Comments on: Some notes on Weyl quantisation

By: Lars Hormander « What’s new

Lars Hormander « What’s new — Fri, 30 Nov 2012 19:53:10 +0000

[...] to multiplying a phase space distribution by the symbol of that operator, as discussed in this previous blog post. Note that such operators only change the amplitude of the phase space distribution, but not the [...]

By: Marcelo de Almeida

Marcelo de Almeida — Mon, 05 Nov 2012 02:08:16 +0000

Reblogged this on Being simple.

By: ��Ԫ

��Ԫ — Sat, 20 Oct 2012 11:35:53 +0000

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By: Sam Sachdev

Sam Sachdev — Thu, 11 Oct 2012 01:34:57 +0000

Totally fascinating. Thanks for taking time to make notes on Weyl quantisation.

By: degosson

degosson — Tue, 09 Oct 2012 11:22:20 +0000

Perhaps I missed something, but it seems that you don’t mention a characteristic property of Weyl pseudodifferential operators: they are the only members of the Shubin class that have the property of symplectic/metaplectic covariance (Stein, Wong) . Also, the fact that Weyl operators is the “perfect” quantization procedure in QM is debatable, as has been pointed out by many physicists. In view of Schwartz’s kernel theorem “everything” which is continuous S–>S’ can be written as a Weyl operator, hence any quantum observable could be “dequantized”, but this is physically not true: there are quantum observables which have no classical analogue. A better (i.e. more physical) quantization procedure might very well be the Born-Jordan scheme, which is not always “dequantizable” (see my latest paper “Symplectic covariance properties for Shubin and Born–Jordan pseudo-differential operators” in the Trans. Amer. Math. Soc, online since last Saturday: http://www.ams.org/journals/tran/0000-000-00/).

By: Steven Stadnicki

Steven Stadnicki — Tue, 09 Oct 2012 00:52:21 +0000

Minor typo: your definition of the position operator is missing the factor of ; presumably it’s meant to be .

[Corrected, thanks - T.]

By: Terence Tao

Terence Tao — Mon, 08 Oct 2012 19:46:45 +0000

Yes, this would work, although for the purposes of Section 1, one could also proceed without the SL_2 invariance just by writing , and working with A’ and B’ throughout; this does not significantly affect the length of the presentation in this simple case, but is helpful when working with more than two variables as alluded to at the end of Section 1. To me it is a matter of taste – if one wants to prove some identity about finite-dimensional symplectic vector spaces, for instance, does one work in a coordinate free fashion, or does one apply the linear Darboux theorem first to place the symplectic form in a normal form? Both approaches are useful. (Of course, strictly speaking I did not work in a coordinate free fashion in my notes, since I did write everything in terms of A and B, but as mentioned above it is not too difficult to phrase things in the coordinate free formalism.)

By: Anonymous

Anonymous — Mon, 08 Oct 2012 17:47:25 +0000

missing expository tag?

By: David Speyer

David Speyer — Mon, 08 Oct 2012 13:15:48 +0000

I haven’t finished reading to the end yet, but I think the first section would be much easier to read if you started with the observation that Weyl quantization has a symmetry. Namely, let be the two dimensional vector space generated by and ; let be the tensor algebra and let be the ring generated by the operators and . If is in , and and are operators from , then . So acts on , and the map commutes with the action.

The Weyl quantization is to lift an element of to the unique preimage in which lies in . This choice of lifts manifestly respects the symmetry.

Therefore, in section 1, there is no need to carry around the and terms. Just use the symmetry to reduce to the case where one operator is and the other is . Moreover, if you want to make life even simpler, you can work with just and and restore the powers of at the end by homogeneity.

By: Fred Lunnon

Fred Lunnon — Mon, 08 Oct 2012 11:53:58 +0000

Section 1 penultimate sentence: F^*(z) definition contradictory?

[Sorry, I don't see what the problem is here. Note that the use of the asterisk in does not quite denote adjoint or complex conjugation, although it is of course related to those two concepts. -T.]