Changing the rig
Here for posterity is a conversation I had with John Baez about using non-standard rigs (rings without negatives). There's nothing I like more (well not much anyway) than this kind of chatting. I haven't put in all the initials to indicate who's speaking, but most the time it's pretty clear.
DC: Now for a problem for future students if you carry on ideas from your Fall 03 Quantum gravity seminar:
Litvinov in this paper points out that corresponding to the Fourier transform for C, there is the Legendre transform for the rig R_max (R union {-infinity}, max ,+, -infinity, 0).
JB: I think the Legendre transform more directly generalizes the Laplace transform. In fact, as Jim Dolan explained but I never got around to retelling in the Fall '03 seminar, the Legendre transform ("finding the minimum of energy") is the temperature -> 0 limit of the Laplace transform ("summing over states weighted by exp(-E/kT)").
In other words, classical statics, where we minimize energy, is the temperature -> 0 limit of statistical mechanics. (Litvinov is maximizing instead of minimizing, but that's no big deal.) And, the ultimate reason this works is that the rig (R union {+infinity}, min, +, +infinity, 0) has a one-parameter deformation, where the deforming parameter is temperature.
When we let this parameter become complex we get quantum mechanics and Fourier transforms....
DC: What is the corresponding construction for the rig of truth values?
JB: Probably something like "finding the possible outcomes". Finding what's possible to do is a simplified version of "finding the least energetic thing to do", which is in turn a simplified version of "doing everything, but doing something of energy E with probability proportional to exp(-E/kT)".
All this needs to be explained very clearly to the world, so everyone will realize how cool it is!
DC:
> When we let this parameter become complex we get quantum mechanics and
> Fourier transforms....
and it seems that the Mellin transform is not far away:" the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject." http://en.wikipedia.org/wiki/Laplace_transform
were you hinting this in week 217?
JB: I wasn't thinking about that; I was just reassuring people thatall the transforms I listed are related by simple changes of variable,so you shouldn't feel ignorant of one if you know about another.
>>> What is the corresponding construction for the rig of truth values?
>> Probably something like "finding the possible outcomes". Finding what's
>> possible to do is a simplified version of "finding the least energetic
>> thing to do", which is in turn a simplified version of "doing
>> everything, but doing something of energy E with probability
>> proportional to exp(-E/kT)".
> So maps X -->{T,F} are subsets of X, and you find out whether it's possible
> to get from x to y within X. Right: and this is a kind of "path integral".
We can compute the "least action for a path from x to y" as an integral in a rig where addition is minimization.
We can compute the "amplitude for a path from x to y" as an integral in a rig where addition is addition.
We can compute the "possibility of a path from x to y" as an integral in the rig of truth values, where addition is "or".
> Sounds a lot like homotopy theory to me.
Yes, I guess any topological space gives a boolean-valued 2-variable function "can you get from x to y along a path?"
> But then the equivalence relationship of path connectedness is reflecting that it's a
> groupoid enriched over {T,F}, or was that impoverished?
Right! The really interesting 2-variable function associated to a topological space X is "the space of paths from x to y". If X is a homotopy n-type, this function will take values in homotopy (n-1)-types. If we think of a homotopy n-type as an n-groupoid this function is just hom(x,y)! But, we can decategorify hom(x,y) down to a homotopy -1-type, aka a truth value, which is "true" if hom(x,y) is nonempty and "false" if it's empty.
David writes:
> So if there is a categorifying chain of values for paths between x and y
> which runs: truth values, set, ..., n-groupoid,...
> are there other chains like
> cost, cost of passing between paths,...
> or probability, probability of passing between paths,.....
or amplitude, amplitude of passing between paths....
Good point! You're shooting ahead of me here, and it's a bit embarrassing, because as you note:
> this just seems to be pointing to things like your higher-gauge theory.
