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Anel and Catren in their introduction to New Spaces in Physics claim that Lagrangian submanifolds are category-theoretic “points” of a symplectic manifold, morphisms from the trivial symplectic manifold in Weinstein’s symplectic category.
Is this accurate?
Anonymous
That was me, by the way.
Without looking up the article:
The statement in #1 seems to be immediate from the would-be definition of the Weinstein symplectic category, whose morphisms are meant, by definition, to be symplectic submanifolds in the product of the domain and codomain manifold. If the domain is the point, this seems to yield what you are quoting.
The notorious subtlety is to make the composition operation in the symplectic category well-defined. Is this what you are worried about?
Isn’t this a tautology in Zakrzewski-Weinstein category ?
Edit: I see Urs responded in the meantime.
Thanks. Yes, I could have checked that.
Perhaps more interesting is what they make of it: The study of symplectic and Poisson geometry shows the classical seeds within quantum physics, allows the latter to be seen as a continuous extension of the former (reference to Urs’s Chap. 5 on prequantum physics), rather than a radical break.
Then the idea of Lagrangian submanifolds encoding the classical seeds of quantum indeterminacy, so that we can speak of quantum particles as localized at ’points’, so long as we take these points as category-theoretic, i.e., morphisms from the terminal object.
David, the notion that Lagrangian submanifolds should be thought as quantum points was widely present idea at the end of 1980s. Just the reasoning was different, not the conclusion. It was often mentioned not only by geometers but also in the context of quantum groups.
As far as ideas on seeds of quantum phenomena, take into account strong results in symplectic geometry like Gromov’s nonsqueezing theorem (wikipedia) from 1987.
Maybe you would be interested to read a related semantic analysis also in the current context of Zakrzewski-Weinstein-Ševera category
Linear logic was described by Girard as a logic of dynamic interactions. On the other hand, Girard suggested an analogy between LL and quantum theory. Following these two intuitions we give an interpretation of linear logic in the language, which is common for both dynamical systems and quantization. Thus, we propose a denotational semantics for multiplicative linear logic using the language of symplectic geometry.
We construct a category of coherent phase spaces and show that this category provides a model for MLL. A coherent phase space is a pair: a symplectic manifold and a distinguished field of contact cones on this manifold. The category of coherent phase spaces is a refinement of the symplectic “category” introduced by Weinstein. A morphism between two coherent phase spaces is a Lagrangian submanifold of their product, which is tangent to some distinguished field of contact cones. Thus, we interpret formulas of MLL as fields of contact cones on symplectic manifolds, and proofs as integral submanifolds of corresponding fields. In geometric and asymptotic quantization symplectic manifolds are phase spaces of classical systems, and Lagrangian submanifolds represent asymptotically states of quantized systems. Typically, a Lagrangian submanifold is the best possible localization of a quantum system in the classical phase space, as follows from the Heisenberg uncertainty principle. Lagrangian submanifolds are called sometimes “quantum points”.
From this point of view we interpret linear logic proofs as (geometric approximations to) quantum states and formulas as specifications for these states. In particular, the interpretation of linear negation suggests that the dual formulas A and A⊥ stand in the same relationship as the position and momentum observables. These two observables cannot simultaneously have definite values, much like the case of two dual formulas, which cannot simultaneously have proofs.
Edit: I added the reference at Weinstein symplectic category.
Branimir Jurčo gave a talk at ESI on a richer version of the Weinstein’s symplectic category of correspondences suited to organize BV-theories into a category (work in progress).
We propose an enhancement of (odd) sympelctic category (in the spirit of Severa and Weinstein) and discuss its relation to homological perturbation theory.
Re #7, in view of recent interest here in linear logic, in the guise of linear dependent type theory, perhaps an interesting project, ’symplectic semantics’:
By no means should it be understood that the model that we discuss in this paper can be taken as some definitive final word. Rather we think of it as a first step in the direction of symplectic semantics and we would like to believe that it may serve as a building block (one among many) for future developments. We interpret only a very poor fragment of Linear Logic and the model itself is very elementary and can be modified in many ways. Apparently a more structure should be added in order to accommodate other connectives.
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