### Canonical Measures on Configuration Spaces

#### Posted by Urs Schreiber

Quantum theory crucially involves - and maybe is all about - subtle summations over all “histories” – or all “configurations” – of a physical system: the *path integral*.

The spaces of all these histories, or these configurations, usually carry a natural category structure on them:

Distinct but physically indistinguishable configurations are related by isomorphisms addressed as *gauge transformations*.

Even though the categorical (or groupoid) nature of configuration spaces is often not made explicit, dealing with the path integral in the presence of such gauge isomorphisms is a topic that occupies physicists a lot, as is evidenced by the large number of surname initials that go into the names of the theories handling these: BRST, FV, BV, BFV-BRST.

The entire kinematics and dynamics of quantum theory can nicely be conceived in terms of natural push-forward or colimit operations involving the category of configurations – an observation apparently going back to Dan Freed.

Or almost. Curiously, it appears as if the highest dimensional level of a quantum theory always requires a non-canonical choice: a measure on the space of histories.

While on the one hand this measure seems to stand out from the rest of the structure in its non-canonicalness, it is at the same time the source of much of the subtlety – and the richness – of quantum theory. Not the least, the lack of a rigorous handle on this measure, in general, is what kept and keeps large parts of physics from being accessible, and being accessed, by mathematically acceptable methods.

One may try to accept this as a sad fact of life – or one might take it as a hint for a hidden structure that still needs to be properly identified.

This post here is for those daring $n$-Café readers who are prepared to not shy away from attempting to chase and hunt down natural abstract structures that might possibly, and secretly, govern our current theories of this world. Or that might not.

I’ll try to summarize and wrap up some of the things that we talked about in the long discussion going on in Isham on Arrow Fields. It’s not that I think this discussion has reached a saturation point where everything interesting has been mentioned and just needs to be archived. Quite the opposite. I expect that there might be some really interesting aspects lurking right around the corner. But I feel that it would be helpful to draw a coherent picture of what we were discussing so far.

The obvious first step is to be serious about configuration spaces, and spaces of histories, as really being categories, whose morphisms encode gauge equivalences or other relations.

Chris Isham coined the phrase *quantization on a category* for this perspective. I want to emphasize, though, that we don’t want to talk about *another* notion of quantization. Rather this is supposed to *refine* our understanding of ordinary quantization. The refinement lies in the fact that we make explicit a structure that is otherwise often swept under the carpet, but is present nonetheless: a morphism space over the space of configurations, or of histories.

We had been discussing this point of view from various angles here at $n$-Café. Parallel to that, and seemingly unrelated, a lot of interest revolved around a Café table where discussion was about a curious function, called a *weighting on a category*, that Tom Leinster found was canonically associated to the spaces of objects of certain (finite) categories: its sum over all objects gives the Euler characteristic of a category.

It was Bruce Bartlett who suggested that this weighting is in fact the canonical measure to use on categories of configurations of physical systems when doing the path integral.

Bruce wrote:

[Chris Isham] mentions that “it is necessary to develop a proper measure on the set [of histories]”.

Indeed! That is precisely what all of us at the $n$-cafe were talking about in the discussion of Tom Leinster’s Euler characteristic of categories.

I am not sure if anybody else from “all of us” actually had made that connection. But Bruce is an expert on Dijkgraaf-Witten theory. So he knew the following fact:

**How Dijkgraaf-Witten theory is governed by the Leinster-measure**

In Dijkgraaf-Witten theory, the “worldvolume” is the fundamental groupoid $\mathrm{worldvol} = \Pi_1(M) \,,$ of some manifold $M$.

Target space is a *finite* group $G$, regarded as a category
$\mathrm{tar} = \Sigma G$
with a single object.

The space of histories $\mathrm{hist} = [\mathrm{worldvol},\mathrm{tar}]$ is the category of functors from the fundamental groupoid into the gauge group. Each such functor can be thought of as representing a $G$-bundle with connection on $M$. Since, for $G$ finite, there is really only a unique connection on every such bundle, this is the same here as just $G$ bundles on $M$.

The morphisms in $\mathrm{hist}$ are natural transformations between these functors. These describe gauge transformations relating the respective bundles.

In particular, the the space of histories, $\mathrm{hist}$, here happens to be a groupoid!

