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By Dirk J. Struik

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Thus, energy levels of the 1-dimensional harmonic oscillator corresponding to level sets for which a particular value of is admissible are given by E = n . As one can read in any textbook on quantum mechanics, the actual quantum energy levels are E = (n + 1/2) . The additional 1/2 can be explained geometrically in terms of the non-projectability of the classical energy level curves, as we shall soon see. 39 Prequantum bundles and contact manifolds Prequantizability can be described geometrically in terms of principal T bundles with connection over T ∗ M .

An advantage of this generalization will be to allow a calculus which is more clearly invariant under changes of coordinates, unlike the previous Fourier transform picture, which requires linear structures on p- and q-space. 7 Let M, B be smooth manifolds, and ι let pM : B → M be a smooth submersion. Dualizing the inclusion E = ker(pM ∗ ) → T B gives rise to an exact sequence of vector bundles over B ι∗ 0 ← E ∗ ← T ∗B ← E ⊥ ← 0 where E ⊥ ⊂ T ∗ B denotes the annihilator of E. The fiber-derivative of a function φ : B → R is the composition dθ φ = ι∗ ◦ dφ, and its fiber critical set is defined as Σφ = (dθ φ)−1 ZE ∗ .

We then turn to the semi-classical approximation and its geometric counterpart in this new context, setting the stage for the quantization problem in the next chapter. 2). 32 The classical picture The hamiltonian description of classical motions in a configuration space M begins with the classical phase space T ∗ M . A riemannian metric g = (gij ) on M induces an inner product on the fibers of the cotangent bundle T ∗ M , and a “kinetic energy” function which in local coordinates (q, p) is given by kM (q, p) = 1 2 g ij (q) pi pj , i,j where g ij is the inverse matrix to gij .