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Stream: learning: questions

Topic: isotropic relations

Cole Comfort (May 29 2021 at 13:54):

The category of (affine) Lagrangian relations over a field gives a semantics for certain classes of electrical circuits, and even stabilizer circuits, given the appropriate field.

There is a cateogry of isotropic relations, where the morphisms are isotropic subspaces of the product space of the domain and codomain, and whose composition is the relational one.

Does anyone know if there are interesting things in electrical circuits or quantum theory which are be modelled by (co)isotropic subspaces, but not Lagrangian subspaces, in general?

John Baez (May 30 2021 at 06:12):

An affine Lagrangian subspace of a symplectic vector space corresponds to a "maximal allowable specification of commuting observables" in the corresponding quantum theory. For example if our symplectic vector space is $\mathbb{R}^{2n} \ni (q_1,p_1, \dots , q_n,p_n)$ we can specify the momenta $p_1, \dots, p_n$, and that picks out an affine Lagrangian subspace. Alternatively we can specify the positions $q_1, \dots, q_n$. But trying to specify both $p_i$ and $q_i$ violates the uncertainty principle. And so on.

A coisotropic subspace corresponds to an "allowable specification of commuting observables".

Guillaume Boisseau (Jun 03 2021 at 01:50):

What's an isotropic subspace?

Guillaume Boisseau (Jun 03 2021 at 01:52):

Diagrammatically I mean

Cole Comfort (Jun 03 2021 at 08:14):

I will show you today :)