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.

**Read Online or Download Lectures on the geometry of quantization PDF**

**Similar geometry and topology books**

**Differential Geometry. Proc. conf. Lyngby, 1985**

The Nordic summer time institution 1985 offered to younger researchers the mathematical features of the continuing study stemming from the examine of box theories in physics and the differential geometry of fibre bundles in arithmetic. the amount contains papers, usually with unique traces of assault, on twistor equipment for harmonic maps, the differential geometric elements of Yang-Mills conception, complicated differential geometry, metric differential geometry and partial differential equations in differential geometry.

**Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 1986–87**

This can be the 3rd released quantity of the court cases of the Israel Seminar on Geometric elements of useful research. the massive majority of the papers during this quantity are unique study papers. there has been final 12 months a powerful emphasis on classical finite-dimensional convexity idea and its reference to Banach house thought.

**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.

- Comprehensive Introduction to Differential Geometry: Sold Only As Individual Volumes See Isbns 0914098845/0914098853 (Volumes 1 and 2)
- Introduction to the Physical Principles of Differential Geometry
- Elementary topology.A first course
- Combinatory Topology of Convex Regions
- Algebraic Geometry: A New Treatise On Analytical Conic Sections

**Additional info for Lectures on the geometry of quantization**

**Example text**

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 .