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Stream: theory: category theory

Topic: Abstract Quantum Mechanics

John Onstead (Jan 17 2025 at 12:18):

I recently read an article about how, analogous to the "unreasonable effectiveness of math in the physical sciences", there's an apparent "unreasonable effectiveness of quantum mechanics in mathematics". This got me thinking about how I'd personally try to abstract the methods and reasoning of quantum mechanics to use its machinery to describe general mathematical objects. To do this, I came up with an idea I called "abstract quantization" which, in a sense, "quantizes" a category. I'll try to think through this on this thread, I don't know if it will go anywhere but I hope it will be interesting! Here's an example of my thinking:

Take the category Set. Now, consider maps from the point into some set- each one lands on a particular element and can be thought of as "picking out" that element. The "space" of all possible selections for such a function is, of course, the set itself. Now, let's "quantize" this situation by adding the concept of a "superposition" into the mix. This gives us a new set of morphisms between the point and the set- but instead of "boring old" functions that pick out a single element, each one of these new morphisms are functions that exist in a superposition of having picked out all possible elements of the set at the same time. Then, just as the "space" of possible functions from the point to the set is the set itself, the "space" of these quantized morphisms from the point to the set is a Hilbert space along with a Hermitian operator on that Hilbert space. The eigenstates of this Hermitian operator correspond to "definite" states where the function only picks out a single element like in the "classical", non-"quantized" situation. In fact, the original set can be recovered as the eigenbasis for this Hermitian operator, and so it "embeds" into the Hilbert space in a "nice way".

John Onstead (Jan 17 2025 at 12:20):

More generally, if one considers functions from the two element set, then this can be thought of as like a "composite system". The resulting quantization will give a Hilbert space that's like if you "tensored" together two of the Hilbert spaces for the one element set. In this way, one can "quantize" the entire category Set into a much larger category of sets and "quantized" functions between them. As we saw above, the original category Set embeds (not fully faithfully) into this one, since "classical" functions correspond to the "definite" quantized functions. Furthermore, this new category is also enriched over Hilbert spaces with the hom functor into Hilb doing the construction of the Hilbert spaces as discussed above.

While this might not sound very useful at first, I have thought of a few directions to take the idea. One is to explore what metric spaces are like in this "quantum" world. For instance, if one considers a three dimensional Euclidean metric space, then its quantizations for maps from the one (or two, three, etc.) element sets will correspond to the usual (composite) quantum Hilbert spaces with the usual position space basis for actual quantum particles. It will also be interesting if we quantize sections of fiber bundles in the category of smooth manifolds, perhaps this could lead to some sort of a quantum field theory. Plus, earlier we saw how sets embed "nicely" into their quantizations- perhaps quantizing other mathematical objects will transport us to a setting that provides a better and much richer setting for understanding those mathematical objects (kind of like a Yoneda embedding)?

Anyways, I have a lot of other ideas on where this can go, and I'll get to those in due time. But I'll stop here for now!

Morgan Rogers (he/him) (Jan 17 2025 at 17:12):

Do you end up with "just" a category of vector spaces or Hilbert spaces this way?

John Baez (Jan 17 2025 at 18:34):

There have been fitful attempts to reconcile mathematics at a very deep level with the empirical observation that the universe is described by quantum mechanics. My advisor Irving Segal thought that someday there would be 'quantum set theory'. At least in the old days, logicians generally reacted negatively to 'quantum logic', or at least to taking it seriously as the 'true logic of the world', due to the lack of a well-behaved internal implication. The few papers I've seen on 'quantum topos theory' seem naive to me. But there's been a big surge in work using symmetric monoidal categories as a general framework going beyond cartesian categories, incorporating ideas on quantum mechanics.

Part of my take on all this is here. I've also been been trying to show that a lot of quantum physics can be redone using spans of finite sets, and my student Jeffrey Morton showed that finite sets labeled by elements of $\mathrm{U}(1)$ can do even more.

John Onstead (Jan 17 2025 at 19:57):

Morgan Rogers (he/him) said:

Do you end up with "just" a category of vector spaces or Hilbert spaces this way?

I did think about this interestingly enough. If we "quantize" $\mathrm{Set}$ as described above, then $\mathrm{Hom}(*, X)$ is very similar to taking a free vector space on the set, but with the added structure of the Hilbert space on top. However, the morphisms are not the same as in vector spaces or Hilbert spaces. A function $* \to X$ in the quantized $\mathrm{Set}$ is an element in the Hilbert space $\mathrm{Hom}(*, X)$ rather than a linear operator on it.

