Quantization and Cohomology (Week 27)
Posted by John Baez
In our final class on Quantization and Cohomology I gave a summary of the trip we’d been on, and a brief description of where we could have gone next — if only there’d been time:

Week 27 (June 5)  Review and prospectus. Classical and quantum mechanics from categories equipped with action or phase functors. Subtleties: path integrals and anafunctors. Geometric quantization of symplectic manifolds. Categorification, to obtain the classical and quantum mechanics of strings. Categorifying the theory of connections as anafunctors.
Supplementary reading:
 John Baez and Urs Schreiber, Higher gauge theory.
 John Baez, Higher gauge theory, higher categories.
 Urs Schreiber, Talks at "Higher Categories and their Applications".
Last week’s notes are here.
It was an exhausting quarter, and I didn’t cover nearly as much as I’d hoped to in this class — in part because I never had enough time to prepare.
Still, Derek Wise took very nice notes for the whole year’s course. Apoorva Khare is turning them all into LaTeX, and Christine Dantas has been preparing figures in electronic form. My new student Alex Hoffnung hopes to polish this material into a readable paper or book. Doing this will take a lot of work, since right now the notes are rambling — I never reached various mountaintops I’d been aiming for. But, since this material is the subject of Alex’s thesis, I hope he’ll be motivated to put in the necessary work!
Re: Quantization and Cohomology (Week 27)
One thing I need to still better understand:
The way you conceive the situation, the inner product always involves the path integral (as described in week 18). I am thinking that this amounts to adopting the point of view of quantum mechanics of constrained systems from the outset, where the path integral acts as the projector onto the physical Hilbert space, i.e. where the physical inner product
$\langle \phi,\psi \rangle_{\mathrm{phys}}$ arises from an inner product $\langle \cdot,\cdot \rangle$ on an auxiliary kinematical Hilbert space from something like a “group averaging procedure” $\langle \phi,\psi \rangle_{\mathrm{phys}} = \int_G \langle \phi, U(g)\psi \rangle \; d\mu_G \,,$ where $G$ is some gauge group and $U(g)$ a representation of it on our kinematical Hilbert space.
For systems like the relativistic particle and the like, the group in question is that of diffeomorphisms of the 1dimensional parameter space, and we get an expression roughly of the form $\langle \phi,\psi \rangle_{\mathrm{phys}} = \int_0^\infty \langle \phi, e^{t H}\psi \rangle \; d t \,,$ where $H$ is the Hamiltonian constraint. This, in turn, should be, then, thought of as coming from just the path integral with boundary conditions specified by $\psi$ and $\phi$.
I have the impression that it is this kind of physical inner product for constrained systems which you have in mind when you consider the inner product as involving the path integral. Is that right?
Recently I understood, from this comment of yours, that this relation between the path integral and the inner product is the way you think of the $\mathrm{Hom}$ as generalizing the inner product.
On the other hand, I had been thinking of the inner product as the pairing $(\psi : \mathbb{C} \to H, \phi: \mathbb{C} \to H ) \mapsto \mathbb{C} \stackrel{\psi}{\to} H \stackrel{\phi^*}{\to} \mathbb{C} := \langle \phi, \psi \rangle \,,$ which I think of as the image of $\mathrm{Id}_{H}$ under $\mathrm{Hom}(\phi^*,\psi) \,.$
I am guessing that in the end the relation between these two points of views comes down to pretty much the relation between ordinary unconstrained quantum theory and that of constrained systems. But I would want to understand this more cleanly.