November 18, 2007

Geometric Representation Theory (Lecture 12)

Posted by John Baez In 1925, Werner Heisenberg came up with a radical new approach to physics in which processes were described using matrices of complex numbers. What makes this especially remarkable is that Heisenberg, like most physicists of his day, had not heard of matrices!

It’s hard to tell what Heisenberg was thinking, but in retrospect we might say his idea was that given a system with some set of states, say $\{1,\dots,n\}$, a process $U$ would be described by a bunch of complex numbers $U^i_j$ specifying the ‘amplitude’ for any state $i$ to turn into any state $j$. He composed processes by summing over all possible intermediate states: $(V U)^i_k = \sum_j V^j_k U^i_j .$ Later he discussed his theory with his thesis advisor, Max Born, who informed him that he had reinvented matrix multiplication!

In 1928, Max Born figured out what Heisenberg’s mysterious ‘amplitudes’ actually meant: the absolute value squared $|U^i_j|^2$ gives the probability for the initial state $i$ to become the final state $j$ via the process $U$. This spelled the end of the deterministic worldview built into Newtonian mechanics.

More shockingly still, since amplitudes are complex, a sum of amplitudes can have a smaller absolute value than those of its terms. Thus, quantum mechanics exhibits destructive interference: allowing more ways for something to happen may reduce the chance that it does!

Heisenberg never liked the term ‘matrix mechanics’ for his work, because he thought it sounded too abstract. Similarly, when Feynman invented a way of doing physics using what we now call ‘Feynman diagrams’, he didn’t like it when Freeman Dyson called them by their standard mathematical name: ‘graphs’. He thought it sounded too fancy. Can you detect a pattern?

But no matter what we call matrix mechanics, its generalizations are the key to understanding how invariant relations between geometric figures give intertwining operators between group representations. And that’s what I talked about this time in the Geometric Representation Theory seminar.

• Lecture 12 (Nov. 6) - John Baez on matrix mechanics and its generalizations. Heisenberg’s original matrix mechanics, where a quantum process from a set $X$ of states to a set $Y$ of states is described by a matrix of complex “amplitudes”: $F: X \times Y \to \mathbb{C}$ We can generalize this by replacing the complex numbers with any rig $R$, obtaining a category $Mat(R)$ where the objects are finite sets, and the morphisms from $X$ to $Y$ are $R$-valued matrices $F: X \times Y \to R$ $Mat(R)$ is equivalent to the category of finitely generated free $R$-modules. For example, $Mat(\mathbb{C})$ is equivalent to the category of finite-dimensional complex vector spaces, $FinVect_{\mathbb{C}}$ . If $\{0,1\}$ is the rig of truth values with “or” as addition and “and” as multiplication, $Mat(\{0,1\})$ is equivalent to the category with finite sets as objects and relations as morphisms, $FinRel$.

There’s an obvious map from $Mat(\{0,1\})$ to $Mat(\mathbb{C})$, which lets us reinterpret invariant relations as Hecke operators. But this is not a functor, so we don’t get a functor $FinRel \to FinVect_{\mathbb{C}}$. To fix this, we can consider $Mat(\mathbb{N})$, where $\mathbb{N}$ is the rig of natural numbers. This is equivalent to $FinSpan$, the category where morphisms are isomorphism class of spans between finite sets. The theory of spans as categorified matrix mechanics.

Posted at November 18, 2007 5:16 PM UTC

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Re: Geometric Representation Theory (Lecture 12)

Mat({0,1}) is equivalent to the category with finite sets as objects and relations as morphisms, FinRel.

Hmm. I’m a little confused by this. Why aren’t finitely-generated free $\{0, 1\}$-modules the same as Boolean algebras which are power sets of finite sets? Then a matrix of 0s and 1s of size $|X| \times |Y|$ acts on a column of 0s and 1s representing a subset of $Y$ to give a subset of $X$, just as a matrix of complex numbers acts on an element of $\mathbb{C}^{|Y|}$.

To get sets as freely generated modules, I thought we needed the generalized ring known as the ‘field with no elements’, i.e., the one associated with the identity functor.

The Kleisli category of the powerset monad is Rel. But then we’re not dealing here with Kleisli categories, are we? They have mappings from $X$ to $P Y$, for some functor $P$. I thought these matrices you’re after are mappings from $P X$ to $P Y$, so that transposes take you in the opposite direction.

On the other hand, a map from $X$ to $P Y$ does generate a map from $P X$ to $P Y$. And as we are dealing with ‘linear’ mappings from $P X$ to $P Y$, a mapping from $X$ to $P Y$ can be recovered.