Part of what's been bugging me a lot about higher gauge theory is that I don't understand how Lagrangians fit into it. Usually people write down a Lagrangian as a function of some fields, which lets you compute an action, and minimizing the action give you equations of motion. You can do this in higher gauge theory too. BUT, usually the action for an ordinary gauge theory is required to be INVARIANT UNDER THE GROUP OF GAUGE TRANSFORMATIONS, since then gauge transformations will map solutions of the equations of motionto solutions. In higher gauge theory we have a 2-GROUP of gauge transformations. What does it mean for an action to be "invariant" under a 2-group? 2-groups really want to act not on a mere set, but on a CATEGORY - and the proper notion of "invariance" is "weak invariance", i.e. invariance up to a specified isomorphism satisfying some (understood) coherence laws.This suggests that actions in higher gauge theory should really take values in a category. And so, presumably, should Lagrangians. But, what category or categories??? Some categorification of the real numbers, maybe.
The problem is, I don't see the physics pointing me towards any particular choice. Probably I'm just being dumb. It's especially galling because I already think I know what one*result* of path-integral quantizing a higher gauge theory mightbe: a 2-Hilbert space of states! I wrote a paper on 2-Hilbert spaces once....
Hmm, this suggests that the appropriate "categorified transition amplitudes" lie not in C but in Hilb!!!
> Maybe it would be good to think how the fundamental groupoid arises through Lagrangian reasoning. In the path connectedness case, we have a space X, and a Lagangian map from X to truth values, i.e. a subset of X. Then for any path in X, there is an action formed by integrating L along it. This tells you whether the path lies wholly in X. Now you form the integral over all paths with the same endpoints. This just sees whether there is any path in X between those two points.
> Up a dimension, we're looking for a set of homotopy classes of paths:
Let me postpone thinking about this, even though it sounds really cool.
> Hmm. I have a feeling this ought to be slicker. Also the first 'integrations'
> in each case were over truth values and yet were ANDs rather than ORs.
I just want to say something about *this*. This actually seems right to me. In physics, a path integral is an integral over paths of the EXPONENTIAL of the action, which in turn is obtained by integrating a Lagrangian along the path. The exponential turns addition into multiplication. In fact, it's often good to think of the "exponential of the action", as more fundamental than the action. It has a clearer meaning. In quantum physics, the exponentiated action exp(iS/hbar) tells you the RELATIVE AMPLITUDE for taking that path. In statistical mechanics there's a version where you get the RELATIVE PROBABILITY.
In the situation you're talking about, the exponentiated action is a truth value saying whether the path is continuous - i.e., the POSSIBILITY of following that path. Now about that "first 'integration'" that's bugging you. The exponentiated action is given by a "product integral" along the path. I don't know if you've thought about product integrals, but they're just like integrals with + replaced by times. I reinvented them when I was a kid so I have a certain fondness for them. Normally you get them by multiplying lots of numbers that are really close to 1, instead of adding lots of numbers that are really close to zero... but normally you can reduce them to ordinary integrals using "exp" and "ln".
You however are doing product integrals in the rig of truth values: you are computing the possibility of a certain path as a product ofpossibilities of lots of little paths! And in this case there's no "exp" and "ln" to save us - unless there's some logical operation nobody every told me about, that converts "or" to "and". It's also neat to think about product integrals in the rig of costs:the rig R^{min} = (R union +infinity, min, +infinity, +, 0). Here we compute the cost of a path as an ORDINARY integral along the path...but the ordinary integral uses +, which is really MULTIPLICATION inthe rig of costs. So, it's again a case of a product integral. And again there's no "exp" and "ln" to save us.
> Presumably homotopy theory is treating a space as though it's
>infinitely cheap to go through the space, and infinitely expensive
> to go outside. So is there a cheap way to get from x to y? Yes,
>so long as there's a path [OR] along which [AND] all points are cheap.
Right! I like to think of truth values as a funny version of the rigof costs where the only two prices are "free" and "you can't afford it". Anyway, now I should go back to your categorified version of the wholesetup:
> Up a dimension, we're looking for a set of homotopy classes of
>paths: We have a space X, paths and paths between paths.
