolds, so I ended up spending more time with a book by Woodhouse.  I guess I will give Varadarajan another look based on your recommendation, but for now all my books have disappeared into storage. -lethe talk [ +] 22:06, 11 July 2006 (UTC)
thread_title:  moving on 

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thread_title:  External links modified 

message: The section on Lie algebra has Y = Z as two basis elements. As a basis (linear algebra) consists of distinct elements, correction is necessary. — Rgdboer (talk) 21:34, 13 January 2018 (UTC)
replies: text: Y and Z are not the same vector.  Sławomir Biały (talk) 02:37, 14 January 2018 (UTC)
thread_title:  Algebra basis 

message: The opening paragraph is as follows:
"In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin{pmatrix}
 1 & a & c\\
 0 & 1 & b\\
 0 & 0 & 1\\
\end{pmatrix}
under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group")."
But in English the word "the" as it appears in "the Heisenberg group" implies unambigously that there is only one group that is the Heisenberg group — which is untrue.
Rather, the paragraph should *perhaps* describe what a Heisenberg group is.
Here is a better idea: Since there in fact is a group that is usually called the Heisenberg group — it is the group where the "commutative ring" is taken as the field of complex numbers — it would be better to define that group as "the" Heisenberg group, and then in a later section of the article, define "generalization" where the defintion  can be applied to an arbitrary commutative ring with identity.
I hope someone knowledgeable on this subject will make such a change in the article.71.8.202.138 (talk) 01:11, 13 October 2020 (UTC)
thread_title:  Contradiction in opening paragraph 

message: Another nonsensical thing about the article, as of this writing, is that although "the Heisenberg group" is defined as a 3-dimensional group (with entries for a, b and c), the next section considers "the 3-dimensional case".
Since this article ought to make sense, the 3-dimensional case should be the main definition of this group, and then higher-dimensional cases can be defined under a later section that is titled Generalizations.
I hope that someone knowledgeable on this subject can implement such a change.71.8.202.138 (talk) 01:17, 13 October 2020 (UTC)
replies: text: No. Please read the intro more carefully; it says More generally, one can consider Heisenberg groups associated to n-dimensional systems... Just because something is difficult to understand does not make it "nonsense". 67.198.37.16 (talk) 19:15, 10 November 2020 (UTC)
thread_title:  Another nonsensical thing about the article