OK, so what are ‘linear’ mappings between ‘field with no elements’-modules, i.e., between sets? Oh, they’re just functions, aren’t they?

Good, I think confusion is over.

Posted by: David Corfield on November 19, 2007 12:53 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

David wrote:

John wrote:

Mat({0,1}) is equivalent to the category with finite sets as objects and relations as morphisms, FinRel.

Hmm. I’m a little confused by this. Why aren’t finitely-generated free $\{0, 1\}$-modules the same as Boolean algebras which are power sets of finite sets?

They are! We discussed this in the video.

Say $X$ is a finite set. Elements of the free $\mathbb{C}$-module on $X$ are ‘wavefunctions’

$\psi: X \to \mathbb{C} \;,$

assigning to each state in $X$ an ‘amplitude’.

By analogy, the class jokingly decided that elements of the free $\{0,1\}$-module on $X$ should be called ‘possibility wavefunctions’, since they’re functions

$\psi: X \to \{0,1\}$

that assign to each state in $X$ a ‘possibility’ — 0 for impossible, 1 for possible.

Of course, a possibility wavefunction is just a fancy name for a subset of $X$.

Anyway, $\{0,1\}$-valued matrices act as linear operators on these possibility wavefunctions, just as complex-valued matrices acts on ordinary wavefunctions. But $\{0,1\}$-valued matrices are just a fancy name for relations.

Here’s an example. Say it’s possible for you to be in Madrid. Then say we apply to your possibility wavefunction the relation ‘can fly from $X$ to $Y$ on British airways’. We get a new possibility wavefunction for you, which now says it’s possible for you to be in London.

So, we’re getting a crude caricature of full-fledged quantum mechanics, where our matrices have entries that are just truth values.

But truth values are just $(-1)$-groupoids! In class we then went on to discuss matrices whose entries are sets — that is, $0$-groupoids. This gives a much better approximation to quantum mechanics. Later we’ll move on to matrices whose entries are $1$-groupoids. These do incredibly well at simulating quantum mechanics. This is ‘groupoidification’.

Posted by: John Baez on November 19, 2007 7:27 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

This will naturally make people wonder what matrices whose entries are 2-groupoids might do for us.

About what else I said in my comment, am I right in thinking that although the Kleisli category looks contrived, really all it’s encoding are linear maps between free algebras?

Posted by: David Corfield on November 20, 2007 9:00 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

David wrote:

This will naturally make people wonder what matrices whose entries are 2-groupoids might do for us.

Right! But, it’s probably easier to jump straight to $\infty$-groupoids, otherwise known as topological spaces. Spans of spaces are very interesting, because they let us perform ‘pull-push’ operations. It’s all very much like what we’ve been doing with spans of groupoids.

In particular, spans of algebraic varieties play an important role in algebraic geometry. They’re called correspondences. In our Tale of Groupoidification, we’ve already seen some of these when we looked at Hecke operators such as the Radon transform.

So, while it’s very exciting to go from spans of $(-1)$-groupoids to spans of $0$-groupoids, and then to spans of $1$-groupoids, at this point I tend to get a bit jaded and want to jump straight to spans of $\infty$-groupoids, aka spans of spaces. It’s like the title of that great book by Gamow: One, Two, Three, … Infinity.

About what else I said in my comment, …

Yeah, sorry — I wanted to answer your comment in a very simple way, rather than getting enmeshed in jargon. But it also deserves a more technical answer.

About what else I said in my comment, am I right in thinking that although the Kleisli category looks contrived, really all it’s encoding are linear maps between free algebras?

Yes! And that’s precisely why I hate the term ‘Kleisli category’ — it makes a very simple idea sound scary and technical!

Any monad $T: C \to C$ has a category of algebras. To make category theory seem like an esoteric and obscure subject, its practitioners call this the Eilenberg–Moore category of $T$.

Similarly, any monad $T: C \to C$ has a category of free algebras. To make category theory seem like an esoteric and obscure subject, its practitioners call this category of algebras the Kleisli category of $T$.

For example, take any rig $R$ (for example a ring). We’ve been using $R[X]$ to mean the free $R$-module on a set $X$: it consists of all finite $R$-linear combinations of elements of $X$.

This gives a functor

$R[-]: Set \to R Mod$

which is left adjoint to the forgetful functor

$U: R Mod \to Set$

The composite

$T = U \circ R[-] : Set \to Set$

is a monad sending any set $X$ to the underlying set of the free $R$-module on $X$.

The category of algebras for $T$ is just the category of all $R$-modules, namely $R Mod$. To scare and confuse people, call it the Eilenberg–Moore category of $T$.

