Download Cohomology, geometric quantization and quantum information.

Cohomology, geometric quantization and quantum
information.
Thesis subject of Juan Pablo Vigneaux
Thesis advisor: Daniel Bennequin ([email protected])
Laboratory: Institut de Mathématiques de Jussieu - Paris Rive Gauche.
Key words: geometric quantization, Poisson geometry, Lie-Rinehart cohomology, monad, operad, information topology.
Description
Methods of homological algebra are more and more used in the literature of
mathematical physics, specially for quantum physics. For instance, classical gauge theory and its quantization was clarified by BRST (Becchi-RouetStora-Tyutin) co-homology, then further clarified by BV (Batalin-Vilkoviski)
theory, and both theories are rooted in homotopical algebra (cf. Stasheff [1]).
Quantization and unitary representation of Lie groups also rely on homological constructions, and the interplay between these two notions is a rich source
of problems (cf. Vergne [2], Knapp-Vogan [3]); for example, those that turned
around the solution of Guillemin-Sternberg conjecture [2], that “symplectic
reduction commutes with quantization”. Following a suggestion of Gabriel
Catren [4], Daniel Bennequin and Gabriel Catren (to be published) gave a
construction of the quantum Hilbert space associated to a maximal commutative algebra of observables on an integral symplectic manifold, as being the
BRST cohomology of a natural involutive algebra in a Poisson algebra. For
that they exploited well known constructions in symplectic geometry and geometric quantification, as invented by Souriau, Kirillov and Kostant, and more
recent constructions used for the solution of Guillemin-Sternberg conjecture
(Meineken, Sjamaar, Tian, Zhang, Paradan). Moreover, this definition of
quantum states can be extended to more general situations, in particular replacing the action of an abelian algebra by any Poisson map (generalized
momentum) between two Poisson manifolds, and using Lie-Rinehart cohomolgy (cf. Kjeseth [5]).
The aim of this thesis will be to develop this co-homological approach in
a wide sense, to get a better geometrical understanding of the overall relation
between the mathematical models of classical and quantum observations, degree of freedoms of states, observable algebras, and specially functoriality. A
first problem is to define the co-homology which is well adapted in this generalized situation of Poisson maps, using tools as monads (or triple) as in Beck
[6] or operads (Loday-Vallette), and study the operations in this co-homology,
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giving the expected natural structures, as hermitian products. The known
categories of generalized quantum canonical transformations (Hörmander,
Sato-Kawai-Kashiwara, Weinstein) have already shown the kind of objects
encountered in this context. Hopefully the new study conducted in the thesis
would permit to obtain new results on open problems; in particular, computing the case of a pair of Lie groups H ⊂ G by this real method, could
give results on explicit basis of representations that are not accessible by a
complex analytic method. We also hope to include a presentation of the
infinite dimensional setting necessary to study examples in Quantum Field
Theory, at least in a formal way.
On another side, several recent works have applied co-homological constructions in Information Theory in a wide sense, for instance Gromov [7, 8],
Baudot and Bennequin [9]. The basic structures in these approaches are families of decompositions or disjoint subsets and families of measures of widths
or sizes for the pieces, and their outputs are entropies or generalization of
them, which can be seen as generalized notions of degrees of freedom for
a system of states. Also in this case, classical and quantum settings can
be characterised. These studies describe structures of observation and localization of observations. One aspect of this thesis will be to connect this
point of view with the quantization problem; in particular, to relate the cohomology theories which appears in both contexts. A special attention will
be put on the decompositions associated to families of disjoint Lagrangian
sub-manifolds in symplectic manifolds. Also, concrete examples of information structures and their topology will be investigated. Explicit computations
of information co-homology groups should be performed.
For instance, quantum information can be constrained by any compact
sub-group of the unitary group, this subject deserves to be further studied,
and relations with symplectic reduction and geometric quantization deserve
to be explored.
The following question will be examined: how to reconstruct the classical
symplectic data from the quantum states? The basic example is the unitary
n + 1 dimensional irreducible representation of SU2 having spin equal to n/2;
in this case there is a normal rational SU2 -invariant curve C in the projective
space of dimension n, where the momentum of the diagonal action gives the
Lagrangian decomposition of C in n + 1 pieces. The goal is to generalize this
example.
Our point of view is that the classical data can be recovered from the
quantum data. This is true for the physical entropy, as defined by Gibbs,
due to the definition of quantum entropy of Von Neumann. We expect an
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extension of this result, in particular for co-adjoint orbits of Lie groups, or
symplectic manifolds playing important role in field theory, like moduli spaces
of flat connections over Riemann surfaces.
References
[1] Jim Stasheff. Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraic topics related to Alan’s interests. The Breadth of Symplectic and Poisson Geometry - Progress in Mathematics Volume 232,
2005, pp 583-601.
[2] Michèle Vergne. Quantification géométrique et réduction symplectique.
Séminaire BOURBAKI, 53ème année, 2000-2001, n. 888.
[3] Anthony Knapp & David Vogan Jr. Cohomological Induction and Unitary
Representations (PMS-45). Princeton University Press, 1995.
[4] Gabriel Catren. On the Relation Between Gauge and Phase Symmetries.
Foundations of Physics, December 2014, Volume 44, Issue 12, pp 13171335.
[5] Lars Kjeseth. A homotopy Lie-Rinehart resolution and classical BRST
cohomology. Homology Homotopy Appl. Volume 3, Number 1 (2001),
165-192.
[6] Jonathan Beck. Triples, algebras and cohomology. Reprints in Theory and
Applications of Categories, No. 2, 2003, pp. 1–59.
[7] Misha Gromov. Morse Spectra, Homology Measures, Spaces of Cycles
and Parametric Packing Problems. Last update April 16, 2015. In
www.ihes.fr/ gromov/PDF/Morse-Spectra-April16-2015-.pdf.
[8] Misha Gromov. In a Search for a Structure, Part 1: On Entropy. Last
update June, 2013. In www.ihes.fr/ gromov/PDF/structre-serch-entropyjuly5-2012.pdf.
[9] Pierre Baudot & Daniel Bennequin. Topological forms of information. Max-Planck-Institut für Mathematik in den Naturwissenschaften,
Leipzig, Preprint no.: 112, 2014. and an extended and amended version
to appear in the journal Entropy.
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