Lectures on the geometry of quantization by Sean Bates, Alan Weinstein

By Sean Bates, Alan Weinstein

Those notes are in line with a direction 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 one prerequisite for the path wanted is a data of the fundamental notions from the idea of differentiable manifolds (differential types, vector fields, transversality, etc.). the purpose is to offer scholars an advent 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 math of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). those notes are supposed to functionality as a consultant to the literature. The authors confer with different assets for plenty of info which are passed over and will be bypassed on a primary examining.

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Lectures on the geometry of quantization

Those notes are in response to a path entitled "Symplectic Geometry and Geometric Quantization" taught by means of Alan Weinstein on the collage 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 elemental notions from the speculation of differentiable manifolds (differential kinds, vector fields, transversality, and so on.

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A half-density a on L determines functions A(q) and B(p) such that (πL−1 )∗ a = A |dq|1/2 (πp−1 )∗ a = B |dp|1/2 and A(q) = |S (q)|1/2 B(p(q)). For each q, we must now compare the Maslov half-density J (L, ι, τ, a) = (2π )−1/2 ei(pq+T (p))/ B(p) dp |dq|1/2 R with that obtained by pull-back: I (L, ι, φ, a) = eiS/ A |dq|1/2 . To this end, we set k(q) = T (p(q)) and apply the principle of stationary phase (see Appendix B). e. where p = S (q) = p(q). Hence, ei(pq+T (p))/ B(p) dp = (2π )1/2 e−iπ·sgn(k)/4 eiS(q)/ |k(q)|−1/2 B(p(q)) + O( 3/2 ).

19 If L is a lagrangian submanifold of a symplectic manifold P , then the restricted tangent bundle TL P is a symplectic vector bundle over L, and T L ⊂ TL P is a lagrangian subbundle. 6). In general, the automorphism group of a symplectic vector bundle E does not act transitively on the lagrangian subbundles of E. Nevertheless, a pair of transverse lagrangian subbundles can be related as follows. 20 Let E → B be a symplectic vector bundle and suppose that L, L are lagrangian subbundles such that Lx is transverse to Lx for each x ∈ B.

For simplicity, we will assume that the additive constants in φ and τ are chosen so that φ = τ + ι∗ (qp). 1] . 42 where p(q) = S (q). From this relation, it follows easily that T (p(q)) = −(S (q))−1 . A half-density a on L determines functions A(q) and B(p) such that (πL−1 )∗ a = A |dq|1/2 (πp−1 )∗ a = B |dp|1/2 and A(q) = |S (q)|1/2 B(p(q)). For each q, we must now compare the Maslov half-density J (L, ι, τ, a) = (2π )−1/2 ei(pq+T (p))/ B(p) dp |dq|1/2 R with that obtained by pull-back: I (L, ι, φ, a) = eiS/ A |dq|1/2 .

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