The category of free algebras for $T$ is just the category of free $R$-modules. To scare and confuse people, call it the Kleisli category of $T$.

As always, we can think of a morphism in this ‘Kleisli category’ in two equivalent ways. We can think of it as $R$-linear map

$R[X] \to R[Y]$

between free $R$-modules, or we can think of it as a function

$X \to U R[Y]$

These are really the same thing, because $U$ is right adjoint to $R[-]$.

Next: in the special case when $Y$ is finite, we have an isomorphism

$U R[Y] \cong R^Y$

In other words: finite $R$-linear combinations of elements of $Y$ are the same as functions from $Y$ to $R$. So, in this case, a morphism in the Kleisli category is the same as a function

$X \to R^Y$

But this is the same as a function

$X \times Y \to R$

Such a function is usually called a matrix: an $X\times Y$-shaped matrix with entries in $R$.

So, that’s a formal explanation of all the tricks I’m using.

Posted by: John Baez on November 21, 2007 5:57 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

We spoke in effect about the ‘collapse’ of the ‘possibility wavefunction’ here.

DC: Presumably a module over the rig of truth values has to look something like a vector of truth values. Imagine the vector answering the two questions: Are you a parent? Are you an aunt/uncle? Any individual can be in one of 4 states. For any two unrelated people as far as you know they could be in any one of 16 states. Finding out about one of them doesn’t help you with the other. But for two siblings (with no other siblings) they can only be in 4 states, and finding out about the state of one tells you about the other.

JB: Okay, very sensible. Now I can translate what you said into math lingo. We only need (for now) to think about FREE modules of the rig $R$, namely those of the form $R^n$. And, with any luck, the tensor product of $R^n$ with $R^m$ is always $R^{nm}$, where you tensor two vectors by multiplying them entrywise to get a rectangular array. And, “entangled states” are those that aren’t expressible as a tensor product of two vectors. For $R$ the rig of truth values, you’ve got entanglement whenever you’ve got a rectangular array $A_{ij}$ of truth values that’s not of the form ($B_i$ and $C_j$). There are lots of these, but as usual, they’re always expressible as a sum - an “or” - of unentangled states.

Posted by: David Corfield on November 20, 2007 10:30 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

Some more night thoughts. So your bicategory of the categories Mat($R$), $R$ a rig, is a (full) bicategory of the categories Mat($\Sigma$), as Durov writes it for generalized rings $\Sigma$.

In the larger bicategory the initial object is Mat($Id$), for the identity monad, which are matrices of 1s and 0s, with only one 1 per row, or in other words functions.

With these generalized rings which are not rigs we don’t have the nice $X \times Y \to R$ characterisation, but they are $\Sigma(n)^m$, often matrices with constraints on the rows, as with the probabilistic functions with their row normalised positive real entries.

Mappings between monads give functors between matrix categories.

Why do I keep sensing that species ought to be making an appearance? Is there an algebraic monad sending a set $X$ to the set of all $X$-labelled species?

Posted by: David Corfield on November 21, 2007 9:11 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

David wrote:

Why do I keep sensing that species ought to be making an appearance?

Because you’re right?

Actually, what’ll really make an appearance is not structure types, also known as species, but a slight generalization: stuff types. Next quarter we’ll talk a lot about multivariable stuff types, stuff operators, and their $q$-analogues.

Posted by: John Baez on November 22, 2007 12:16 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

One last comment to which you’re under no obligation to reply.

So we can aim for juicier and juicier rigs as ways of giving more exciting free rig-modules. Spaces is one very juicy rig. Then one can aim for things like Vect. Free Vect-modules giving K-V 2-vector spaces.

Then there’s the idea of going for polynomials. As a first step, one can map a finite set to the set of complex polynomials in that many variables. This type of construction, I think, is a Durov generalized ring. Up the ladder, we’ll start looking for stuff types, etc.

A couple of thoughts, then.

1) Is there anything up the ladder from the probability monad? I mean as matrix entries turn into things like sets, vector spaces, etc., have we lost the opportunity to row normalize?

2) Instead of mapping out of Finset to finite $R$-modules, we want to map out of (essentially finite) groupoids. Is there then a theory of algebraic monads in this context, so that we can have Durov-ian generalized somethings?

Posted by: David Corfield on November 22, 2007 9:36 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 12)

More to remind myself than anything else, let me refer to some notes by Joachim Kock on polynomial functors. They’re still be written, so we will need to revisit to find out what he has to say about species/structure types.

Posted by: David Corfield on November 24, 2007 11:35 AM | Permalink | Reply to this

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