Harrison's Geometric Calculus | The n-Category Café

May 18, 2008

Harrison’s Geometric Calculus

Posted by Urs Schreiber

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Jenny Harrison
Lectures on Geometric Calculus
arXiv:math-ph/0501001

Morris Hirsch # is quoted as having said the following:

A basic philosophical problem has been to make sense of “continuum”, as in the space of real numbers, without introducing numbers. Weyl [5] wrote, “The introduction of coordinates as numbers… is an act of violence”. Poincaré wrote about the “physical continuum” of our intuition, as opposed to the mathematical continuum. Whitehead (the philosopher) based our use of real numbers on our intuition of time intervals and spatial regions. The Greeks tried, but didn’t get very far in doing geometry without real numbers. But no one, least of all the Intuitionists, has come up with even a slightly satisfactory replacement for basing the continuum on the real number system, or basing the real numbers on Dedekind cuts, completion of the rationals, or some equivalent construction.

Harrison’s theory of chainlets can be viewed as a different way to build topology out of numbers. It is a much more sophisticated way, in that it is (being) designed with the knowledge of what we have found to be geometrically useful (Hodge star, Stokes’ theorem, all of algebraic topology,…), whereas the standard development is just ad hoc – starting from Greek geometry, through Newton’s philosophically incoherent calculus, Descarte’s identification of algebra with geometry, with additions of abstract set theory, Cauchy sequences, mathematical logic, categories, topoi, probability theory, and so forth, as needed. We could add quantum mechanics, Feynman diagrams and string theory! The point is this is a very roundabout way of starting from geometry, building all that algebraic machinery, and using it for geometry and physics. I don’t think chainlets, or any other purely mathematical theory, will resolve this mess, but it might lead to a huge simplification of important parts of it.

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The key idea of chainlets

As Harrison discusses on p. 10, the distinguishing key ingredient and starting point of her approach is a certain new choice of norm with which to complete the vector space of polyhedral chains in $ℝ^{n}$ (definition of those on p. 15).

She calls this norm the natural norm – in muscial contrast to the flat and sharp norms on chains in $ℝ^{n}$ introduced by Whitney in the old days.

The elements in this vector space of chains completed with the natural norm are her chainlets. The precise definition of these is hidden on p. 32.

Hint for readers of the lecture notes. Maybe something went wrong in the text composition: when reading this start with chapter 1 on p. 15, which is called “Chainlets” but does not once mention the term. It ends with the definition of the natural norm on polyhedral chains at the bottom of p. 17 and with the observation that the obvious boundary operator is a bounded operator in the natural norm – in prop. 1.1 on p. 18.

Skip the standard discussion of standard differential forms in chapter 2 (unless you really don’t know yet what a differential form is) and skip for the moment also the beginning of chaper 3 and jump to corollary 3.6 on p. 32. The sentence below is the definition of chainlets. We are reminded that on them the boundary operator is continuous, which is the first goal to be achieved by the natural norm. The next paragraph states the second goal: Stokes’ theorem for the pairing of chainlets with suitable forms. Take that as the motivation and jump back to p. 30 now, where the “suitable forms” are defined as those having in turn bounded norm of certain kind (the notation for mass norm and derivative used there were defined on p. 16.)

All the norm gymnastics then ensures that the pairing appearing in Stokes theorem is well defined, with the structure of the proof being essentially trivial. Go back to the table on p. 10 to check how the natural norm is thus an improvement over the sharp and the flat norm.

A closer look at the space of chainlets is then taken from p. 33 on. A crucial statement is that on top of p. 36 (corollary 3.13), which gives an abstract characterization of the natural norm in terms of crucial properties.

The point of the natural norm and of chainlets

The point is, as far as I understand, that the natural norm cures the deficiency of Whitney’s sharp and flat norms and admits the crucial ingeredients found in ordinary differential geometry: boundary, Hodge star, coboundary to exist on chains and their duals. The crucial theorems, such as Stokes’s and Green’s (p. 55) theorem, then hold and the crucial operators, such as curl, divergence, Laplace operators (both for cochains as well as on chains, see p. 48) exist essentially by construction.

But with that in hand, it turns out that the space of chainlets and their duals is much larger than that of the smooth chains and their smooth duals. For instance, the examples on p. 33 say that chainlets may be fractals or graphs or $L^{1}$ -functions. This is then the topic of section 4: Calculus on fractals.

Infinitesimals

A crucial observation is, p. 43, that there are chainlets which are “infinitesimal”: the sequence of $k$ -dimensional chains which are linear multiples of factor $(2^{ℓ})^{k}$ with a $k$ -dimensional cube of side length $2^{- ℓ}$ does converge to a chainlet as $ℓ \to ∞$ .

This chainlet behaves like a “Dirac delta $k$ -form”, or rather the dual to that. And it is not just the limit of cubes becoming infinitesimal, but of any $k$ -volume becoming infinitesimal (prop. 4.2, p. 44).

Using such Dirac delta-like chainlets, Harrison suggests a formalism of discrete calculus in the last section, starting on p. 75. I have not yet fully absorbed that, but I need to quit for now.

