Deformation quantization: pro and contra

Pseudo-differential operators and deformation quantization

“…Usually it is based on a partition of unity subordinate to a coordinate covering of X and at the rst glance has nothing to do with the canonical construction of deformation quantization for general symplectic manifolds 3,4 ] . Many times I was asked by my colleagues about the meaning of the canonical deformation quantization of 3,4] for the case of do but was not able to give a satisfactory answer. So, I decided to study this question more carefully.…”

Index Theory and Adiabatic Limit in QFT

“…Of course these are heuristic remarks only, motivated on the geometric interpretation of Haag, suggesting that in general realistic situation there should exist a commutative subalgebra A cl in the algebra of detectors A whose spectrum elements are parameters with immediate physical interpretation. 16 In order to check the consequences of the above postulate (suitably supplemented) one have to introduce (natural) analytic structures allowing concrete computations. We shell describe only some first steps towards this goal, based on the (rigorous) micro-local perturbative approach of Brunetti and Fredenhagen and formulate its connection to local algebraic approach of Haag in terms of formal index theorem of Fedosov and asymptotic representations (generalizing the asymptotic representations of Fedosov).…”

“…But first we remind that the analog 17 (A, D, H) of the Connes' spectral triple for pseudo-riemannian manifold, as proposed by Strohmaier [11], is given by a pre-C * -algebra A with involution * acting as an algebra of bounded operators not in the ordinary Hilbert space but in a Krein space 18 [14] H. The involution is represented by taking the Krein adjoint, the Dirac operator D is selfadjoint in the Krein sense. Important role is played by the so called fundamental symmetries of the Krein space H. These are operators: J : H → H, such that: J 2 = 1 and (·, J·) = (J·, ·), where (·, ·) is the indefinite inner product in the Krein space H. With the help of J, one can 16 In general we cannot, however, expect that the spectrum of A cl will be sufficient to designate all points of A, for example the representation of A cl induced by an irreducible representation of A is not irreducible in general if A cl does not lie in the center of A. Algebraically speaking: possibly many different localizations are needed to reconstruct the algebra A and its relevant spectrum, giving different types of coincidence arrangements of detectors. Most of all we should be interested in the coincidence arrangements of detectors encountered in particle physics, of course.…”

“…Now we pass to the existence problem for the adiabatic limit, which in the formulation of Dütsch and Fredenhagen is equivalent to the following question: under what (accessible) conditions the formal seres are convergent, and thus when the formal representation of the deformed algebra turns into an actual representation of an actual (C * -)algebra in an ordinarily Hilbert space? But on the other hand Fedosov [16] proved an interesting theorem in the theory of deformations of symplectic (or even Poisson) manifolds. Namely he showed that the deformed algebra admits a so called asymptotic operator representation in ordinary Hilbert space iff his (Fedosov') formal index fulfills some integrality conditions.…”

On the Dequantization of Fedosov's Deformation Quantization

“…. , where B r , r ≥ 1, are differential operators on M. The existence and classification problem for deformation quantization was first solved in the non-degenerate (symplectic) case (see [7], [21], [11] for existence proofs and [12], [19], [8], [2], [23] for classification) and then Kontsevich [18] showed that every Poisson manifold admits a deformation quantization and that the equivalence classes of deformation quantizations can be parameterized by the formal deformations of the Poisson structure. It turns out that all the explicit constructions of star-products enjoy the following property: for all r ≥ 0 the bidifferential operator C r in (1) is of order not greater than r in both arguments (the most important examples are Fedosov star-products on symplectic manifolds and the Kontsevich star-product on R n endowed with an arbitrary Poisson bracket).…”