>The Lagrangian takes paths to truth values. When we integrate
>the Lagrangian along a path between paths it tells us whether
>we can do this all within X. Then for a given path f, we integrate
>[form set union] over all paths with the same end points,
> collecting all those homotopic to f. Now we integrate [form
>set union] over all paths f, forming the union of homotopy classes.
and think of the final result as an ordinary integral of a product integral. Btw, the "state sum models" in TQFT are all done by multiplying anamplitude for each labelled simplex and then summing over labellings, so it's the same sort of deal.
> About the transformation between quantum and classical, the
> trouble I'm having is that according to your lectures Sets
> and Relations are already on the quantum side.
Yeah! But that's GOOD.
In today's talk I explained how for any rig R there's a PROP whose morphisms from x^n to x^m are n x m matrices with entries in R, with the "tensor product" of morphisms being direct sum of matrices. In other words,"finitely generated free R-modules, made into a symmetric monoidal category using direct sum"This lets us do "matrix mechanics" in the manner we've been discussing, and when R is the rig of truth values we get "finite sets and relations, made into symmetric monoidal category using disjoint union" But we can also use the rig of costs....
> Oh, I see that I'd already got the point. But isn't that
> all the same odd to call anything matrix-like 'quantum'?
Well, FinRel is a symmetric monoidal category where the product is not cartesian, and it's a *-category, so in many ways it more closely resembles Hilb than Set or FinSet. The superposition principle is a bit stunted given that the rig of truth values has just one nonzero element, but don't let that fool you.
> Putting it naively, the things in the sets of Rel are
> perfectly classical. If the mere fact that things are
> related is enough to make them quantum, isn't that a sign
> that my twins entanglement idea is right - that a chunk of
> the weirdness of entanglement is little more exciting than
> that a twin marrying 12000 miles away makes you instantly
> an in-law. Or is your worry here that this is just about
> information? But then Fuchs, Cave et al want to say this is
> really all EPR experiments are doing.
Well, the sexier features of QM probably require a more interesting rig.
I'm not sure this is the right analogy; in quantum mechanics entanglement is about tensor products of vector spaces, so in FinRel we should be looking at tensor products of modules over the rig of truth values....
Hmm, it might turn out that you're right, and that the analog of a "entangled state" boils down to a pair of sets with a relation between them. But this is something one just needs to calculate. For example, maybe a relation f: S -> T can be dualized to give a relation g: 1 -> S* tensor T just like a linear operator f: V -> W gives a linear operator g: C -> V* tensor W - which is the same as a state in V* tensor W. The identity operator f: V -> V gives a maximally entangled state g: C -> V* tensor V, so maybe we can do the same thing with relations.
But, first I'd need to figure out if there really is a tensor product of "modules over the rig of truth values" (probably), and what it is.
> 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.
Okay, very sensible. Now I can translate what you said intomath 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 wheneveryou've got a rectangular array A_{ij} of truth values that's notof 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.
DC: Now for a problem for future students if you carry on ideas from your Fall 03 Quantum gravity seminar:
Litvinov in this paper points out that corresponding to the Fourier transform for C, there is the Legendre transform for the rig R_max (R union {-infinity}, max ,+, -infinity, 0).
JB: I think the Legendre transform more directly generalizes the Laplace transform. In fact, as Jim Dolan explained but I never got around to retelling in the Fall '03 seminar, the Legendre transform ("finding the minimum of energy") is the temperature -> 0 limit of the Laplace transform ("summing over states weighted by exp(-E/kT)").
In other words, classical statics, where we minimize energy, is the temperature -> 0 limit of statistical mechanics. (Litvinov is maximizing instead of minimizing, but that's no big deal.) And, the ultimate reason this works is that the rig (R union {+infinity}, min, +, +infinity, 0) has a one-parameter deformation, where the deforming parameter is temperature.
When we let this parameter become complex we get quantum mechanics and Fourier transforms....