Posted at May 18, 2008 1:23 PM UTC

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Re: Harrison’s Geometric Calculus

Hello Urs,

Thanks for your interest. The paper you are reading was just a draft and is now obsolete. New vistas opened up as soon as it was written. I am working hard on a monograph to reflect the current state of the theory which has gone quite a bit further. If you can hold on a little while longer, I will be pleased to produce it, or at least the first few chapters.

Regards,

Jenny Harrison
Professor of Mathematics
University of California, Berkeley

Posted by: Jenny Harrison on May 18, 2008 11:13 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Jenny!

It’s been a long time. Great to see you here!

I’m excited to see Urs has picked up your old papers again and I am really looking forward to seeing your latest stuff.

If we can learn anything from this blog, it is that you don’t need to have all the details perfectly worked out before sharing thoughts with the world. I hid in a closet for 6 years trying to work out my own stuff and then when I finally graduated and invited Urs in, we cranked out everything in a matter of weeks. I hope you really share your stuff with us. I, for one, would love to see it even in incomplete draft form!

Cheers!

Posted by: Eric on May 19, 2008 2:07 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Hi Eric,

Greetings to you, too! You were supportive of my ideas early on and I thank you for that. I have had to work quietly for the past few years to figure out several large steps, both backwards to a spring source and forwards to applications. It has been an interesting sojourn as I did not know just where the research was going, only that it was going somewhere. It felt like I was following the scent of a honeysuckle vine in vernal bloom, hidden just around the next bend. Now everything is coming together and you should not have to wait too long for a rigorous and reasonably polished rendition.

All the best,

Jenny

Posted by: Jenny Harrison on May 19, 2008 6:13 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi,

thanks a lot for your reply!

You write:

The paper you are reading was just a draft and is now obsolete.

When you say “obsolete”, what precisely do you have in mind? I understand that you have been refining your ideas since then – but do I need to entirely forget the content of Lectures on Geometric Calculus?

I have seen that you also have a more recent review: Lectures on Chainlet Geometry.

I quickly looked at that one trying to grasp some key ideas. But for that purpose I felt I had more luck with the Lectures on Geometric Calculus.

I am working hard on a monograph to reflect the current state of the theory

Good to hear! Please include an introductory section where you describe what is new about your approach, where you informally describe the key ideas and the key tools and the key motivations.

For instance, do you roughly agree with what I extracted as the key points of your approach above? If not, I’d be grateful for comments.

If you had to describe the essence of the idea behind your geometric calculus/chainlet geometry in two sentences, what would these be?

Posted by: Urs Schreiber on May 19, 2008 8:11 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Urs,

Yes, the work has evolved in significant ways since those notes were posted on the arxiv. Thank you for this opportunity to tell you what I can while preparing the monograph. (My poetic outburst to Eric was like a champagne toast to an old friend, and I will keep to mathematics henceforth :-).)

In two sentences: The idea is to work with spaces predual to differential forms, rather than dual spaces as has been the custom in the study of distributions and currents. Everything comes down to finding a suitable topology $τ$ on the space of “chainlets” (of order zero and dimension $k$ ) defined as free sections of the trivial bundle $V \times Λ_{k} (V)$ where $Λ_{k} (V)$ is the $k$ -th exterior power of the exterior algebra of a vector space $V$ .

You can think of these as formal sums of geometric Dirac deltas, supported in finitely many points of V. Each point has assigned to it a k-vector, uniquely determined by its k-direction, mass, and orientation. (There are also higher order chainlets that represent dipoles, quadrupoles, etc.)

There are recently discovered “axioms of calculus” which say just what properties the topology $τ$ should have. Once you find such a $τ$ , and examples do exist, you end up with a locally convex, complete topological vector space, predual to differential forms. Its elements are called “differential chains”. One of the axioms says that the spaces of differential chains and differential forms together form a pairing with just the right properties to give us a well behaved integral. Using other axioms, you can quickly deduce major theorems of calculus for domains of differential chains and integrands of differential forms in smooth manifolds. For a full theory, you need some additional structure such as a metric. The subspace of chainlets is dense in the space of differential chains and the restricted calculus is a discrete calculus that could well be called “quantum exterior calculus”.

There are nontrivial representation theorems following from the axioms that show that singular chains, self-similar fractals, dipole surfaces such as soap films, and vector fields are canonically represented by differential chains. Since vector fields are not canonically represented by differential forms, you can see that something different is going on. The predual approach is not simply a rehash of the classical theory of distributions and currents. The representation theorems show us that the resulting calculus unifies various viewpoints, notably the Cartan and quantum exterior calculi.

You ask if you need to forget the posted arxiv notes? There are lots of good ideas in those notes. Some of them were developed in my Berkeley courses, and the best are being folded into the monograph. However, some early choices led to blind alleys or unnecessary difficulties, and were later abandoned. I am happy to give you feedback on any of it, and will try to comment in detail on your initial posting later on today. For example, it is much easier not to start with polyhedral chains. They cause many headaches that vanish when you replace them with chainlets. Another useful tool is the Koszul complex which you can see was being rediscovered in the arxived notes. The beautiful algebra of this complex has a nice geometrical interpretation when interpreted as chainlets.