And it is equivalent to a *finite* groupoid. For instance, we can simply replace the fundamental groupoid of $M$ with its skeleton, which is then nothing but a disjoint union of (suspended) fundamental (finite) groups, one for each connected component. (This works here, but not in general gauge theory, because there is no smoothness condition or anything like that. All categories and functors are internal just to $\mathrm{Set}$ and so every connected groupoid is equivalent to any of its vertex groups.)

As a finite groupoid, its Leinster measure $d\mu$ is defined and coincides with the Baez-Dolan prescription: $d\mu : x \mapsto \frac{1}{|s^{-1}(x)|} \,,$ where the denominator is the number of morphism with source $d\mu$.

Moreover, our groupoid is equivalent to its skeleton, which is just a disjoint union of 1-object groupoids. On this skeleton, the above measure reduces to $d\mu : x \mapsto \frac{1}{|\mathrm{Aut}(x)|} \,,$

This measure is indeed precisely the measure that enters the path integral (which here is just a finite sum) in Dijkgraaf-Witten theory.

You can find this, for instance, in equation (2.1) on p. 9 of

Daniel S. Freed, Frank Quinn, Chern-Simons Theory with Finite Gauge Group

or in equation (4.25) and (4.31) on p. 68 of

Bruce H. Bartlett, Categorical Aspects of Topological Quantum Field Theories.

**How integrals of set-valued functors are governed by the Leinster measure**

Bruce went on to claim that

When [the category of histories] is a poset, it’s also well known what measure to use (see Tom Leinster’s paper).

This clearly involves a generalized notion of “well known”, it seems. In any case, when I first read this statement I was sceptical. Similar reservations were later also voiced by Eric Forgy # and Tim Silverman #.

But then I was converted from a sceptic to a true believer. This happened after I went back to Tom Leinster’s explanation of the way he found his weighting. Suddenly everything seemed to fall into place, and and rather beautiful general picture suggested itself.

So consider this:

While summation of numbers is something invented by man, direct summation of sets is something given to us from the gods.

Better yet, while integration of functions is something invented by man, integrals (colimits) of set-valued functors is something given us from the gods.

Given a category $X$, and a functor $F : X \to \mathrm{FinSet}$, the colimit $\mathrm{colim}_X F$ of $F$ over $X$ is, heuristically, the disjoint union of all the sets associated by $F$ to all the objects of $X$, divided out by all the identitfications implied by the value of $F$ on the morphisms of $X$.

One way to think of the natural numbers is as names of isomorphism classes of finite sets. In fact, in many texts on foundations, this is *the* way to think of natural numbers.

So let’s do that. If we have a finite set $S$, its equivalence class is its cardinality $|S|$. What is the equivalence class of the colimit of a set-valued functor, i.e. what is $\int_X F := |\mathrm{colim}_X F| \;\;\;?$

There is maybe no general answer to that. But when $F$ *acts freely* in a precise sense (and if $X$ is finite) then, Tom Leinster found, there is (see his paper for the detailed conditions and qualifications) a function
$d\mu : \mathrm{Obj}(X) \to \mathbb{R}$
such that
$\int_X F = \sum_{x \in \mathrm{Obj}(X)} |F(x)| \; d\mu(x)
\,.$

This function is precisely that *weighting on the category* $X$.

In other words: if we were only willing to accept that (natural) numbers are really labels of sets (and really they are!), and that number-valued functions are then really decategorifications of set-valued functors – then we are lead to
*regard the Leinster measure on a category as a canonical measure induced on a space of objects by a space of morphisms over these*.

In fact, there are other indications that we would ultimately need to do precisely that: regard the numbers that appear in quantum theory as placeholders for sets. While I don’t want to get into that right now, it does give me additional impetus to take this connection serious.

**How to test if the Leinster measure on configuration categories possibly governs the path integral**

Given that, I am prepared to speculate: *maybe the correct identification of the spaces of morphisms over the spaces of configurations that we encounter in physics will induce, via the weighting on that category, the right measure for the path integral over these*.

We already know that this *is* the case for Dijkgraaf-Witten theory.

It would be good to test this in other examples.

One problem is that, at this point, the Leinster measure is only understood, and defined, for finite categories.

This means that if we want to check our speculation on ordinary quantum mechanical systems, like the free quantum particle – and this is what I would like to do – we need to find suitable discrete approximations to their spaces of configurations.

