By Sean Bates, Alan Weinstein

Those notes are in keeping with a path entitled "Symplectic Geometry and Geometric Quantization" taught through Alan Weinstein on the college of California, Berkeley (fall 1992) and on the Centre Emile Borel (spring 1994). the single prerequisite for the direction wanted is a data of the fundamental notions from the idea of differentiable manifolds (differential varieties, vector fields, transversality, etc.). the purpose is to provide scholars an creation to the tips of microlocal research and the similar symplectic geometry, with an emphasis at the function those rules play in formalizing the transition among the maths of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). those notes are supposed to functionality as a advisor to the literature. The authors consult with different resources for plenty of info which are passed over and will be bypassed on a primary interpreting.

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39 Prequantum bundles and contact manifolds Prequantizability can be described geometrically in terms of principal T bundles with connection over T ∗ M . It is customary to make the following definition. 6 For fixed ∈ R+ , the prequantum T bundle associated to a cotangent bundle (T ∗ M, ωM ) consists of the trivial principal bundle QM, = T ∗ M × T together with the connection 1-form ϕ = −π ∗ αM + dσ. Here, σ denotes the multiple-valued linear variable in T and π : QM, → T ∗ M is the bundle projection.

In general, the restriction of the projection U × Rm → U to Σϕ is non-injective, and thus the image of Σϕ is a non-projectable lagrangian submanifold. From the point of view of the WKB method, this generalization amounts to replacing Maslov’s ansatz (2π )−n/2 ei( p,q +T (p))/ a(p) dp |dq|1/2 Rn for the solution of Schr¨odinger’s equation by the more general form (2π )−m/2 eiφ(q,θ)/ a(q, θ)|dθ||dq|1/2 , Rm 46 where θ is an auxiliary variable in Rm which may have nothing to do with the variable p dual to q.

Then there exists a compatible complex structure J on E satisfying JL = L . Proof. Let J0 be any compatible complex structure on E. Since L and J0 L are both transverse to L, we can find a symplectomorphism T : L ⊕ L → L ⊕ J0 L which preserves the subbundle L and maps L to J0 L. A simple check of the definition then shows that J = T −1 J0 T is a compatible complex structure on E which satisfies JL = L . 21 If E is a symplectic vector bundle over M , then any pair L, L of lagrangian subbundles of E define a cohomology class µ(L, L ) ∈ H 1 (M ; Z) as follows.