I hope this helps as a start.

Posted by: Jenny Harrison on May 19, 2008 1:49 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Jenny,

One of the reasons why I’ve been looking forward to seeing your stuff is that I have a gut feeling that it is somehow related to what Urs and I were working on.

In our paper, we focused on what we called “ $n$ -diamonds” that are topically $n$ -cube complexes. However, that was more for simplicity. The key insight due to Urs was the relation between dimensionality and the incidence number of directed edges. An $(n + 1)$ -dimensional manifold requires $n + 1$ edges emanating from each node in order for the discrete calculus to have a nice continuum limit.

I’ve “guessed” (“conjectured” is too strong a word for what I do) that for any smooth manifold $M$ , the spacetime version $M \times R$ can be “diamondated”, where the precise meaning of “diamondated” is left to be defined by someone smarter than me :)

Roughly what I mean by “diamondated” is that if you begin with any $(n + 1)$ -dimensional smooth manifold $M \times R$ and sprinkle it with “nodes”, then you can connect those nodes with directed edges in such a way that each node has exactly $n + 1$ directed edges emanating away from it and that the resulting discrete calulcus (which is defined once you have specified the nodes and directed edges) is $(n + 1)$ dimensional.

I’m hoping there is some nontrivial relation between this and what you’re doing even if only in the continuum limit.

Since we talked, I committed a grave sin and moved to Wall Street. I now work in asset management *gasp!* To atone for my sins, when I am rich and famous, I promise to sponsor a conference on n-categories. I’m working on Urs’ research trust fund now!

Best regards

Posted by: Eric on May 19, 2008 3:29 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Hi Eric,

It is good to hear your ideas have developed since we last spoke. I look forward to learning about your work with Urs, but it will have to wait until the monograph is finished. If your “topically n-cube complexes” are simplicial complexes, then they are, indeed, represented as differential chains. From the viewpoint of the latter, the topology is the `glue’ that holds everything together.

Best regards,

Jenny

Posted by: Jenny Harrison on May 19, 2008 4:54 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Oops. “Topically” was supposed to be “topologically”.

I think our motivations were aligned from the beginning but we both took different routes. I for one have no doubt I’ll learn a lot by a side-by-side comparison.

The topology on n-diamonds arises kind of mysteriously (to me anyway!). As I noted, you begin with a collection of nodes and directed edges (in an abstract sense with no embedding in any other manifold) connecting nodes. This defines a discrete (graded) calculus and the precise manner in which the connections are made defines an n-dimensional topology. In effect, you do not put a topology in by hand explicitly. Rather, it emerges from the calculus. A bit backward from standard ways of thinking of it. What comes first, the calculus or the topology?

I had absolutely no luck explaining this to Dennis Sullivan over dinner even though the construction is so simple (completely my fault). I speak “engineering” and he spoke “mathematics”. Two completely different dialects of English and only a few have mastered both :)

Posted by: Eric on May 19, 2008 6:05 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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What comes first, the calculus or the topology?

Yes, that describes the difference in the two ways of thinking very well.

I still need to better understand the application of chainlet calculus to the discrete setup. I wouldn’t be too surprised if one can go from there to here. So I am hoping to find in the chainlet concept the unifying picture which allows me to move from discrete to continuous without having to travel by foot anywhere in between. Which I guess is one of the main motivations of the development of chainlets in the first place.

One other difference in approaches is:

one might or might not want to consider the limit of the sequence appearing on top of p. 76. In the “truly” discrete setup one wouldn’t take the limit. It’s the difference between taking the derivative of a delta-distribution and taking the finite difference of two of them.

Posted by: Urs Schreiber on May 19, 2008 6:20 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Dear Urs,

You wrote,

“So I am hoping to find in the chainlet concept the unifying picture which allows me to move from discrete to continuous without having to travel by foot anywhere in between.”

I am not sure what you mean by having to travel by foot, but the concept you may be looking for is the representation theorem for a k-cell. We say that a differential $k$ -chain J “represents” a $k$ -cell $σ$ if the chainlet integral of any smooth $k$ -form $ω$ over $J$ is the same as the Cartan-Riemann integral of $ω$ over $σ$ :

(1)

\int_{J} ω = \int_{σ} ω

The obvious guess works, although the proof is not trivial. We have known this for some time using the norms of the direct limit topology, but I only just worked it out from the more general axiomatic approach a few days ago.

You also wrote, “one might or might not want to consider the limit of the sequence appearing on top of p. 76. In the “truly” discrete setup one wouldn’t take the limit. It’s the difference between taking the derivative of a delta-distribution and taking the finite difference of two of them.”

Here is a place where the obsolescence of my old paper shows. Now we know how to use the tensor product of the symmetric and exterior algebras to express the limit on p. 76 as a truly discrete object without taking a limit. It is just $v \otimes σ$ , supported at $p$ . (One can formalize this using bundle terminology.) Think of $σ$ like a vector which can be approximated by secants, or has a purely discrete version. Our $v \otimes σ$ happens to have a symmetric aspect to it. If you want a picture of what is going on, you can take a limit of differences of $k$ -elements, as on p. 76. If you want a clean, algebraic expression, use the tensor product.

