at quantum observable there used to be just the definition of geometric prequantum observables. I have added a tad more.
]]>added to Poisson manifold a subsection
which states the form of the Poisson bracket on presymplectic manifolds and then discusses how this is isomorphic as a Lie algebra to the infinitesimal quantomorphisms, the infinitesimal symmetries of any prequantization of the presymplectic manifold.
I have written that as one subsection for the exposition at geometry of physics – prequantum geometry, but I suppose it serves well to have it right there in the entry on Poisson brackets itsef, too.
]]>I have tried to expand the Idea-section at orbit method a little.
]]>I have started something at Bohr-Sommerfeld leaf, but need to continue later when I have more time and energy
]]>gave Lagrangian cobordism an Idea-section added references related to the Fukaya category and cross-linked with relevant entries.
]]>brief note on deformation quantization of the 2-sphere
(to go along with the existing geometric quantization of the 2-sphere)
]]>We are in the process of finalizing an article on prequantum theory in higher geometry. An early version of our writeup I have now uploaded. It needs a few more cycles of polishing, but I thought I’ll provide this here on the nForum right now as a kind of explanation for the sheer drop in the amount of noise that I have been making around the nLab as of lately ;-:
Domenico Fiorenza, Chris Rogers, Urs Schreiber,
Regulars here will recognize various things that I/we have been talking about for a good while now. Finally they are materializing in a more pdf-kind of incarnation…
]]>Added to Maslov index and to Lagrangian Grassmannian the following quick cohomological definition of the Maslov index:
The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers
The generator of this cohomology group is called the universal Maslov index
Given a Lagrangian submanifold of a symplectic manifold , its tangent bundle is classified by a function
The _Maslov index of is the universal Maslov index pulled back along this map
]]>started a minimum at hard Lefschetz theorem and Lefschetz decomposition
]]>started Guillemin-Sternberg geometric quantization conjecture
So far just the brief Idea and a few commented references.
]]>I want to collect, for expositional purpose, in one place all the ingredients that go into the story of geometric quantization of the 2-sphere, a simple and archetypical example of geometric quantization.
So far I have included everything (I think) pertaining to the prequantum line bundles, the polarization and the spaces of quantum states. Next I’ll add discussion of the angular momentum quantum operators.
]]>added to Noether theorem a brief paragraph on the symplectic/Hamiltonian Noether theorem
]]>I am starting Wick algebra. So far I have an Idea-section, references, and a discussion of the finite dimensional case, showing how the traditional “normal ordered Wick product” is the Moyal star product of an almost-Kähler vector space.
]]>microformal morphism a la Theodore Voronov.
]]>It is a classical fact that a formal deformation quantization of a Lie-Poisson structure is provided by the universal enveloping algebra of the corresponding Lie algebra. Remarkably, this statement generalizes to some extent to more general (polynomial) Poisson algebras. In particular it holds for every such up to degree three in ! This is due to Penkava-Vanhaecke 00.
I have added a quick summary of this theorem to deformation quantization in a new subsection: Existence – Deformation by universal enveloping algebras. I also gave this an entry on its own at polynomial Poisson algebra.
This is maybe remarkable, since there is possibly no physical measurement known which could detect contributions of higher than third order in . Though I’d need to check. This is subtle because order in is different from the usual loop order that is commonly stated (which is order in the coupling constant) and the relation between the two is complicated.)
Also (and that’s how I came across this article) at least in special cases this gives a way to quantize just by universal constructions on Lie algebras, hence this might potentially tell us something about the quantization of Poisson bracket Lie n-algebras (for which no analog of the corresponding Poisson algebra, i.e. with an associative product around, is known).
]]>I gave Fedosov deformation quantization its own entry, so far with an Idea-section putting the construction in perspective, an informal outline of how the method proceeds, and some references.
]]>I have created a new entry
meant as a disambiguation page for the various different kinds of definitions that exists. Presently it points to the entries
that already provide dedicated discussion of special defintiions. In addtion it lists references that have further proposals for defintion which don’t at the moment however have dedicated Lab pages associated with them.
]]>given the concept of Heisenberg Lie n-algebra, there is an evident definition of Weyl n-algebra: its universal enveloping E-n algebra.
I noted that down for reference at Weyl n-algebra. In the process I noticed that Markarian proposed a different definition just a few months back
]]>I have been working on filling genuine content into
The first part Infinitesimal symmetries should be about readable, it starts out plenty expositionary, I hope, but towards the end it is still very terse. I wanted to get much further today, but it didn’t work out that way.
]]>created some bare minimum at symplectic spinors and metaplectic quantization
]]>created a brief entry on quadratic Hamiltonians, just for completeness
]]>am starting an entry on extended affine symplectic group, the restriction of the quantomorphism group of a symplectic vector space to elements that cover elements of the affine symplectic group.
(The name is not great, as the “extended” is too unspecific, but this seems to be close to standard in the literature – or else “extended inhomogeneous symplectic group”, which is not any better.)
]]>