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Then the series expansion can be written in terms of the (normal) operator exponential. In the answer to the lined question, I defined this operator using Fold, but this required knowing the expansion order beforehand.

Then the series expansion can be written in terms of the (normal) operator exponential. In the answer to the lined question, I defined this operator using Fold, but this required knowing the expansion order beforehand.

An example where this is needed would be a pathological function such as f = 1/x^2. In Mathematica 8.0.4, there is no problem doing the required Fourier transforms in star[1/x^2,p] or star[p,1/x^2], independently of the order. But in version 9, you will need the last definition of star. What happens is that the FourierTransform remains unevaluated for star[1/x^2,p]. In that case, the If statement takes the unevaluated result, switches the order of the Fourier integration variables. The justification for this is that it produces a result which differs from the opposite order star[p,1/x^2] in the expected way, so it preserves the desired skew symmetry of the Moyal bracket.

An example where this is needed would be a pathological function such as f = 1/x^2. In Mathematica 8.0.4, there is no problem doing the required Fourier transforms in star[1/x^2,p] or star[p,1/x^2], independently of the order, provided you first set $Assumptions=x>0. But in versions 9 and 10, you will need the last definition of star. What happens is that the FourierTransform initially remains unevaluated for star[p,1/x^2] (Mathemtatica tries for a while but then gives up). In that case, the If statement takes the unevaluated result and switches the order of the Fourier integration variables. The justification for this is that it produces a result which differs from the opposite order star[1/x^2,p] in the expected way, so it preserves the desired skew symmetry of the Moyal bracket.

Edit

Edit

The definition of the Moyal product in the question was missing a factor 1/n! under the sum. I used the textbook definition instead. A reference for this definition, without any unnecessary technicalities, is here: Quantum Mechanics in Phase Space (arXiv), see the appendix.

Edit

The definition of the Moyal product in the original question was missing a factor 1/n! under the sum. I used the textbook definition instead. A reference for this definition, without any unnecessary technicalities, is here: Quantum Mechanics in Phase Space (arXiv), see the appendix. Thanks to Rahul Narain for spotting the discrepancy between my textbook definition and the original form of the question.

In the following, I also set the constant h from the question equal to 1.

First part of the solution: polynomials

Edit: More general functions