Symmetrization of the Berezin Star Product and Path-Integral Quantization

Abstract

We propose a new star pruduct which interpolates the Berezin and Moyal quantization. A multiple of this product is shown to reduce to a path-integral quantization in the continuous time limit. In flat space the action becomes the one of free bosonic strings. Relation to Kontsevich prescription is also discussed.

Normalized Weyl-type *-product on Kähler manifolds

“…In conclusion we have found that the Weyl-type product defined in [8] can be decomposed into two kinds of Berezin-type products and introduce a normalization factor into a Weyl-type product since a normalization factor is necessary for each kind of Berezin-type product. Note, in general, we must use f 1 * f 2 = f 1 ⊙ f 2 unlessêh = e −1 h .…”

“…A Poisson bracket appears in the first order ofh, which means the product f 1 ⊙ f 2 is really a Weyl-type product and which was not shown in [8].…”

“…[8] is insufficient in the sense that there is no consideration to the general Kähler manifolds for which A = 1 2 n i,j=1 h ij ∂ i ∂ j log det H is not 0, where the matrix H is a metric.…”

“…We generalize the product defined in [8] so that it is applicable to general Kähler manifolds. Moreover we expand the product perturbatively and show that it really has a property characteristic of a Weyl-type product and that the normalization factor necesarry for a Berezin-type product introduced by Reshetikhin and Takhtajan is unnecessary for a Weyl-type product at least in the first order ofh.…”