DC: What is the corresponding construction for the rig of truth values?
JB: Probably something like "finding the possible outcomes". Finding what's possible to do is a simplified version of "finding the least energetic thing to do", which is in turn a simplified version of "doing everything, but doing something of energy E with probability proportional to exp(-E/kT)".
All this needs to be explained very clearly to the world, so everyone will realize how cool it is!
DC:
> When we let this parameter become complex we get quantum mechanics and
> Fourier transforms....
and it seems that the Mellin transform is not far away:" the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject." http://en.wikipedia.org/wiki/Laplace_transform
were you hinting this in week 217?
JB: I wasn't thinking about that; I was just reassuring people thatall the transforms I listed are related by simple changes of variable,so you shouldn't feel ignorant of one if you know about another.
>>> What is the corresponding construction for the rig of truth values?
>> Probably something like "finding the possible outcomes". Finding what's
>> possible to do is a simplified version of "finding the least energetic
>> thing to do", which is in turn a simplified version of "doing
>> everything, but doing something of energy E with probability
>> proportional to exp(-E/kT)".
> So maps X -->{T,F} are subsets of X, and you find out whether it's possible
> to get from x to y within X. Right: and this is a kind of "path integral".
We can compute the "least action for a path from x to y" as an integral in a rig where addition is minimization.
We can compute the "amplitude for a path from x to y" as an integral in a rig where addition is addition.
We can compute the "possibility of a path from x to y" as an integral in the rig of truth values, where addition is "or".
> Sounds a lot like homotopy theory to me.
Yes, I guess any topological space gives a boolean-valued 2-variable function "can you get from x to y along a path?"
> But then the equivalence relationship of path connectedness is reflecting that it's a
> groupoid enriched over {T,F}, or was that impoverished?
Right! The really interesting 2-variable function associated to a topological space X is "the space of paths from x to y". If X is a homotopy n-type, this function will take values in homotopy (n-1)-types. If we think of a homotopy n-type as an n-groupoid this function is just hom(x,y)! But, we can decategorify hom(x,y) down to a homotopy -1-type, aka a truth value, which is "true" if hom(x,y) is nonempty and "false" if it's empty.
David writes:
> So if there is a categorifying chain of values for paths between x and y
> which runs: truth values, set, ..., n-groupoid,...
> are there other chains like
> cost, cost of passing between paths,...
> or probability, probability of passing between paths,.....
or amplitude, amplitude of passing between paths....
Good point! You're shooting ahead of me here, and it's a bit embarrassing, because as you note:
> this just seems to be pointing to things like your higher-gauge theory.
Part of what's been bugging me a lot about higher gauge theory is that I don't understand how Lagrangians fit into it. Usually people write down a Lagrangian as a function of some fields, which lets you compute an action, and minimizing the action give you equations of motion. You can do this in higher gauge theory too. BUT, usually the action for an ordinary gauge theory is required to be INVARIANT UNDER THE GROUP OF GAUGE TRANSFORMATIONS, since then gauge transformations will map solutions of the equations of motionto solutions. In higher gauge theory we have a 2-GROUP of gauge transformations. What does it mean for an action to be "invariant" under a 2-group? 2-groups really want to act not on a mere set, but on a CATEGORY - and the proper notion of "invariance" is "weak invariance", i.e. invariance up to a specified isomorphism satisfying some (understood) coherence laws.This suggests that actions in higher gauge theory should really take values in a category. And so, presumably, should Lagrangians. But, what category or categories??? Some categorification of the real numbers, maybe.
The problem is, I don't see the physics pointing me towards any particular choice. Probably I'm just being dumb. It's especially galling because I already think I know what one*result* of path-integral quantizing a higher gauge theory mightbe: a 2-Hilbert space of states! I wrote a paper on 2-Hilbert spaces once....
Hmm, this suggests that the appropriate "categorified transition amplitudes" lie not in C but in Hilb!!!
