In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as $(M, \omega)$ where $\omega$ is a sympathetic form on $M$ corresponding to the Poisson bracket. Given a Hamiltonian $H$, its flow is the trajectory of our particle with initial conditions. Now, in local coordinates, we can write $$ \vec{z} = P_t \vec{z}(0) \quad (1) $$ where $P_t$ is the flow evaluated at $\vec{z}(0)$, which is the propagator. Note that there is a differential equation that (1) also satisfies. In particular, just differentiate both side by $t$, then we get an ODE which we can solve the flow $P_t$ if we only know the Hamiltonian vector field. Now, in quantum mechanics, I learned from class that we obtain an similar equation that looks like $$ | \psi \rangle = U_t | \psi_0 \rangle \quad (2) $$ where $U_t$ is unitary and satisfies $U_{t+s} = U_tU_s$ for all $s,t$ defined. And (2) also satisfies a differential equation similar to that of (1). So my understanding is that every thing we did in my quantum class is just a local theory. My intuition tells me that globally, one which to have some configuration space $N$ that is in some sense "diffeomorphic" to the projective space of $l^2(\mathbb{N})$ locally. And the manifold we want to work with is $M:=TN$ whatever $TN$ means in this case. Then our states are just point on $M$, with Hamiltonian defined over $M$ to $\mathbb{R}$. And $U_t$ would be a Hamiltonian flow. \

However, anything that goes infinite dimensional can go crazy. So I am looking for a global picture of quantum mechanics. In particular, a definition for

- What does smoothness mean (or even differentability) for $f: l^2(\mathbb{N}) \rightarrow \mathbb{R}$ or $l^2(\mathbb{N})$.
- definition of locally $l^2(\mathbb{N})$ manifold
- explaination/theorem on the global picture of quantum mechanics.
- Or perhaps none of that makes any sense whatsoever, then what should I be looking into?

Thanks tonnes!

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