Abstract
Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.
Highlights
The problem of quantization, i.e., the transition from classical to quantum physics, may be formulated mathematically as the search for a functor from a “classical” category whose objects are symplectic manifolds to a “quantum” category whose objects are Hilbert spaces.On the classical side, it is useful to include among the morphisms X ← Y symplectomorphisms, which should produce unitary operators uponA solution to the composition problem on the classical side has been given by Wehrheim and Woodward [4], who introduce a category which is generated by the canonical relations, but where composition is merely “symbolic” unless the pair being composed fits together in the best possible sense
It is useful to include among the morphisms X ← Y symplectomorphisms, which should produce unitary operators upon
The morphisms in this category are equivalence classes of sequences of canonical relations which are composable as set-theoretic relations
Summary
The problem of quantization, i.e., the transition from classical to quantum physics, may be formulated mathematically as the search for a functor from a “classical” category whose objects are symplectic manifolds to a “quantum” category whose objects are Hilbert spaces (or more general objects, such as spaces of distributions, Fukaya categories or categories of D-modules). A solution to the composition problem on the classical side has been given by Wehrheim and Woodward [4], who introduce a category which is generated by the canonical relations, but where composition is merely “symbolic” unless the pair being composed fits together in the best possible sense. The morphisms in this category are equivalence classes of sequences of canonical relations which are composable as set-theoretic relations. The purpose of this note is to show that any morphism in the WW category may be expressed as a product of just two canonical relations. We will place the Wehrheim-Woodward construction in an appropriate general setting which is robust enough to apply to categories of operators in which quantization functors may take their values
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