# How does one geometrically quantize the Bloch equations?

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I've just now rated David Bar Moshe's post (below) as an "answer", for which appreciation and thanks are given.

Nonetheless there's more to be said, and in hopes of stimulating further posts, I've added additional background material. In particular, it turns out that a 2003 article by Bloch, Golse, Paul and Uribe “Dispersionless Toda and Toeplitz operators” includes constructions that illustrate some (but not all) of the quantization techniques asked-for, per the added discussion below.

How does one geometrically quantize the Bloch equations?

## Background

From a geometric point-of-view, the Bloch sphere is the simplest (classical) symplectic manifold and the Bloch equations for dipole-coupled spins specifies the simplest (classical) nontrivial Hamiltonian dynamics.

In learning modern methods of geometric quantization — as abstractly described on Wikipedia's Geometric Quantization article for example — it would be very helpful (for a non-expert like me) to see the quantum Hamiltonian equations for interacting spins derived from the classical Hamiltonian equations.

To date, keyword searches on the Arxiv server and on Google Books have found no such exposition. Does mean that there's an obstruction to geometrically quantizing the Bloch equations? If so, what is it? Alternatively, can anyone point to a tutorial reference?

The more details given, and the more elementary the exposition, the better! :)

## Some engineering motivations

It is natural in quantum systems engineering to pullback quantum Hamiltonian dynamics onto tensor network state-spaces of lower-and-lower dimension (technically, these state-spaces are a stratification of secant varieties of Segre varieties).

It should be appreciated too that in this context “quantum Hamiltonian dynamics” includes stochastic unravellings of Lindbladian measurement-and-control processes (per these on-line notes by Carlton Caves). Presenting the unravelled trajectories in Stratonivich form allows the open quantum dynamics of general Lindblad processes to be pulled-back with the same geometric naturality as closed quantum dynamics of Hamiltonian potentials and symplectic forms. This Lindbladian pullback idiom is absent from mathematical discussions of geometric quantization, e.g. the above-mentioned article Bloch et al.. In essence we engineers are using these pullback techniques with good success, without having a complete or even geometrically natural understanding of them.

Pulling back through successive state-spaces of smaller-and-smaller dimensionality, we arrive (unsurprisingly) at an innermost state-space that is a tensor product of Bloch spheres that inherits its (classical) symplectic structure from the starting Hilbert space. Moreover, the Lindblad processes pull back (also unsurprisingly) to classical noise and backaction that respects the standard quantum limit.

For multiple systems engineering reasons, we would like to understanding this stratification backwards and forwards, in the following geometrically literal sense: on any state-space of this stratification, we wish the dual option of either pulling-back the dynamics onto a more classical state-space, or pushing-forward the dynamics onto a more quantum state-space.

Insofar as possible, the hoped-for description of geometric (de/re)quantization will illuminate this duality in both directions. Needless to say, the simpler and more geometrically/informatically natural the description of this duality, the better (recognizing that this naturality is a lot to hope for). :)

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Nice question! I'm keen to see what the answer is to this one.

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I have written an answer to Mathoverflow in which explicit formulas for the classical and quantum Hamiltonians of a spin system (Generators of $SU(2))$ were written explicitely. The classical Hamiltonians are given by means of functions on the two sphere and the quantum Hamiltonians by means of holomorphic differential operators (which act on the sections of the quantum line bundle). For many spin system with a linear Hamiltonian in each spin, one just has a distinct one particle Hamiltonian per spin. Sorry for referring to my own work, but it is by no means original.

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answered Nov 21, 2011 by (4,195 points)
Thank you for this pointer, David! For non-experts (like myself) it's nontrivial to connect your concrete example to (for example) Wikipedia's three-step general recipe for geometric quantization. If you'd care to try your hand at such an exposition, I will instantly rate it as an answer, and moreover, please allow me to encourage you to write it up for publication (since to the best of my knowledge there presently is no such exposition in the literature). And as was mentioned above, the more details, the better! :)

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On reflection, I have accepted this as the answer. Thank you David!

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I have added a link to an article by Bloch, Golse, Paul and Uribe, titled “Dispersionless Toda and Toeplitz operators” (2003).

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