Is the time derivative of the WKB phase globally defined?

+ 4 like - 0 dislike
57 views

Let $Q$ and $\mathcal{L}$ be smooth n-dimensional manifolds and $\iota_t:\mathcal{L}\rightarrow T^*Q$ a time-dependent Lagrangian embedding that is smooth in $t$ and satisfies the Bohr-Sommerfeld quantization condition for a fixed value of $\hbar$. Thus, if $\vartheta$ is the canonical 1-form on $T^*Q$ and $m_{\iota_t}$ is the Maslov class of the embedding $\iota_t$,the real cohomology class $$[\iota_t^*\vartheta]-\frac{\pi\hbar}{2}m_{\iota_t}$$ takes values in $\hbar\mathbb{Z}$.

Fix a $t\in\mathbb{R}$. Because $\iota_t^*\vartheta$ is closed, we can choose an open cover $\{\Lambda_\alpha\}$ for $\mathcal{L}$ such that $\iota_t^*\vartheta=d S_t^\alpha$ on $\Lambda_\alpha$. Because the Bohr-Sommerfeld condition holds, the $S^\alpha_t$ can be chosen such that on $\Lambda_{\alpha\beta}=\Lambda_\alpha\cap\Lambda_\beta$, $$S^\alpha_t-S^\beta_t=\frac{\pi\hbar}{2}m_{\alpha\beta} ~\text{mod}~2\pi\hbar,$$ where the $m_{\alpha\beta}$ are the transition functions for the time-$t$ Maslov principal $\mathbb{Z}$-bundle over $\mathcal{L}$.

Now my question. I believe that, at least in a small open neighborhood of $t$, $m_{\alpha\beta}$ can be regarded as smooth integer-valued functions of time and that the $S^\alpha_t$ can be chosen to be smooth in $t$. I am therefore lead to the conclusion that $\frac{d}{dt}S^\alpha_t$ is a globally defined function on $\mathcal{L}$. Are my beliefs incorrect? Is it true that $\dot{S}_t=\frac{d}{dt} S_t$ is a well-defined function on $\mathcal{L}$?

I have a partial answer already. It turns out that $\dot{S}_t$ is globally defined in the special case where $\iota_t=\phi_t\circ\iota_0$, with $\phi_t$ the flow map of a globally Hamiltonian vector field on $T^*Q$; there is an explicit expression for $\dot{S}_t$ in terms of $\iota_t$ and the Hamiltonian function associated with $\phi_t$.

This post imported from StackExchange MathOverflow at 2015-05-27 22:07 (UTC), posted by SE-user Josh Burby
asked Jan 17, 2014
retagged May 27, 2015

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:$\varnothing\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.