The n-Category Café

“It would be better if I called them ‘operators’. It’s a curious and important fact that these operators are used to describe both processes and also observables. For example, the operator H called the ‘Hamiltonian’ corresponds to the observable called ‘energy’, but the operators exp(−itH) correspond to the processes called ‘time evolution’.”

This is the aspect I have never been told about. Physicists I talked with about quantum mechanics tended to say: “The Universe works like this: anytime you want to measure a position of electron, you ask the Universe, and He gives you one of eigenvectors of apropriate operator. That’s the way things are and nobody knows more”

But I knew there must be something more to this! Actually, there was time when I supposed that the state after measurement is a state that comes from action of the appropriate operator on the state before measurement. But this is not the way things go.

However, now I see that it was not that stupid! You seem to say that these observables are indeed somehow connected with changing states. I’d love to hear more about this. Are references above still appropriate?

sirix wrote:

Actually, there was time when I supposed that the state after measurement is a state that comes from action of the appropriate operator on the state before measurement. But this is not the way things go.

It is the way things go if your operator is a projection: a self-adjoint operator $p$ with $p^2 = p$). These correspond to ‘filters’, like a polaroid filter that only lets light of a specific polarization through. You can read about this in Feynman’s book.

But, it’s not the way things go for a general self-adjoint operator.

I too was very puzzled by this when first learning quantum mechanics. I think everybody who actually thinks about the subject goes through this puzzlement; some people give up thinking about it, while others keep worrying about it. What do operators in quantum mechanics really mean? Why do we use them to describe both processes and observables — and what about the case when the process is an observation?

Are the references above still appropriate?

Yes, you can learn a lot about this from reading Feynman and also the history of what Heisenberg did! But, you’ll have to do a lot of thinking on your own — nobody seems to come right out and give a formal theory relating ‘operators as observables’ and ‘operators as physical processes’, except for the special case of how self-adjoint observables $H$ give rise to one-parameter groups of unitary processes $exp(-i t H)$.

I’m intrigued by these connections with discrete calculus, which I first met in your previous posts and which I now understand a bit better thanks to the explanations of my friend and colleague Misha Feigin. He also pointed me towards some papers of Novikov and Dynnikov about graph Laplacians, discrete Schrodinger, etc, such as this.

When Rota started developing Mobius inversion for posets, he pointed out that it could be understood as a common generalization of two different things: (i) Mobius inversion in number theory, and (ii) solution of difference equations. Everyone remembers (i), of course, but (ii) is the relevant one here. And we want to replace the usual number line by a general directed graph (especially one underlying a category).

That’s all a bit vague, but I hope to go to a seminar on Thursday that will sharpen my thinking on this.

Allow me to add my three cents worth to this discussion about arrow algebras, and vector space of states!

Cent 1 : Taking the category algebra (or arrow algebra) $\mathbb{C}[X]$ of a groupoid $X$ is cool, but in my opinion in many cases it is unecessary and obscures the problem.

For instance : If $G$ is a group, one defines the loop groupoid $\Lambda G$ as the category

where ‘$\Sigma$’ of a group refers to viewing it as a one object category ($\Lambda G$ is a category Urs and I often talk about ). Thus $\Lambda G$ is just $G$ acting on itself by conjugation. It has objects $x \in G$, and morphisms $\stackrel{g}{\leftarrow} x$ for each $g, x \in G$, with composition given by

Anyhow, Freed was the first to realize that the Hopf algebra which quantum algebraists call the ‘Drinfeld double’ of $G$ is nothing but the groupoid algebra of $G$:

Suppose we start talking about representations. Then its clear that

that is, the representation category of the algebra $D(G)$ is the same as the representation category of the groupoid $Rep (\Lambda G)$. (By the way, this stuff is all done here).

To me, the groupoid picture is much simpler! For a representation of $\Lambda G$ is manifestly just a vector space $V_x$ sitting above each $x \in G$, with an linear map $V_x \rightarrow V_{gxg^{-1}}$ sitting above $x \rightarrow gxg^{-1}$ in $\Lambda G$.

Of course, a reprsentation of the algebra $D(G)$ can also be thought of in this way - but only once you’ve applied projection operators, etc. Why bother?

Its the same as with representations of quivers. Nobody I know likes to think of a representation of a quiver $Q$ as a representation of the quiver algebra $\mathbb{C}[Q]$ - rather, they think of it as a representation of the free category associated to the quiver $Q$.