John Baez said:

But there's been a big surge in work using symmetric monoidal categories as a general framework going beyond cartesian categories, incorporating ideas on quantum mechanics.

That's quite interesting! I don't understand too much about these more logical approaches to abstracting the essence of quantum mechanics. It's something I hope to learn one day!

John Onstead (Jan 17 2025 at 20:06):

John Baez said:

Part of my take on all this is here. I've also been been trying to show that a lot of quantum physics can be redone using spans of finite sets, and my student Jeffrey Morton showed that finite sets labeled by elements of $U(1)$ can do even more.

That looks very cool! I admit didn't think of direct categorification when I was wondering how to deal with quantum mechanics via category theory, but it seems very effective. What really struck me is how Fock spaces can be easily categorified into a combinatorial framework- which seems very distant from the original setting. Further, that the non-commutativity in the creation and annihilation operators can be expressed in a combinatorial way that uses "classical" intuition, particularly that there's more ways to put something into a box and take it out than take one out and then put one in.

Though to be honest, I'm having trouble wrapping my mind around how the combinatorial categorification actually works. Combinatorics is a classical mechanical phenomena, and we know that quantum mechanics does not obey classical mechanics or intuition. So how is it that quantum mechanics can be categorified into such a classical framework?

John Onstead (Jan 17 2025 at 20:22):

Also, here's two further questions: is the category $\mathrm{Hilb}$ monadic over either $\mathrm{Set}$ or $\mathrm{Vect}$? You would think this to be a well known fact but I can't seem to find any reference anywhere as to if there's even a "Lawvere theory of hilbert spaces". Maybe I'm just not a good researcher!
And for vector spaces, given the free functor $F: \mathrm{Set} \to \mathrm{Vect}$ and sets $X$ and $Y$, is there any "neat" way to prove that $F(X \times Y) \cong F(X) \otimes F(Y)$?

John Baez (Jan 17 2025 at 21:30):

John Onstead said:

Though to be honest, I'm having trouble wrapping my mind around how the combinatorial categorification actually works. Combinatorics is a classical mechanical phenomenon, and we know that quantum mechanics does not obey classical mechanics or intuition. So how is it that quantum mechanics can be categorified into such a classical framework?

That's an interesting question.

First, we have to separate "classical mechanics" (the study of physics using F = ma, Lagrangians and Hamiltonian) from "classical logic" and other "classical" things. I really don't think combinatorics is a classical mechanical phenomenon!

Second, a certain chunk of quantum mechanics is about symmetric monoidal dagger-categories, and the bicategory of groupoids, spans and maps of spans is a symmetric monoidal dagger-bicategory, so when we study combinatorics using the groupoid of finite sets and various groupoids of structured finite sets, we are getting fairly close to quantum mechanics. This shows up very clearly in the combinatorics of Feynman diagrams.

John Onstead (Jan 19 2025 at 00:38):

John Baez said:

I really don't think combinatorics is a classical mechanical phenomenon!

To me, combinatorics seems very classical- you are dealing with definite quantities with properties that are always definite, and there's no "fuzziness" so to speak. For instance, the statement "there's more ways of putting in and taking out than taking out and putting in" does not depend on the number of elements in the set already, and certainly does not imply that the number of elements in the set is indeterminate and in some sort of a superposition.

However, things are much different in the quantum world. There, noncommutativity actually does directly imply superposition. For instance, position and momentum do not commute, meaning knowledge of exact momentum fundamentally precludes knowledge of exact position. This directly implies that it is a valid state of being for a particle to be in "multiple places at the same time". For the creation and annihilation operators, the noncommutativity suggests that it's possible for a field to be in a superposition of having multiple different particle numbers at the same time. This is unlike in our classical example where the fact that "there's more ways of putting in and taking out than taking out and putting in" has no bearing on the number of elements of our system.

This is the source of my doubt that quantum mechanics can be entirely explained combinatorially. Sure, combinatorics might give some classical intuition for what it means for operators to fail to commute. But ultimately, the noncommutativity needs to imply superposition, which is something that does not exist in combinatorics and only exists in the domain of quantum mechanics.