Posted by: Jenny Harrison on May 19, 2008 7:50 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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One final remark and I have to stop for the day: There is also a theorem taking us from $k$ -cells down to $k$ -elements by renormalized shrinking. So we can move continuously, in and out, from the quantum discrete to the smooth continuum of cells. (I have not yet justified the use of the word `quantum’ here, but let’s leave that for another time.)

Posted by: Jenny Harrison on May 19, 2008 7:56 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Thanks, Eric, for the typo correction. It sure stopped me in my tracks!

The beauty of having a complete topological vector space is that any dense subset becomes viable for an approximation theory. Perhaps diamonds generate a dense subspace of differential chains.

The flexibility in choice of topology is useful as we eventually want to represent exponential vectors $e^{v}$ as differential chains. The axioms lead to “calculus theories” much like the Eilenberg-Steenrod axioms lead to “homology theories”. Before this development the calculus and topology were tied together.

Posted by: Jenny Harrison on May 19, 2008 7:20 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

What comes first, the calculus or the topology?

It’s obvious: Calculus.

For that question alone there ought to be a track-back between this thread and one of the various “comparative smootheology” threads.

Posted by: Andrew Stacey on May 20, 2008 8:26 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Hi Andrew,

I probably have not been able to convey enough about this new geometric calculus to explain how topological vector spaces are fundamental to the calculus. Before we declare something to be `obvious’, let us get our terms and our history straight. This is not `topology’ in the sense of algebraic topology, homotopy theory or differential topology. Of course, these rely on calculus!

Look towards distributions and you will see that Schwartz’ discovery of the LF topology on the space D of functions with compact support is what made distributions work. Without the topology, D is just a vector space and we cannot talk about its continuous dual, i.e., distributions. Our goal in geometric calculus is to form a predual to functions and we cannot do this without a topological vector space. Once we have a predual, we can ask if its topological dual is the vector space of functions we started with. Then, and only then, can we start developing calculus with the pairing that results.

There is a fascinating account in Schwartz’ autobiography (p. 226 of A Mathematician Grappling with His Century) of a meeting with de Rham and Cartan over hot chocolate in a beautiful tea room in Royat. “De Rham told us that he was hoping for a more general theory, because his present theory of currents was visibly defective. His space of currents had no natural topology, whereas all the vector spaces useful in analysis come equipped with a topology, in which they are usually complete, and also with a useful weak topology.” De Rham went on to describe the wonderful topology of the space of Radon measures and its useful properties. “He had not managed to obtain any results of this kind… We thought about it together. He made this marvelous prediction: `It’s not for us, it will be found by the next generation’.” Schwartz then relates the story of how he discovered a topology for D two years later. His topology was the ansatz for the theory of distributions and was extended to currents by de Rham. Schwartz was awarded the Fields Medal soon after this in 1950.

Regards,

Posted by: Jenny Harrison on May 20, 2008 1:27 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

[Note: Throughout this comment, the words “calculus” and “topology” are appropriate, but may refer to things slightly more abstract than the usual terms imply, e.g. “calculus” refers to a set of rules for an abstract differential (graded) algebra that may have no relation to any continuum concept. Likewise, “topology” is generally finitary and does not necessarily invoke a continuum.]

Hi Andrew,

Sorry that the point of my question was not clear.

The question of what comes first, topology or calculus, was meant to highlight an apparent difference between Jenny’s approach and the approach that Urs and I worked on. One of the curious aspects of our work was that the calculus is, indeed, more fundamental (or at least equally fundamental) than the topology. In other words, given a topology, the calculus is uniquely defined (in our framework, which goes back to Dimakis and Mueller-Hoissen). However, the opposite is true as well. Given a calculus, you can deduce what the topology must have been in order to arrive at that calculus.

The story of how Urs and I started working together was kind of fun. I somehow piqued his interest enough for him to give it some thought. I had written some papers on how a particular form of noncommutativity between 0-forms and 1-forms gives rise to the algebraic rules of stochastic calculus without ever introducing randomness or stochastic processes (which applies equally well to stochastic approaches to quantum mechanics via Nelson). Then, when it was clear that a given topology leads to a unique calculus, we asked the question of whether the reverse could be true. Given a calculus, can you deduce the topology that gives rise to that calculus? Urs then went on a bike ride across Europe and when he came back, we had both independently deduced what the topology should be in order to arrive at the (discrete) stochastic calculus.

I am pretty sure I can lay claim to being the first person to apply noncommutative geometry to mathematical finance :)

During the time we were working on this, I was exchanging emails with Jenny and was (and still am!) really excited about the work she is doing. Once we are able to get our hands on her latest papers, we’ll be able to ascertain how the two approaches may be similar and how they might differ. In both cases, it will be a learning experience and I’m looking forward to it.

Cheers

PS: As far as comparative smootheology is concerned, I’m way out of my depth there and can’t say anything about it.