Its kind of trivial, but its important, because in the groupoid picture all sorts of geometrical intuition is immediately available. One can talk about cocycles, twistings, sections, etc. which brings me to…

Cent two : John spoke about the space of states $H(X)$ as the vector space of linear combinations of the objects of $X$. I would venture that perhaps its nicer to think of $H(X)$ as the space of sections $\Gamma (X)$ of the trivial line bundle with trivial connection over $X$. Again, all these ideas are borrowed from this.

Cent three : When one thinks about how the arrows act on the states, I am reminded of some stuff I read by Chris Isham : “A new approach to quantising space-time : III. State vectors as functions on arrows”. I think his ideas are cool, and definitely relevant to the stuff we talk about here! (Also his other papers.) One of the interesting things he defines is an arrow field on a category $X$ : this is just a selection $A(x)$ of an arrow $x \stackrel{A(x)}{\rightarrow}$ for each object $x \in X$. Note that you can compose two arrow fields, so that the collection of arrow fields form a monoid $Arr(X)$ - which is a group if $X$ is a groupoid.

Anyhow, clearly $Arr(X)$ acts on $H(X)$… just something to think about. Another thing which acts on $H(X)$ is of course $Aut(X)$ - which is more relevant when $H = \Gamma (X)$ for a non-trivial connection on $X$.

Since when do quantum system come born with a special operator $Z$, which means ‘do all possible processes and add them all up’? They don’t.

They do: the path integral! :-)

In this entry I tried to describe a way to obtain the quantum propagator using something very similar.

I’ll expand on that a little:

Consider a system with a set $$\mathrm{conf}$$ of configurations, which, in each time interval $$\tau$$ may jump along any one of a collection of oriented edges $$E = \{ (a \to b), (c\to d), \ldots\}$$ from one configuration to another.

(Alternatively, if you understand enough topos theoretic synthetic differential algebra or something along these lines, consider this internal to a suitable topos such that it defines the infinitesimal evolution of our system. But this shall not concern me here.)

For simplicity, let’s assume that there is at most one edge with given source and target configuration in $E$.

Let there be a “phase”, (“cost of transition”) associated with each such edge. More precisely, let there be some abelian category $C$ and a graph map $$\mathrm{tra} : E \to C \,.$$

“Graph map” just means: like a functor, but without any condition on composition (since we have no composition in $E$). If we pass from $E$ to the graph category it generates, then $\mathrm{tra}$ becomes a functor on that.

Now, let’s quantize. What should that mean? Somehow, we should “sum $\mathrm{tra}$ over everything in sight”.

So consider the morphism $U$ in $C$ which is defined as follows (think of the “image” of the adjacency matrix of $E$ under $\mathrm{tra}$):

$U$ is an endomorphism $$U : \oplus_{x \in \mathrm{conf}} \mathrm{tra}(x) \to \oplus_{x \in \mathrm{conf}} \mathrm{tra}(x)$$ of the direct sum of objects in $C$ that $\mathrm{tra}$ assigns to each point in configuration space.

So $U$ is entirely specified by giving all its components $$U(x,y) : \mathrm{tra}(x) \to \mathrm{tra}(y) \,,$$ between the fibers over all points in configuration space. Let’s simply set $$U(x,y) = \left\{ \array{ 0 & \text{if}\;x \stackrel{\gamma}{\to} y\; \text{ does not exist} \\ \mathrm{tra}(x \stackrel{\gamma}{\to} y) & \text{otherwise} } \right. \,.$$

So $U$ is supposed to be something like “the sum of phases over all one-step evolution paths of our system”. Where “sum” is some blend of “direct sum” and something else.

(All the effort of that entry of mine was to realize this as an honest colimit of something.)

Notice now various things:

- This $U$ is much like the image under $\mathrm{tra}$ of what John called the zeta operator, or what Eric Forgy and myself used to call the graph operator. It has a couple of interesting properties.

- This $U$ is essentially $$U = U(\tau)$$ the “quantum propagator” over time $\tau$ of our system.

This is maybe better seen by considering its powers $$U(n\tau) := (U)^n \,.$$ The $(x,y)$-component of this morphism is the morphism between the fiber over $x$ to that over $y$ that is the sum of phases over all possible paths of length $n$ between $x$ and $y$.

Up to a measure on the space of such paths, this is indeed the (discrete) path integral representation of the quantum propagator!

One might imagine that this measure comes in more or less naturally as we take the continuum limit of the above context, somehow.

When I first wrote about this, I found this very simple observation pretty cool. As you recall, I was trying to understand Freed’s philosophy that “taking the path integral is ‘the same’ as taking the space of sections” concretely in simple examples.

