Let Q and L be smooth n-dimensional manifolds and ιt:L→T∗Q a time-dependent Lagrangian embedding that is smooth in t and satisfies the Bohr-Sommerfeld quantization condition for a fixed value of ℏ. Thus, if ϑ is the canonical 1-form on T∗Q and mιt is the Maslov class of the embedding ιt,the real cohomology class [ι∗tϑ]−πℏ2mιt takes values in ℏZ.
Fix a t∈R. Because ι∗tϑ is closed, we can choose an open cover {Λα} for L such that ι∗tϑ=dSαt on Λα. Because the Bohr-Sommerfeld condition holds, the Sαt can be chosen such that on Λαβ=Λα∩Λβ, Sαt−Sβt=πℏ2mαβ mod 2πℏ, where the mαβ are the transition functions for the time-t Maslov principal Z-bundle over L.
Now my question. I believe that, at least in a small open neighborhood of t, mαβ can be regarded as smooth integer-valued functions of time and that the Sαt can be chosen to be smooth in t. I am therefore lead to the conclusion that ddtSαt is a globally defined function on L. Are my beliefs incorrect? Is it true that ˙St=ddtSt is a well-defined function on L?
I have a partial answer already. It turns out that ˙St is globally defined in the special case where ιt=ϕt∘ι0, with ϕt the flow map of a globally Hamiltonian vector field on T∗Q; there is an explicit expression for ˙St in terms of ιt and the Hamiltonian function associated with ϕt.
This post imported from StackExchange MathOverflow at 2015-05-27 22:07 (UTC), posted by SE-user Josh Burby