Posted by: Eric on May 20, 2008 3:48 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

I am pretty sure I can lay claim to being the first person to apply noncommutative geometry to mathematical finance :)

So that’s how the stock market crashed! $pic$

Posted by: Bruce Bartlett on May 20, 2008 6:07 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Don’t look at me! :D

Posted by: Eric on May 20, 2008 7:08 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Eric: Are we on the same page in terms of what we mean by topology? Are you referring to a topological vector space? If so, what is the vector space? Is the topology that generates your calculus uniquely determined? What happens if you weaken or strengthen it? Do you lose anything?

Andrew: I would be interested to know what you meant by topology in your message.

I see we are both graduates of Warwick! I will be giving a seminar there on quantum exterior calculus next month.

Posted by: Jenny Harrison on May 20, 2008 6:17 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Jenny,

I’m sure you’ll remember my obvious lack of math skills, so you probably won’t be too surprised if I don’t have good answers for you. Hopefully, if I say enough, Urs can swoop in and clarify things (although he is understandably very busy with other things!).

If you haven’t had the pleasure of interacting much with Urs yet, you might enjoy this article by Bruce:

What I learned from Urs

If I could form a dream team, I’d put you and Urs together in a room for a couple of hours and have no doubt that some cool stuff would emerge.

I am suddenly having flashbacks from the dreadful experience at dinner with Sullivan, “Mathematics is like a language and you just gave me a sentence without a verb.” Ouch. That is a direct quote :)

I suck at explaining this and whenever I try, I end up resorting to an explanation of what we did. Essentially, we proclaim that the rules

$d (ab) = (da) b + (- 1)^{∣ a ∣} a (db)$

and

$d^{2} = 0$

are Cartan (or any other deity of your choosing… Leibniz?) given and see where this leads us. If you apply these Cartan-given rules to a directed graph after specifying a fairly obvious product of nodes and directed edges, then you get essentially our paper (on the Arxiv), whose basic idea is essentially attributed to Dimakis and Mueller-Hoissen.

Depending on the way the directed edges recombine, the Cartan-given rules tell you that certain higher dimensional objects vanish while others do not. Consequently, this calculus “could” tell you about the topology and you would get topological vector spaces from that I suppose, but we did not go down that road. In the typical roadrunner fashion, Urs worked out details I had spent years (unsuccessfully) on and moved onto other things leaving me in a cloud of dust *beep beep!* :)

What I can say is that the topology is finitary and is unique for a given calculus and strengthening it or weakening will effectively alter the corresponding calculus. The two are essentially inseparable.

I have no doubt what I just described is mysterious and completely devoid of any mathematical rigour, but I hope it conveys some information that might help.

Posted by: Eric on May 20, 2008 8:05 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Eric,

You want stability, so if you tweak the topology a little, the calculus should only change a little. In chainlets, this shows up as slightly different continuity of the integral with respect to both domain and integrand, as well as slightly different completions. A weaker topology gives more domains. You do not want such a weak topology that the space of domains (chains) is so large that the dual space of forms is tiny and thus of little use. What kinds of domains do you find in your topology? Our chains include charged particles, singular chains, dipoles, vector fields, fractals, graphs of $L^{1}$ functions, Moebius strips, Cantor sets, submanifolds with corners or cusps, and triple branched soap films. You also want the main operators, such as boundary and geometric Hodge star, to remain intact. Do you find this happening in your theory? Finally, why did you choose diamonds? Did this come from the light cone structure? What do you get that is not possible through other methods? What do you have to give up, if anything?

I met John Baez for the first time today and got a chance to talk over a cup of coffee. I have learned so many things from him through his blog and numerous other postings. He was the first blogger, was he not? He gave a nice talk today at UC Berkeley on groupoids with a new commutation relation with physical realism.

Posted by: Jenny Harrison on May 21, 2008 2:48 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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You want stability, so if you tweak the topology a little, the calculus should only change a little.

Yep yep :) Although I am horrid at maths, I was a fairly decent engineer whose expertise was numerical analysis, so stability is certainly of utmost important to me. Both stability of any numerical results as well as stability of the general formalism. I believe that our stuff satisfies this requirement. A slight change in topology will induce only a slight change in the calculus.

In chainlets, this shows up as slightly different continuity of the integral with respect to both domain and integrand, as well as slightly different completions. A weaker topology gives more domains. You do not want such a weak topology that the space of domains (chains) is so large that the dual space of forms is tiny and thus of little use.

Again, I can’t wait to compare notes :)

What kinds of domains do you find in your topology?

The domains we considered in our paper are what we called “n-diamonds”. They are essentially “directed n-cubes”. To visualize a simple 2-d case, consider a square grid, i.e. checkerboard, with each horizontal edge directed to the right and each vertical edge directed up. Then rotate the grid 45 degrees. What you have is a “binary tree” whose cells look like a bunch of little mini light cones.

A 3-diamond is a “directed 3-cube” where the directed edges represent a flow across the diagonal of the cube. And so on and so forth. An n-diamond is basically a “directed n-cube”.

I am about 65% sure that any smooth manifold of the form $M \times R$ can be decomposed into n-diamonds like this, but I lack the skills to prove or disprove it.