So I posed myself the following

**Excercise:** *Find a finite category, $\mathrm{tar}$, that decently models a $d$-dimensional pseudo-Riemannian spacetime. Find another finite category, $\mathrm{par}$, that decently models the abstract worldline of a relativistic particle. Then determine the Leinster measure on the category of histories
$\mathrm{hist} = [\mathrm{par},\mathrm{tar}]
\,.$
thought of as a model for the category of paths traced out by the particle in target space.
*

This measure – or rather a suitable limit of it as we take the size of our categories to tend to infinity – would be the canonical candidate for the measure that enters the path integral of the relativistic particle. I would like to check if it comes anywhere close to the expected result.

**First observations on possible models of pseudo-Riemannian spacetimes by finite categories**

The exercise posed is vague. It is one of those typical vague problems one encounters in physics, of the kind: “find a good mathematical model for a given physical situation”.

On the other hand, some people have thought hard about pretty much this exercise before, in a very related context, but without the light of the Leinster measure to guide them.

Pseudo-Riemannian structures of signature $(-++\cdots+)$ have a very notable property: they are, under mild conditions, equivalent to *light cone structure with measure*:

if, on some manifold, you know at every point all tangent vectors that are lightlike, and at the same time know the volume density at that point, then you can reconstruct the corresponding peudo-Riemannian metric.

In the general context of efforts of coming to grips with quantum gravity, concerned with understanding the nature of the general relativistic theory of gravity in light of the fact that physics is, fundamentally, quantum physics, it was mainly Rafael Sorkin who emphasized that this equivalent reformulation of pseudo-Riemannian structures might be helpful:

Rafael Sorkin, Causal Sets: Discrete Gravity .

The fact that a light cone structure introduces the structure of a poset on the set of spacetime points brought categories into the picture. On the other hand, my impression is that the suggestion that we think of spacetime – in its incarnation as the configuration space of the single particle – as a *category*, has not been seriously thought to its end yet.

In any case, motivated by these ideas, I suggested (after some trial and error) to approach the above exercise as follows:

Suppose we model a fixed pseudo-Riemannian spacetime in terms a finite category by interpreting the space of objects as a finite approximation to the collection of spacetime points, by interpreting the morphisms as future-pointing light-like paths between these points (thus as defining a light-cone structure) and by interpreting the Leinster-measure on this category as the volume density on this spacetime: which categories will correspond to which pseudo-Riemannian structures?

A large class of examples of suitable finite categories comes from categories freely generated from finite directed graphs $(V,E)$ without loops. The vertices of these graphs would model the spacetime points, while the edges would model the infinitesimal lightlike paths between these.

It seems that I finally convinced myself of the following #

**Proposition**: *The Leinster measure $d\mu$ on the category $X$ freely generated from a finite directed graph $(V,E)$ without loops is*
$d\mu : x \mapsto 1 - |s^{-1}(x)|
\,.$
Here the second term denotes the number of edges in $E$ emanating from the vertex $x$.

Proof: Use induction. The statement is clearly true when the vertex has no outgoing edges. So suppose the vertex has a positive number of outgoing edges, and for all vertices reachable by following paths of emanating edges the statement has already been shown. Then it follows that the entire tree of edges sitting above each of the edges in $s^{-1}(x)$ contributes one unit. This implies the statement for the vertex $x$.

This is potentially helpful, in that the classifying spaces of these categories are nothing but (the union of the edges of) the graphs they come from. See the last paragraph on “interpretation” for a comment on that.

**A couple of simple examples**

The Leinster measure on the category freely generated from the graph $\array{ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+4,m) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ (n,m+1) &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ \downarrow && \downarrow && \downarrow \\ (n,m+2) &\to& (n+1,m+2) &\to& (n+2,m+2) \\ \downarrow && \downarrow \\ (n,m+3) &\to& (n+1,m+3) \\ \downarrow \\ (n,m+4) }$ is $\array{ -1 &\to& -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow \\ -1 &\to& 1 \\ \downarrow \\ 1 }$

Maybe more interesting than the graph of the above shape, for the applications that we (that is: myself and whoever wants to join in) have in mind, is a Lorentzian cylinder, i.e. piece of a cylindrical 2-dimensional Minkowski spacetime:

$\array{ && && (n,m) &\to& \\ && && \downarrow && \downarrow \\ && (n+1,m-1) &\to& (n+2,m-1) &\to& (n+1,m+1) &\to& \\ && \downarrow && \downarrow && \downarrow \\ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+2,m+2) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ && \downarrow && \downarrow \\ && && (n+2,m+2) }$ Notice the periodic identification. The Leinster-measure of (the category generated by) this is still of the now familiar form, -1 everywhere except for the future boundary: $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

So this Minkowski 2-cylinder has proper volume -6.

