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Quantization as a Kan Extension

This entry is about the article

• Johan Alm, Quantization as a Kan extension

Summary

The article examines in terms of finitized toy examples of quantum field theories the suggestion that path integral quantization of a σ-model to a functorial quantum field theory

$$Z_{\nabla }:\Sigma \to \mathrm{Vect}$$Z_\nabla : \Sigma \to Vect

is effectively the Kan extension of the differential cocycle $\nabla$ on a target space $X$ that describes the background field; the Kan extension being along a bundle $A\to \Sigma$ over parameter space $\Sigma$ (the “worldvolume”) which is such that sections $\Sigma \to A$ are the fields of the theory.

$$\begin{array}{ccccccc}& & A& \to & X& \stackrel{\nabla }{\to }& \mathrm{Vect}\\ & {}^{\varphi }\nearrow & \downarrow & & & {\nearrow }_Z\\ \Sigma & \stackrel{\mathrm{Id}}{\to }& \Sigma \end{array}\begin{array}{ccccccc}& & \mathrm{ext}.\mathrm{configuration}\mathrm{space}& \to & \mathrm{target}\mathrm{space}& \stackrel{\mathrm{background}\mathrm{field}}{\to }& \mathrm{space}\mathrm{of}\mathrm{phases}\\ & {}^{\mathrm{field}\mathrm{config}.}\nearrow & \downarrow & & & & {\nearrow }_{\mathrm{quantum}\mathrm{theory}}\\ \Sigma & \stackrel{\mathrm{Id}}{\to }& \mathrm{parameter}\mathrm{space}\end{array}$$\array{ && A &\to& X &\stackrel{\nabla}{\to}& Vect \\ &{}^{\phi} \nearrow & \downarrow &&& \nearrow_{Z} \\ \Sigma &\stackrel{Id}{\to}& \Sigma } \;\;\; \;\;\;\;\;\;\;\;\;\;\;\; \array{ && ext. configuration space &\to& target space &\stackrel{background field}{\to}& space of phases \\ &{}^{field config.}\nearrow & \downarrow &&&& \nearrow_{quantum theory} \\ \Sigma &\stackrel{Id}{\to}& parameter space }

References

The suggestion originates in the trivial remarks

• The first part of the story of quantization by pushing to a point

where the observation is highlighted that the space of states of the ordinary charged quantum particle – associated by its quantum propagation functor $Z:{\mathrm{Bord}}_1^{\mathrm{Riem}}\to \mathrm{Vect}$ assigns to the point – is nothing but the psuh-forward (i.e. Kan extension) of the object-part of the fiber-assigning parallel transport functor $\nabla :X\to \mathrm{Vect}$ that characterizes the bundle with connection that describes the background field that the particle is charged under.

Simple as it is, this was an attempt to find a formalization of the indications on the n-categorical physics behind the path integral as suggested in

• Dan Freed, Higher Algebraic Structures and Quantization (arXiv)

as nicely surveyed in section 5.4 of

• Bruce Bartlett, Categorical Aspects of Topological Quantum Field Theories (arXiv)

The suggestion that this indicates a more general relation between quantization and push-forward/Kan extension was later formulated vaguely as

• The n-Cafe Quantum Conjecture

It then became clear that in the process of a full quantization by some kind of abstract-nonsense push-forward the Leinster measure should arise automatically and provide (at least in finite QFT models) the right weight in the path integral

• Canonical Measures on Configuration Spaces

Indeed, the Leinster measure produces not only the right path integral measure for 3-dimensional Dijkgraaf-Witten theory but also for its 4-dimensional sibling, the Yetter model.

This proposition was stated here

• comment to Dijkgraaf-Witten and its categorification by Martins and Porternd its Categorification by Martins and Porter).

Since Tom Leinster motivated his measure from considerations of push-forwards of functors to a point, as described by him at

• comment by Tom Leinster at This Week’s Finds in Mathematical Physics (Week 244)

this seemed like a natural, albeit still vague, indication for the validitiy of the $n$-Café quantum conjecture . In Johan Alm’s computation of the Kan extension the Leinster measure appears automatically as the path integral weight, and in the expected way when applied to Dijkgraaf-Witten theory.

A little later more indications accumulated that the push-forward appearing in the $n$-Café quantum conjecture is the push-part in a pull-push operation, where a quantum σ-model with target space $X$ and background field $\nabla :X\to V$ classifying a bundle $P\to X$ assigns to a cobordism cospan ${\Sigma }_{\mathrm{in}}\to \Sigma \leftarrow {\Sigma }_{\mathrm{out}}$ the pull-push operation through the span $[{\Sigma }_{\mathrm{in}},P]\leftarrow [\Sigma ,P]\to [{\Sigma }_{\mathrm{out}},P]$.

A toy example supporting this idea was described as

• An Exercise in Groupoidification

and then developed into a framework workable in full differential nonabelian cohomology using infinity-stacks with concrete details for Dijkgraaf-Witten theory worked out in

• Nonabelian cocycles and their sigma model QFTs ,

whose treatment of QFT is structurally similar to that discussed in the context of

• geometric infinity-function theory.

Johan Alm’s developing work on quantization as a Kan extension suggested that there ought to be a reformulation of this pull-push quantization equivalently in terms of Kan extensions. A simple observation for how that might occur was given in

• Relation of Kan extension to Pull-Push

In his latest version Johan Alm now presents an idea for a full proof of this suggested relation.

While Johan Alm was preparing his notes the article

• Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (arXiv)

appeared, which in its section presented ideas about an abstract nonsense formulation of path integrals in finite theories not unsimilar to the ones discussed here.

For instance the simple example of 1-dimensional Dijkgraaf-Witten theory appearing on page 3 there is reproduced naturally as a special case of Johan Alm’s general abstract Kan extension quantization.

Blog discussion on Johan Alm’s work is here:

• Alm on quantization as a Kan extension

For those non-experts who are interested in working through Alm’s paper step-by-step, see:

• An Exercise in Kantization

category: reference