Our chains include charged particles, singular chains, dipoles, vector fields, fractals, graphs of L 1 functions, Moebius strips, Cantor sets, submanifolds with corners or cusps, and triple branched soap films.

I understand that one of the many features of your stuff is that it can naturally handle fractal domains. That is interesting. Our stuff naturally encompasses stochastic processes, which trace out fractal paths. That is another reason I suspect some interesting links between the two approaches.

You also want the main operators, such as boundary and geometric Hodge star, to remain intact. Do you find this happening in your theory?

Yep yep :) This is the old “wish list” you and I discussed so many years ago. All the pieces are there in our approach. The Hodge is one place that deviates from earlier work by Dimakis et al. The idea is quite beautiful in my opinion and due 100% to Urs. You start with a bare bones Hodge that is more topological in nature, but the truer geometrical Hodge becomes a simple deformation. These deformations are very interesting and spawned all kinds of interesting discussions with Urs on the “String Coffee Table”, which was a pre-cursor to the “n-Category Cafe”, that went far beyond this discrete geometry stuff.

Finally, why did you choose diamonds? Did this come from the light cone structure?

The choice of diamonds fell on our lap. Urs noticed that in order for the discrete calculus to have a nice continuum limit, you need $n$ edges directed away from each node. At first, I didn’t like this because I was thinking of a splitting of spacetime into space and time. In 2-d, I thought the edges should be along the space axis and along the time axis separately. If Urs was right and if edges were aligned the way I thought, it would introduce anisotropy in the grid because all space-like edges are pointing in the same direction. The “EUREKA” moment occurred when we thought to rotate the grid so that time flowed along the diagonals. The anisotropy was gone and the edges took on an interpretation as being light rays.

What do you get that is not possible through other methods?

We were trying to develop a “discrete” calculus that satisfied the Cartan-given rules

$d (ab) = (da) b + (- 1)^{∣ a ∣} a (db)$

and

$d^{2} = 0$

prior to taking the continuum limit and which agreed with the continuum versions upon taking the continuum limit. Diamonds are the only shapes that satisfy this requirement as far as I know. It wasn’t a choice, but a consequence.

What do you have to give up, if anything?

If my “guess” is right and any smooth manifold of the form $M \times R$ can be “diamondated”, then our framework handles a huge chunk of general differential geometry and we aren’t really giving up anything that I can tell.

It is interesting that our approach naturally forces “time” into the equation. We didn’t set out for it to be that way. Rather it was a natural consequence of the “axioms” we began with. One way to think of it is that to integrate a 1-form requires one click of the clock. Integration and motion are inseparable concepts, which seems slightly profound.

I suspect that one of these days Urs is going to come out with some grand sweeping blog post where he recasts everything as some completely natural application of n-categories. That day I will be very happy :)

I met John Baez for the first time today and got a chance to talk over a cup of coffee. I have learned so many things from him through his blog and numerous other postings. He was the first blogger, was he not? He gave a nice talk today at UC Berkeley on groupoids with a new commutation relation with physical realism.

Cool. *Psst* Don’t tell him this, but I’ve always considered John to be one of the greatest scientists on the planet. When I write my memoirs someday, I’ll recall the night my wife and I took him out for drinks. “Me?! I had drinks with John Baez?!” :)

I’m glad for his sake Helen came along because she is a lot more entertaining than I am :)

Posted by: Eric on May 21, 2008 6:17 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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I finally thought of a simple one line description of what chainlets are all about.

Chainlets are the study of distributions without the test functions.

They rely on a geometric/algebraic definition of Dirac deltas and its derivatives which are used as the basic elements, the building blocks, of the calculus. They rigorize the atoms of the Greeks, the monads of Leibniz, and fluxions of Newton. They encode algebraic structures at a single point which enrich the smooth continuum. Formally, they are defined using the tensor product of the symmetric and exterior algebras at each point in space. This is what I had told Hirsch about that led to his marvelous quote cited at the start of this thread.

The role of weak derivative in distribution theory is replaced by a geometric chainlet operator called `prederivative’. This dualizes to Lie derivative. The boundary operator $\partial$ in chainlets is a derivation and satisfies $\partial^{2} = 0$ , so we retrieve the Cartan calculus on restriction to the subspace of singular chains.

Any dense subspace can be used for approximations. There are many choices, including fractal chains and simplicial chains. (This means some fractal-based creatures on another planet might devise a calculus with domains consisting of curves that are arcs in the van Koch snowflake and end up with Stokes’ theorem, the divergence theorem and all the rest of the Cartan theory as a special case. They might not ever discover smooth domains at all, or think of them as contrived if they did!) Your diamonds should be there, too, and useful because of the light cone.

Bye for now,

Posted by: Jenny Harrison on May 21, 2008 1:32 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Thanks, that’s very helpful. Sounds nice.

Sorry for not having replied more here since I started the thread. Too many distractions.