> Maybe it would be good to think how the fundamental groupoid arises through Lagrangian reasoning. In the path connectedness case, we have a space X, and a Lagangian map from X to truth values, i.e. a subset of X. Then for any path in X, there is an action formed by integrating L along it. This tells you whether the path lies wholly in X. Now you form the integral over all paths with the same endpoints. This just sees whether there is any path in X between those two points.
> Up a dimension, we're looking for a set of homotopy classes of paths:
Let me postpone thinking about this, even though it sounds really cool.
> Hmm. I have a feeling this ought to be slicker. Also the first 'integrations'
> in each case were over truth values and yet were ANDs rather than ORs.
I just want to say something about *this*. This actually seems right to me. In physics, a path integral is an integral over paths of the EXPONENTIAL of the action, which in turn is obtained by integrating a Lagrangian along the path. The exponential turns addition into multiplication. In fact, it's often good to think of the "exponential of the action", as more fundamental than the action. It has a clearer meaning. In quantum physics, the exponentiated action exp(iS/hbar) tells you the RELATIVE AMPLITUDE for taking that path. In statistical mechanics there's a version where you get the RELATIVE PROBABILITY.
In the situation you're talking about, the exponentiated action is a truth value saying whether the path is continuous - i.e., the POSSIBILITY of following that path. Now about that "first 'integration'" that's bugging you. The exponentiated action is given by a "product integral" along the path. I don't know if you've thought about product integrals, but they're just like integrals with + replaced by times. I reinvented them when I was a kid so I have a certain fondness for them. Normally you get them by multiplying lots of numbers that are really close to 1, instead of adding lots of numbers that are really close to zero... but normally you can reduce them to ordinary integrals using "exp" and "ln".
You however are doing product integrals in the rig of truth values: you are computing the possibility of a certain path as a product ofpossibilities of lots of little paths! And in this case there's no "exp" and "ln" to save us - unless there's some logical operation nobody every told me about, that converts "or" to "and". It's also neat to think about product integrals in the rig of costs:the rig R^{min} = (R union +infinity, min, +infinity, +, 0). Here we compute the cost of a path as an ORDINARY integral along the path...but the ordinary integral uses +, which is really MULTIPLICATION inthe rig of costs. So, it's again a case of a product integral. And again there's no "exp" and "ln" to save us.
> Presumably homotopy theory is treating a space as though it's
>infinitely cheap to go through the space, and infinitely expensive
> to go outside. So is there a cheap way to get from x to y? Yes,
>so long as there's a path [OR] along which [AND] all points are cheap.
Right! I like to think of truth values as a funny version of the rigof costs where the only two prices are "free" and "you can't afford it". Anyway, now I should go back to your categorified version of the wholesetup:
> Up a dimension, we're looking for a set of homotopy classes of
>paths: We have a space X, paths and paths between paths.
>The Lagrangian takes paths to truth values. When we integrate
>the Lagrangian along a path between paths it tells us whether
>we can do this all within X. Then for a given path f, we integrate
>[form set union] over all paths with the same end points,
> collecting all those homotopic to f. Now we integrate [form
>set union] over all paths f, forming the union of homotopy classes.
and think of the final result as an ordinary integral of a product integral. Btw, the "state sum models" in TQFT are all done by multiplying anamplitude for each labelled simplex and then summing over labellings, so it's the same sort of deal.
> About the transformation between quantum and classical, the
> trouble I'm having is that according to your lectures Sets
> and Relations are already on the quantum side.
Yeah! But that's GOOD.
In today's talk I explained how for any rig R there's a PROP whose morphisms from x^n to x^m are n x m matrices with entries in R, with the "tensor product" of morphisms being direct sum of matrices. In other words,"finitely generated free R-modules, made into a symmetric monoidal category using direct sum"This lets us do "matrix mechanics" in the manner we've been discussing, and when R is the rig of truth values we get "finite sets and relations, made into symmetric monoidal category using disjoint union" But we can also use the rig of costs....
