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New entry semiclassical approximation. It requires a careful choice of references. The ones at the wikipedia article are catastrophically particular, 1-dimensional, old and non-geometric and hide the story more than reveal. Stub Maslov index containing the main references for Maslov index.
I added more references, including to Szabo's inspirative book on equivariant localization for path integrals, supersymmetry and Witten index at semiclassical approximation and more references and link to Ranicki's Maslov index page at Maslov index.
I have expanded and have tried to polish the Idea-section at semiclassical approximation.
Also, I am splitting off the previous redirect WKB method as a separate entry. But nothing much there yet.
The entry is redone quite in couple of major points to the contrary to my knowledge on the subject (and this is one of the topics I studied most of my student years) and I think a couple of main sentences have to be corrected to the previous content. Splitting what is essentially a synonym WKB as a separate entry is adding to the confusion.
First of all, semiclassical approximation (outside physics also known as stationary phase approximation) is to look at asymptotic expansions (of fast oscillating integrals) to any finite order in Planck constant (more generally, small parameter), by no means only to the first order. Semiclassical expansion is to any finite order. The WKB is more or less the synonym, though more often used in 1-dimensional quantum mechanics (the difference in conventions being much smaller than between the BRST method (without antifields) and BV method (with antifields) which have a single entry in Lab). All the major figures in the field, far the greatest being Maslov and Hormander, take this point of view. The basic equation – for eikonal has been generalized by those to multidimensional case. Eikonal of 1d WKB becomes the Maslov canonical operator of the multidimensional case. The theory of pseudodifferential operators (PDO) and more general Fourier integral operators came out from the semiclassical approximation for Fourier integrals. It is not about quantum mechanics only, it is used in optics (the geometric optics approximation), in PDO theory etc.
It is different from the formal deformation quantization which studies formal power series without analytic questions; semiclassical approximation is, on the contrary, the study of asymptotic series, with asymptotic convergence, Maslov index, Stokes phenomenon, Stokes rays etc. The semiclassical approximation is in the heart of harmonic analysis and is sharply different from the deformation quantization where all such asymptotic questions are neglected.
I see a call to nonexisting entry rotating phase approximation. Much more often it is called stationary phase approximation and it is a synonym to semiclassical approximation, in more abstract context of mathematics. The Feynman integral case is infinite-dimensional case; while the ordinary WKB in -dim QM employs the finite-dimensional integrals.
Not sure what the disagreement is. I did add explicitly the statement that semiclassical is first order or any finite order. This wasn’t in your original text even.
About splitting off WKB: sure, if you don’t like it, we don’t have to. But I think there are apsects of semiclassical approximation which cannot reasonably be attributed to WKB.
Zoran,
really, I think what I did was just to take your text and try to polish it a bit, as I found it a bit rough going.
Thanks. The entry says
The semiclassical approximation or quasiclassical approximation to quantum mechanics is the restriction of this deformation to just first order in ℏ.
This did not seem to include the finite order, that is why I complained. Plus, I do not consider asymptotic expansions “restrictions” of formal, because one needs to discuss the remainder. Simply, some formal expansions do not qualify as asymptotical, so they can not restrict.
But I think there are apsects of semiclassical approximation which cannot reasonably be attributed to WKB.
If you restrict the subject to tetxbooks for physics students. But it is not essential, that is what I mean, just a slight inclination in terminology, rather than fundamentally different terms for Lab. Anyway, I need to leave now for next few days, due emergency.
Okay, I have added another “or any finite order”, where it was missing previously.
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