So it seems you have a theory of “distributions” with products, is that right? Ordinary distributions not having a good notion of products is regarded as one of incarnations of the mystery of quantum field theory. So if you have a notion with products…

Posted by: Urs Schreiber on May 21, 2008 6:10 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Urs,

Thanks for coming back! It has been very helpful to try and explain this to newcomers to the theory who hang out in this cafe. My main audience has been a close circle of confidantes for quite some time. Now, with the new axioms in place, I have restarted the monograph, yet again, but this time the entry door is clear and the reader does not have to slog through 70 pages of a newfangled example of a “theory of calculus” to get somewhere interesting. This was an understandable obstruction to many people, but it did give me space and time to continue to quietly develop an independent viewpoint. (Creative ideas come more easily on my own.) The theory has never stood still. Each development led to new ones, day after day. It has taken deconstructing each and every definition and mantra of mathematics and physics, and then reconstructing it all with the pieces that were natural (in the sense of category) and physically meaningful (in some intuitive sense). This has taken many years and was not the result of one moment of epiphany.

If this all hangs together in the way it seems to be going, it will be related to other important topics that others have been developing. I very much want to read your papers right away so that I can respond to your comments, but I am trying hard to finish the first part of the monograph for a review panel and lectures I will soon give in England. I did not anticipate this public display of affection :-)

There is no product on distributions, but there is a product on an interesting subspace of distributions, namely the space Ch of chainlets of arbitrary order and dimension. These represent Dirac deltas and their derivatives, but give them additional dimensional structure not usually seen.

There is a spectral triple in this theory. Ch has an inner product, and with a suitable choice of topology, the Hilbert space Ch is a subspace of differential chains. Ch has a product and possibly a coproduct. (I have drafted a proof of the latter, but not lived with it long enough to make an announcement. So this is not meant to be a claim, by any means. It seems strange to write something here that is so open and I don’t know the unwritten rules. Generally, I try to be extremely careful to use rigorous mathematics to back everything up.)

The operator algebra O is generated by five fundamental operators of calculus $(e_{v}, e_{v}^{*}, P_{v}, F_{*}, f)$ where $e_{v}$ is exterior product, or extrusion, $e_{v}^{*}$ is projection, $P_{v}$ is prederivative, $F_{*}$ is pushforward and $f$ is multiplication by a function. The axioms say that these are all continuous in the topology. The algebra O contains many other useful operators (e.g., boundary $\partial$ , perp $⊥$ and their combinations) and subalgebras (e.g., Clifford algebra, the noncommutative algebra generated by pushforward operators). There is a natural Dirac operator, but variations of it might be of use.

This is a sample of what is going on, and I hope you will bear with me while I write it up carefully.

Best regards,

Jenny

p.s. Eric, Please reserve one of your diamond chainlets for me!

Posted by: Jenny Harrison on May 21, 2008 9:57 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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the space Ch of chainlets of arbitrary order and dimension. These represent Dirac deltas and their derivatives, but give them additional dimensional structure not usually seen.

Good. So $Ch$ is the space of “delta-distributed $n$ -poles”, right? Point charges, point dipoles, point “tripoles”, etc.

There is a spectral triple in this theory. Ch has an inner product, and with a suitable choice of topology, the Hilbert space Ch is a subspace of differential chains.

Oh, I see. Is that Hilbert space separable? Probably not, right?

What’s the inner product between two monopoles sitting at different points? It vanishes I suppose?

For physical purposes non-separable Hilbert space are a bit dangerous, I think.

It seems strange to write something here that is so open and I don’t know the unwritten rules.

I don’t either. My personal rule is that I like exchange about math topics and that here I get it. If you ever type anything which you later feel you would rather have removed due to whatever reason, just drop me an email and I delete your comment.

Generally, I try to be extremely careful to use rigorous mathematics to back everything up.

Sure. But talking about it may help people understand the ideas behind the rigour.

There is a natural Dirac operator,

Yes, the “Hodge-Kähler-deRham-orwhateveronecallsit” Dirac operator which on cochains reads $d \pm d^{†}$ .

In much of physics what is important is the “chiral” version of this guy, where one chooses a decomposition of the exterior bundle into the tensor product of two spin bundles (if possible) and then discards one of the factors, leaving just an “ordinary” Dirac operator. That’s where much of the all-important anomaly business comes from. Did you think about that from within your formalism?

I hope you will bear with me while I write it up carefully.

I certainly will. I am grateful for having the chance to talk with you about this stuff here. I am very interested in your chainlets (diamond or not :-) but only beginning to appreciate the technical details.

Posted by: Urs Schreiber on May 22, 2008 11:38 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Urs,

The main reason I started looking at an axiomatic approach was so that the choice of topology would become more flexible. The one space that we had spent so much time on (a predual to differential forms $ℬ$ ) contained the inner product space of chainlets, but not the entire Hilbert space $Ch$ . The goal was to weaken the topology just enough to capture $Ch$ , but not so much as to lose too many differential forms. If such a topology $τ$ is found (and I have a good candidate in mind), then the Hilbert space $Ch$ is separable in $τ$ .

The topology $τ$ is the `glue’ that keeps things together so that this theory does not result in a free qft without `communication’ between the particles. (This is not meant in some hand wavy fashion as it may sound.) The inner product provides an extra structure that is useful for an $ℓ^{2}$ theory which kicks in at small distances, assuming this new topology works as planned. We also need the Hilbert space $Ch$ to be contained in the topological space of $τ$ to solve certain problems, such as solutions to differential equations and in spectral theory.