So I tried to see if the above $U$ is the colimit of something nice: because it incorporates the right kind of sum over objects and over morphisms at the same time.

Meanwhile, I feel I have a good understanding of Freed’s prescription at the level of objects #.

But I still feel the need to incorporate the above construction more thoroughly in a general conceptual understanding of “quantization”.

`Don’t come with a special operator $Z$ …’

That’s asking for trouble :D

I don’t quite know what it means,…

Right, it’s not only that I did not try to prove something here, I did not even try to make something precise enough for it to admit anything like a proof! :-)

But maybe we can figure it out together.

I think I was imagining something like: assume that we take any two morphisms with the same target as “having the same angle enclosed between them”.

Then deficit angles translate into numbers coincident morphisms, which is, in turn, what determines those “weightings”.

One aspect that makes this a little subtle is that you are talking about categories where most of the literature on deficit angles and the likes talks about graphs.

For that reason it might be very helpful to better understand what your construction implies in the special case that the category in question is that generated by a graph.

Is it surprising that if you take any two categories with the same number of objects and same number of arrows between pairs, but different multiplication between arrows, then the Euler characteristics of their classifying spaces (if well-defined) are equal?

Is it even obvious for the Klein 4-group and the cyclic 4-group that these Euler characteristics should be the same?

To try to understand this localness issue, why not take Gelfand’s advice and look at the simplest non-trivial example? So what’s the classifying space of your poset $C = a \to b$?

Or is this not sufficiently non-trivial?

By saying that the formula is “local” I just meant that it arises as the sum over all vertices of some quantity assigned to each vertex, i.e. that it has the form $$\xi(X) = \sum_{\mathrm{Obj}(X)} f(x) \,.$$ That’s reminiscent of the “local” formulas for the Euler characteristic of manifolds, which look like $$\xi(X) = \int_X \; d\mu(x) \,.$$

So, to borrow M. Hopkin’s words: “the Euler characteristic of $X$ accumulates via integration of local data”.

But you are perfectly right that this quantity assigned to each vertex in Tom’s formula is itself determined by a rather “non-local” prescription.

But so is, for instance, the “deficit angle” : in order to determine it at a given vertex, one has to probe the vicinity of that vertex.

That’s the analogy I had in mind. But John’s comment might suggest that this analogy applies, if at all, not to $X$ but to $B X$.

Anyhow, both these formulas are manifestly local - unlike Tom’s formula.

Hm, so now I am not sure what you mean by “local”: the formula $$\xi(G) = \sum_{x \in \mathrm{Ob} G} \frac{1}{\sum_y \mathrm{card}\mathrm{hom}(x,y)}$$ involves, under the first summation sign, a sum over lots of morphisms in the category, pretty much like the formula determining Tom’s “weighting” does.

What is it that distinguishes these two terms under the summation sign for you?

Bruce wrote:

I don’t get why Tom’s formula is local. Perhaps I’m missing something!

I don’t think it’s local either, for the reasons that you give.

But it’s “sometimes local”. What I mean is that there’s a fairly large class of categories for which a local formula can be given.

For instance, suppose that $C$ is a finite skeletal category in which the only endomorphisms are the identities. Say that a path

in $C$ is nondegenerate if no $f_i$ is an identity. Then $C$ has a unique weighting $w$, given by

I would call this a local formula. Why? Well, for each $x$ we have the coslice category $x/C$, and $w(x)$ is defined entirely in terms of $x/C$. (Specifically, take the full subcategory $C_x$ of $x/C$ consisting of all the objects except the initial object $1_x$; then $w(x) = 1 - \chi(C_x)$.)

A similar statement can be made in the more general case that $C$ has no idempotents except identities (my Theorem 1.4).

I wrote:

(We may equivalently think of this colimit as computing some “space of sections”

Whoops, now that these functors take values in sets instead of vector spaces, I should be more careful here: in particular, it is the space of cosections that would yield the colimit.

What I mean is this:

For $$\mathrm{tra} : C \to \mathrm{FinSet}$$ a functor, its space of (flat) cosections would be its co-push-forward to a point, i.e. the functor $$q(\mathrm{tra})$$ such that $$\mathrm{Hom}(\mathrm{tra},p^*S) \simeq \mathrm{Hom}(q(\mathrm{tra}),S)$$ where $S$ is any set, regarded as a functor $$S : {\bullet} \to \mathrm{FinSet}$$ and where $$p : C \to {\bullet} \,.$$ This yields $$q(\mathrm{tra}) = \mathrm{colim}_C tra \,.$$

Yep, that is exactly what I was looking for.