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Newton series expansion of bosonic operator functions
by Jürgen König, Alfred Hucht
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|Authors (as registered SciPost users):
|Alfred Hucht · Jürgen König
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 and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third 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. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.
Published as SciPost Phys. 10, 007 (2021)
Author comments upon resubmission
we thank all three referees very much for their positive assessment and their useful comments. Two of the referees ask for more examples and illustrations of the supremacy of the Newton over the Taylor expansion. Motivated and stimulated by these requests, we substantially enhanced our manuscript by adding several new aspects that show the usefulness of the Newton expansion and clarify its properties and connections to other fields in physics. The changes are detailed in the section "List of changes" below.
In conclusion, we are sure that we have substantially enhanced our manuscript, the additional text adds up to 6 pages in the draft. We have included more applications and more connections of the Newton expansion scheme to different fields in physics. Therefore, we are confident that our manuscript is now eligible for publication in SciPost Physics.
List of changes
We substantially enhanced our manuscript v1 by adding several new aspects that show the usefulness of the Newton expansion and clarify its properties and connections to other fields in physics. To be more specific:
- As a new application, we have added a complete section 3.2 demonstrating that the Newton series expansion is naturally connected to coherent states. Expectation values of an operator function with respect to a coherent state are easily obtained from the function's Newton series, which provides, e.g., a convenient starting point for analyzing the Husimi distribution of a quantum-statistical system.
- In section 3.1 on the Holstein-Primakoff representation of spins, we have added a discussion illustrating that the Newton expansion allows for an approximative but systematic treatment of the spin operators in the sense that the r-th partial sum of the Newton series for the spin representation yields the correct spin commutation relations within the subspace of up to r Holstein-Primakoff bosons. In contrast, the corresponding r-th partial sum of the Taylor expansion already breaks the spin commutation relations when at least one Holstein-Primakoff boson is present.
- In section 3.3 on photon statistics, we have included a new paragraph that demonstrates more explicitly the supremacy of factorial over raw moments. We show that the expansion of a discrete probability function in terms of factorial moments converges fast, while the corresponding expansion in terms of raw moments bears the problem of bad convergence since raw moments generically grow exponentially with the order k.
- We have added a full subsection 2.4 that clarifies the connection between finite-difference calculus and commutators.
- We have added the new section 4, in which we introduced the "normal-order transform" of an operator function, which is obtained by applying the normal-ordering operator on the formal power series of the function. We were able to represent both the normal-order transform as well as its inverse transform directly in terms of an integral transformation, and reexpressed it in terms of the well-known Mellin transform.
- We have added six new references [4,5,6,31,32,33], updated Ref. , and have modified sentences all over the manuscript to improve the presentation. As requested by one of the referees, we now avoid abbreviations.
- Finally, we have slightly restructured the outline of the manuscript to account for the newly included material.
In conclusion, we have substantially enhanced our manuscript (by 6 pages). We have included more applications and more connections of the Newton expansion scheme to different fields in physics.
Submission & Refereeing History
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