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Alan Weinstein studies the mathematical interface between classical mechanics and quantum mechanics, particularly symplectic geometry. His Berkeley research profile identifies this connection.^[ref]

With Sean Bates, he wrote Lectures on the Geometry of Quantization by Sean Bates and Alan Weinstein. Their treatment of half-densities explains how to integrate products of wave-like objects intrinsically: the product of two half-densities is a density, giving a coordinate-independent inner product and hence a Hilbert space. This is part of the geometric infrastructure needed to relate classical phase spaces to quantum states.^[ref]

On a smooth manifold, a half-density has local expression $\sigma =\psi (q)|d{q}_1\cdots d{q}_n{|}^{1/2}$. Under a change from coordinates $q$ to $Q$,Consequently $\overline{\sigma }\sigma$ is a nonnegative density on a manifold. It can be integrated without choosing an orientation or a volume form, but the integral need not be finite. Square-integrable half-densities are precisely those satisfyingTheir inner product is $⟨\sigma ,\tau ⟩={\int }_M\overline{\sigma }\tau$. Completing the smooth compactly supported half-densities gives an intrinsic Hilbert space; the Jacobian transformation makes its norm independent of coordinates.^[ref]

For instance, $|dq{|}^{1/2}$ on the real line is a smooth half-density with infinite squared norm, whereas $e^{-q^2}|dq{|}^{1/2}$ has finite squared norm. In geometric quantization, half-densities let configuration-space states carry a coordinate-independent inner product.

Sean Bates coauthored Lectures on the Geometry of Quantization by Sean Bates and Alan Weinstein with Alan Weinstein. The manuscript's title page verifies the byline; its preface traces the notes to Weinstein's courses on symplectic geometry and geometric quantization.^[ref]

The lectures connect Hamiltonian mechanics on a symplectic manifold with unitary evolution on a Hilbert space. Their account of half-densities supplies an intrinsic way to define an inner product, avoiding dependence on a chosen coordinate volume element. The contribution is an exposition of the geometry linking the two mathematical descriptions of motion.^[ref]

A symplectic form is a closed, nondegenerate differential two-form $\omega$ on a smooth manifold. Nondegeneracy forces even dimension and converts the differential of a function into a Hamiltonian vector field. For example, with $\omega ={\sum }_{j}d{q}_j\wedge d{p}_j$ and the convention ${\iota }_{X_H}\omega =dH$, the Hamiltonian flow satisfiesThus the geometry of phase space encodes the equations of classical mechanics. The cotangent bundle supplies this structure intrinsically, without a choice of coordinates.^[ref]

The Symplectic Darboux theorem makes every such form locally standard; there is no local curvature invariant analogous to Riemann curvature tensor. Global geometry still imposes strong constraints. A symplectomorphism preserves symplectic volume, while the Gromov non-squeezing theorem obstructs squeezing a ball through a narrow conjugate coordinate plane even when the target has infinite volume. Lagrangian submanifolds, symplectic reduction and J-holomorphic curves describe different ways this global rigidity interacts with dynamics and topology.