John Baez (Jan 19 2025 at 00:55):

The category of finite sets and functions is cartesian closed and you can't add morphisms, so it doesn't allow 'superpositions' and its logic is classical. But the bicategory of finite sets and spans is noncartesian, instead it's a compact closed dagger-category, and you can add morphisms - just like the category of finite-dimensional Hilbert spaces!

It gets even better when you use the bicategory of groupoids and spans. This is what groupoidification is all about. Then you get the canonical commutation relation between annihilation and creation operarators, which relies on being able to add morphisms:

$aa^\dagger \cong a^\dagger a +1$

John Onstead (Jan 19 2025 at 01:17):

Ah, I think I see. Thanks!

John Onstead (Jan 25 2025 at 16:37):

So in this next journal entry of mine I want to cover how abstract "structure" can be thought of in a quantum mechanical way. I think this "abstract quantization" can allow for many kinds of structure to be put on the same footing with many interesting results. Let's start by considering a set $S$ with a structure on it given by some function $f: S \to X$. We will focus on quantized functions of form $* \to S$, but not quantize $f$ itself, as we want to consider a superposition of all possible particle states on a mathematical structure, not a superposition of all possible structures themselves! Since a quantized function from the point is in a superposition of being all possible elements of $S$ (it's an abstract "particle" in $S$), it means that by $f$, the "particle" is also in a superposition of possessing all possible values, elements of $X$ that are in the image of $f$. From this, it seems any function $f$ can be "upgraded" into an operator $\hat f$ on $H(S)$, the Hilbert space for the "particle" in $S$.

Let's see a basic example of this in action by considering a metric space structure. This is a function $f: S \times S \to \mathbb{R}^+$. Now, recall that $\mathrm{Hom}(*, S \times S) \cong \mathrm{Hom}(2, S)$. As covered last time, the Hilbert space for functions from $2$ is the tensor product space of the one you get from the one point set by an abstract "composite system principle". Thus, you can think of a function $* \to S \times S$ as a composite two particle system in $S$, in which the Hilbert space is $H(S \times S) \cong H(S) \otimes H(S)$. One can then imagine the setup as two abstract particles placed in a metric space, in a superposition of having all possible distances (in the image of $f$) from each other. This means we can upgrade $f$ into $\hat f$ on the space $H(S) \otimes H(S)$, which is the "distance operator". Every eigenstate of this operator corresponds to a classical definite distance value between the two particles.

John Onstead (Jan 25 2025 at 16:42):

It's all well and good that we can put many kinds of structures into the setting of Hilbert space as operators, but what good does that do? I've realized that in many cases, imagining physics as playing out on these abstract structures can, in some cases, give information about the actual structure. Here's what I mean with an eye-opening (at least for me) example. Imagine the metric space $\mathbb{R} \times \mathbb{R} \to \mathbb{R}^+$ for ordinary Euclidean space. Now imagine we have a composite system of three particles $A, B, C$ on this space $H(S) \otimes H(S) \otimes H(S)$, which yields an overall distance operator which can be decomposed into a distance operator for each pair. Now consider I start the system in a random state, and not in any eigenstate for any distance operator. This means all three particles exist in a superposition of being every possible distance apart from either of the other two.

Now, imagine this was a real quantum system on this metric space and you performed a projective measurement on the distance basis $(A, B)$. While we now know the distance from $A$ to $B$, we still don't know any of the other distances. $C$ is still in a superposition of all possible distances with respect to $A$, for instance. But now here's the interesting part- if we perform any other projective measurement on a different distance basis like $(B, C)$, then we can just use the metric to calculate what the third distance must be ($f(A, B) + f(B, C) = f(A, C)$, assuming $B$ is between $A$ and $C$). In this sense, doing two measurements fixes the third result without having to independently measure it. This is the direct result of the metric space structure "constraining" possible states, which we showed can be directly detected just by thinking about physics happening on the metric space!

Anyways that's enough for today. Hopefully a lot to think about!

David Egolf (Jan 25 2025 at 19:15):

I hope it's alright to ask a basic question about an earlier part of this thread.

John Onstead said:

Take the category Set. Now, consider maps from the point into some set- each one lands on a particular element and can be thought of as "picking out" that element. The "space" of all possible selections for such a function is, of course, the set itself. Now, let's "quantize" this situation by adding the concept of a "superposition" into the mix. This gives us a new set of morphisms between the point and the set- but instead of "boring old" functions that pick out a single element, each one of these new morphisms are functions that exist in a superposition of having picked out all possible elements of the set at the same time.