Next, consider our 2-diamond cylinder from before #, but now at one spot I have added in just one more edge (the two edges at this point now indicated by a double arrow $\Downarrow$). Here is the resulting graph with its Leinster measure $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& -2 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \Downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

This remains true for arbitrary insertions of an additional edge in the interior: it only affects the Leinster meausure at its source vertex, nowhere else: $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -2 &\Rightarrow& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

And now, a gravitational Leinster-Sorkin wave propagating on a 2-dimensional cylindrical spacetime:

$\array{ && && -2 &\to& \\ && && \Downarrow && \downarrow \\ && -1 &\to& -2 &\to& -1 &\to& \\ && \downarrow && \Downarrow && \downarrow \\ -2 &\Rightarrow & -2 &\Rightarrow& -3 &\Rightarrow& -2 &\Rightarrow& 1 \\ \Downarrow && \downarrow && \Downarrow && \downarrow \\ &\to& -1 &\to& -2 &\to& 1 \\ && \downarrow && \Downarrow \\ && && 1 } \,.$ It originates in a pointlike perturbation at the incoming boundary, that produces two wave crests which propagate at the speed of light in both spatial directions. Both run around the spacelike circle once, pass right through each other once and then hit the outgoing boundary.

Apart from toying around with such examples that can explicitly written down on the back of an envelope, it would be interesting to have a general procedure that takes a globally hyperbolic spacetime of the form $M = \Sigma \times \mathbb{R}$ with $\Sigma$ a Riemannian space.

Eric Forgy has thought about this problem. He presents a way to obtain from a triangulation of $\Sigma$ a light-cone graph on $\Sigma \times \mathbb{R}$ in a systematic way.

Eric Forgy: Light Cone Graphs from Spatial Triangulations.

We should eventually see if we can better understand the volume measure that this procedure induces on $M$.

**Interpretation of these models**

One might want to pause for a moment and reflect on precisely how these examples model what we would like them to model.

I think they do give a consistent picture, if we accept that for these categories to model what we want them to model, the above is telling us that what looks like pseudo-Riemannian volume on large scales is the accumulation of lots of topological defects on tiny scales.

In a way, this is an old quantum-gravitist’s dream: realize flat space(time) on small scales as a very non-classical object which lots of curvature and lots of topological funny stuff going on, such that on sufficiently large scales it looks just like ordinary space.

The fun thing is: I am not dreaming this up! I am just pointing out that given the single one proviso that we accept that it is good to think of integer-valued functions on a graph as actually set-valued functors, then this interpretation forces itself upon us. This is just how such large graphs do appear to us on small and on large scales.

**Now try to complete the above exercise**

Yes, let’s try to do that next…

(But not right now. Local time here is after 2 AM, and counting. In three hours a taxi will (hopefully) pick me up and bring me to the train which will bring me to the bus which will bring me to the plane which will bring me to the train that will bring me back home. Hopefully the above account is not too affected by my level of bleary-eyedness. Any polishing will have to wait until that journey is over.)

## Re: Canonical Measures on Configuration Spaces

Man, Urs. Bleary-eyed or not, this is the most beautiful piece of scientific thinking I have ever seen. I’m happy just to be able to say I know you :) In some wishful fantasy world, I would like to think I nudged you in one or two right directions along the way even though I never possessed the skills to go where you have gone. Time spent batting ideas around with you has been the highlight of my humble scientific career. It is certainly goofy, but my eyes are a bit moist just thinking about the beauty of this. And to think, I remember back when you were just learning differential forms :)

Ok. Enough gushing :)

It would be good to test this in other examples.Quantum mechanics is certainly an obvious place to start, but there is perhaps an even easier testing group that might be a natural next step. That is “Wick rotated” QM, a.k.a. Brownian motion. We already know that a binary tree provides a nice approximation to Brownian motion (and stochastic calculus via discrete calculus of the binary tree).

I could tell you were onto something deep when I spewed the statement here. I know the steps haven’t been completed, but the path seems pretty clear to me (although I could never take the steps myself, at least I can see them).

Fantastic!