Quantization and Coherent States over Lagrangian Submanifolds

A membrane technique, in which the symplectic and Ricci forms are integrated over surfaces in a complexification of the phase space, as well a “creation” connection with zero curvature over lagrangian submanifolds, is used to obtain a unified quantization including a noncommutative algebra of functions, its representations, the Dirac axioms, coherent integral transformations for solutions of spectral and Cauchy problems, and trace formulas. The results obtained in [1] for the phase space R and Gaussian coherent states (see also [2–5]) are developed in this work for curved phase spaces and general coherent states. §1. Complexification. Let X be a manifold endowed with a complex structure J . By (x, y) we denote points from the product X = X × X, and we equip this space with the complex structure J ⊖ J and with the groupoid multiplication (x, y)◦(y, w) = (x,w). The set of units (diagonal) in X is identified with X ≈ diag; the left and the right reductions X π ←− X π −→ X in this groupoid are given by the projections π(x, y) = x, π(y, x) = x. These mappings are, respectively, a morphism and an antimorphism of complex structures, and so X is a complex groupoid corresponding to the manifold (X, J), see [6, 7]. Let Π(x), Π(x) ⊂ X be fibers of π and π over x ∈ X. Following physicists, we denote (x, y) ≡ y|x. Note that one can identify the complexified tangent spaces TxX with the tangent spaces Tx|xX # by using the natural identifications TxX ≈ Tx|x diag and i · TxX ≈ (J ⊖ J)Tx|x diag. Then the eigenspaces of Jx corresponding to the eigenvalues +i and −i coincide with the tangent subspaces Tx|xΠ(x), Tx|xΠ(x) in Tx|xX . Thus, the groupoid X can be regarded as a complexification of the manifold (X, J); the fibers Π(x) can be interpreted as the integral leaves of the complex polarization Π on X corresponding to the complex structure J , and the fibers Π(x) can be interpreted as the leaves of the conjugate polarization Π. The groupoid inversion mapping x|y → y|x can be interpreted as an involution in the complexification. We use z = {z} to denote local holomorphic coordinates on X, and by ∂ we denote the holomorphic differential, ∂ = ∂/∂z. Thus, Π is generated by the vectors ∂/∂z. §2. Membrane amplitudes. Throughout the sequel X is assumed connected and simply connected. We call X a symplectic-Kähler manifold if it is equipped with a symplectic form ω compatible with the complex structure J , and is also endowed Typeset by AMS-TEX 1 2 MIKHAIL V. KARASEV with a Kählerian metric g. Let us choose a lagrangian submanifold Λ ⊂ X and suppose that