It makes sense to me that a function $f:1 \to A$ in $\mathsf{Set}$, where $1$ is a singleton set, picks out a single element of $A$. But I don't understand your definition of the new morphisms you are introducing, that correspond to picking all the elements of set in some kind of superposition. How would one compose such morphisms?

David Egolf (Jan 25 2025 at 19:20):

The first idea that occurs to me is to take formal weighted sums of functions $:1 \to A$ as our new morphisms. But I'm not sure that's what you have in mind.

John Baez (Jan 25 2025 at 19:45):

If we want a category where we can take formal weighted sums of maps between finite sets, with weights in some rig $R$ (think $\mathbb{C}$ for ordinary quantum mechanics), the natural candidate is the category $\mathrm{Mat}(R)$, where:

• objects are finite sets
• a morphism from $X$ to $Y$ is a $Y\times X$ matrix of elements of $R$, i.e. a function $Y \times X \to R$
• composition is matrix multiplication.

This category $\textrm{Mat}(R)$ has been studied a lot, and it's equivalent to the category of finitely generated free $R$-modules. When $R$ is a field, as in our example $R = \mathbb{C}$, this is the category of finite-dimensional vector spaces over $R$. But when $R$ is the boolean rig $\{0,1\}$, $\textrm{Mat}(R)$ is equivalent to the category where morphisms are relations between finite sets.

John Baez (Jan 25 2025 at 19:48):

And when $R = \mathbb{N}$ with its usual + and $\times$, $\mathrm{Mat}(R)$ is equivalent to the category where morphisms are isomorphism classes of spans between finite sets. (Spans start out giving a bicategory, but if we decategorify one notch we get this category.)

Here are some slides of a talk I gave about this:

• Spans in quantum theory.

I explain how these ideas are connected to Heisenberg's 'matrix mechanics' approach to quantum mechanics, and also to the principal of least action in classical mechanics.

David Egolf (Jan 25 2025 at 20:58):

Interesting! I see that each function $:Y \times X \to R$ associates to each $(y,x)$ an $R$-weighting.

I don't think I had realized that you can define $R$-modules for $R$ a rig (not necessarily a ring). It's pretty neat that we can recover a category of relationships as a special case.

David Egolf (Jan 25 2025 at 20:58):

I'm currently interested in the tropical rig. So let me see what happens when we use it as $R$. We get a category where:

• objects are finite sets
• a morphism from $X$ to $Y$ is a function $Y \times X \to R$

In this case, the underlying set of $R$ is $\mathbb{R} \cup \infty$.

Let's see what multiplication corresponds to. By analogy with matrix multiplication, I'm guessing we have for $f:X \to Y$, and $g:Y \to Z$:

$(g \circ f)(z,x) = \sum_y g(z,y)f(y,x)$.

But since addition in the tropical rig is taking the minimum, and multiplication is addition, we get:
$(g \circ f)(z,x) = \min_y g(z,y) + f(y,x)$.

David Egolf (Jan 25 2025 at 21:02):

If we view $f(y,x)$ as the "cost" of going from $x$ to $y$, and $g(z,y)$ as the "cost" of going from $y$ to $z$, then $(g \circ f)(z,x)$ is the minimum cost of getting from $x$ to $z$ while passing through $Y$.

(Ah, I now see this is mentioned in the notes linked above!)

John Baez (Jan 25 2025 at 22:28):

David Egolf said:

I don't think I had realized that you can define $R$-modules for $R$ a rig (not necessarily a ring). It's pretty neat that we can recover a category of relationships as a special case.

People don't talk about rigs enough, and they really don't talk enough about modules of rigs. Some people call them "semimodules", but those are the same people who call rigs "semirings".

One thing that's cool is that we can define a module of a ring to be just a module of its underlying rig. That is, the definition of a module doesn't need to mention negatives or the law

$(-a) x = - a x$

If your rig happens to be a ring, you'll get this law automatically for any of its modules.

John Baez (Jan 25 2025 at 22:30):

David Egolf said:

If we view $f(y,x)$ as the "cost" of going from $x$ to $y$, and $g(z,y)$ as the "cost" of going from $y$ to $z$, then $(g \circ f)(z,x)$ is the minimum cost of getting from $x$ to $z$ while passing through $Y$.

Yes, this is lots of fun. It's really good if you discovered it on your own before looking at my slides.