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Newton series expansion of bosonic operator functions
by Jürgen König, Alfred Hucht
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|Authors (as Contributors):||Alfred Hucht · Jürgen König|
|Arxiv Link:||https://arxiv.org/abs/2008.11139v1 (pdf)|
|Date submitted:||2020-08-26 11:56|
|Submitted by:||Hucht, Alfred|
|Submitted to:||SciPost Physics|
We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms. As a second example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 3 on 2020-10-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2008.11139v1, delivered 2020-10-07, doi: 10.21468/SciPost.Report.2057
Nice introduction/reminder of the Newton expansion for finite-difference calculus. The approach is naturally motivated for operators that take only discrete values. The expansion can be quite useful for square roots of operators, such as appear in the Holstein-Primakoff representation of spins. For this example, the Newton expansion terminates at a finite order, in contrast to the usual Taylor expansion used for spin wave computations. The Newton expansion thus reproduces all matrix elements within the bosonic Hilbert space exactly.
There is also a nice discussion of the application of the method to photon statistics, with an emphasis on the value of the “factorial moments” over the more conventional raw (or ordinary) moments.
It would be nice if there were an example where a key physical result is correctly computed by this method, but not by the more conventional approaches.
This paper presents a clear pedagogical explanation of the use of finite-difference calculus to accurately compute the expectation values of operator expansions that may formally have non-physical points of non-analyticity in the expansion, (e.g. sqrt operator around the origin).
I believe the acceptance criteria have been met.
Anonymous Report 2 on 2020-9-24 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2008.11139v1, delivered 2020-09-24, doi: 10.21468/SciPost.Report.2021
The paper makes an interesting mathematical observation, which could be useful in several fields of theoretical physics.
The paper provides two examples of how the Newton series can be useful, however, they are both somewhat abstract, and it would be nice to see a more concrete example, where this usefulness is evident.
I can already see the report by the other reviewer, which I in many ways agree with. The authors are making an interesting observation, which might be of interest to other researchers working on quantum mechanical problems. It is perhaps hard to identify a very specific result in this manuscript, but it is indeed interesting to note how the Newton series has certain advantages over the standard Taylor expansion. As such, I believe that the paper should be published, as it may stimulate further theoretical developments in many different fields.
A minor suggestion would be not to use abbreviations such as "w.l.o.g.", which might not be known to all readers.
Report 1 by Dirk Schuricht on 2020-9-16 (Contributed Report)
- Cite as: Dirk Schuricht, Report on arXiv:2008.11139v1, delivered 2020-09-16, doi: 10.21468/SciPost.Report.1997
The authors discuss the application of the Newton expansion to define operator-valued functions of the bosonic number operator. They briefly discuss the mathematical background and relation to the usual Taylor series. They present two specific examples: Holstein-Primakov representation of spins and photon statistics. In particular, they point out the advantages of using the Newton series over the usually applied Taylor series.
In my view, the article can be seen as a neat little note pointing out a fairly simple yet usually unknown mathematical method which can be applied in quantum mechanics to simplify and clarify operator-valued functions. As such I find the ideas of the note worthwhile publishing (maybe in SciPost Physics Core). However, I would appreciate some additions or remarks on further applications or generalisations, eg, on the applicability of the approach to operators that do not possess a spectrum of equally spaced real values, fermionic system, or systems with several degrees of freedom.
As a minor remark, I think the notation in (2) is somewhat imprecise regarding to which part of the expression the derivative is applied.