> Oh, I see that I'd already got the point. But isn't that
> all the same odd to call anything matrix-like 'quantum'?
Well, FinRel is a symmetric monoidal category where the product is not cartesian, and it's a *-category, so in many ways it more closely resembles Hilb than Set or FinSet. The superposition principle is a bit stunted given that the rig of truth values has just one nonzero element, but don't let that fool you.
> Putting it naively, the things in the sets of Rel are
> perfectly classical. If the mere fact that things are
> related is enough to make them quantum, isn't that a sign
> that my twins entanglement idea is right - that a chunk of
> the weirdness of entanglement is little more exciting than
> that a twin marrying 12000 miles away makes you instantly
> an in-law. Or is your worry here that this is just about
> information? But then Fuchs, Cave et al want to say this is
> really all EPR experiments are doing.
Well, the sexier features of QM probably require a more interesting rig.
I'm not sure this is the right analogy; in quantum mechanics entanglement is about tensor products of vector spaces, so in FinRel we should be looking at tensor products of modules over the rig of truth values....
Hmm, it might turn out that you're right, and that the analog of a "entangled state" boils down to a pair of sets with a relation between them. But this is something one just needs to calculate. For example, maybe a relation f: S -> T can be dualized to give a relation g: 1 -> S* tensor T just like a linear operator f: V -> W gives a linear operator g: C -> V* tensor W - which is the same as a state in V* tensor W. The identity operator f: V -> V gives a maximally entangled state g: C -> V* tensor V, so maybe we can do the same thing with relations.
But, first I'd need to figure out if there really is a tensor product of "modules over the rig of truth values" (probably), and what it is.
> 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.
Okay, very sensible. Now I can translate what you said intomath 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 wheneveryou've got a rectangular array A_{ij} of truth values that's notof 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.
2 Comments:
Thanks. Very interesting. I will have to think about this. At the moment, I am not sure that I am aware of all the details that the two of you are talking about.
I am wondering, though, if we can incorporate the generalized path integrals that you have in mind into something along the lines of integration in Lawvere's spirit, the way I have tried to indicate above.
If instead of the semi-ring $R$ of non-negative integers which Lawvere uses to describe metric geometry we use the ring semi $R_\mathrm{max}$, where addition is the supremum, then we can set up all the above generalized metric theory with this semi-ring, too.
And (unless I am making some mistake), it appears to me that when using that $R_\infty$ rig instead of $R$, and plugging it into the formula for the categorical integral which I described in the precvious comment on integration (see here) we do obtain precisely formula (1) in Litvinov's paper.
Moreover, here is one observation possibly relating the notion of integral I talked about (using limits of distance funcors over nerve categories).
In my comment on integration I considered the case of integrals over categories modelling 1D spaces, only.
But, if you look at the construction, you see that the definition of the integral is much more general.
We can also compute that categorical limit over the nerve of some category $X$ (the way I described), but throwing out all $n$-tuples of morphisms in the nerve which do not start and end at two specified points.
What the categorical integral spits out in this situation is precisely the path (of morphisms) in $X$ which maximises (if our ring $R$ or $R_\mathrm{max}$, or whatever, is regarded as a monoidal category with $\geq$ as morphisms, otherwise maximization is replaced by something more general) - which maximizes the integral of the "distance function" $d_X : X \to \Sigma(R)$ over that path.
So, I think, this concept of integral does incorporate at least
1) the ordinary one (over 1D spaces)
2) any tropical generalizations of the ordinary one
3) classical mechanics, in the sense of producing paths extremalizing some "actions".
What I cannot see right now (and I am a little bit in a hurry), if there is any lax functor $d_X : X \to S$
which would make that integral also incorporate path integrals as in statistical mechanics and quantum mechanics.
Gotta run now.
I wrote:
"What the categorical integral spits out in this situation is precisely the path (of morphisms) in X which maximises [...] "
Sorry, this is not true in general. In fact, it makes sense only in very simple special cases.
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