A subspace dense in $Ch$ in the $τ$ topology is provided by k-chainlets with rational data – finite sums supported in points with rational coordinates, and k-vectors with rational mass, rational k-direction and with rational directions of infinitesimal displacement.

Yes, there are $n$ -poles – in particular, we have representations of those interesting tripoles that show up in the standard model, as well as monopoles and dipoles. There are bosons and fermions represented, with creation and annihilation operators, dare I add? Some of these operators are continuous in $τ$ and thus extend to differential chains such as submanifolds, soap films, and fractals, but others are not.

The inner product between two monopoles sitting at different points vanishes. We have

delta times delta = delta

for 0-dimensional Dirac deltas if they are of unit mass and supported in the same point. The inner product is defined for the exterior algebra factor of the chainlet using determinant, whereas the inner product for the symmetric algebra factor uses the permanent of a matrix.

What was hard in this theory was to spend sufficient time on the structure at a single point to tie the infinitesimal geometry and algebra together, and then to move into a vector space, blending in analysis and topology with the geometry and algebra in a new way that retrieves the classical theories in the limit. It was hard because so many venerated mantras have pointed in other directions for so many years. It would have been much easier to jump ahead and go to the fringes with some of the new tools. There was a lot of pressure for us to do this. Now the basic tools all seem to be in place, it is not hard at all to generalize everything to various types of manifolds and bundles. So, yes, one can define the covariant (i.e., chainlet) Dirac operator on spinors, and revisit characteristic classes to see if the covariant view can deepen our understanding of obstructions to globalizations.

Thanks for the link which led me to Dan Freed’s paper. It looks very interesting and is now on the top of my `to read’ stack. It reminds me to tell you that there is a homology theory for chainlets that is the same as singular homology in smooth manifolds, but distinguishes the topologist’s sine circle from the smooth circle. It is an extraordinary homology theory, satisfying the Eilenberg-Steenrod axioms with the exception of dimension. (A dipole cycle supported at a point is the key to understanding the topologist’s sine circle.) Perhaps it should be called `differential homology theory”.

We have basically rewritten and simplified large parts of differential topology and functional analysis. There is a chapter on measure theory, too. Most of this has already been taught to three classes of undergraduates and graduates at Berkeley. It is getting to be simple enough with the new axiomatic approach that I believe we can teach it to undergraduates with only linear algebra and basic analysis. (Pedagogy has not been my main goal, but it has been a great thrill to give a rigorous, mathematically complete presentation to beginning math students, taking them from the first definition of a vector space to many of these topics in the course of one academic year. We always tried to find proofs that fit on one blackboard and almost always succeeded.)

It has been a pleasure to meet you here and I am happy to answer any of your questions.

Posted by: Jenny Harrison on May 23, 2008 2:38 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Jenny said:

The operator algebra $O$ is generated by five fundamental operators of calculus $(e_{v}, e_{v}^{*}, P_{v}, F_{*}, f)$ where $e_{v}$ is exterior product, or extrusion, $e_{v}^{*}$ is projection, $P_{v}$ is prederivative, $F_{*}$ is pushforward and $f$ is multiplication by a function.

When you describe the operator $e_{v}$ as an extrusion, is that in an infinitessimal sense or in a finite sense? We didn’t discuss our algebra this way in our paper, but there is a real sense in which our operator algebra may also be interpreted as extrusion.

A 0-diamond $e^{i}$ is (dual to) a point. A 1-diamond $e^{ij}$ is (dual to… I will stop saying “dual to” from now) a directed edge extending from $e^{i}$ to $e^{j}$ .

Here’s where it gets only slightly tricky…

A 2-diamond $e^{ijl}$ is a directed 2-cube extending from $e^{i}$ to $e^{l}$ . What does that mean???

There are 4 nodes in a 2-diamond:

$e^{i}$ , $e^{j}$ , $e^{k}$ , $e^{l}$ .

There are 4 directed edges in a 2-diamond:

$e^{ij}$ , $e^{jl}$ , $e^{ik}$ , $e^{kl}$ .

Note that if you draw a square and label the nodes and directed edges as indicated above, you will see a “flow” around the boundary from node $e^{i}$ to $e^{l}$ . This is what I mean by a “directed 2-cube extending from $e^{i}$ to $e^{l}$ ” and is what we call a “2-diamond” $e^{ijl}$ .

We never made it explicit in the paper, but there are two senses of orientation for $n$ diamonds: 1.) an $n$ -direction, and 2.) an $(n - 1)$ orientation.

The direction of a 1-diamond is obvious. The direction of a 2-diamond is as I described above, which extends to $n$ -cubes in a manner that I think is obvious.

Orientation is only slightly less obvious…

The 0-orientation for a 1-diamond is simply an assignment of +1 or -1 to the edge.

Note that for the 2-diamond there are 2 distinct 1-paths:

1.) $e^{i} \to e^{j} \to e^{l}$

and

2.) $e^{i} \to e^{k} \to e^{l}$

The 1-orientation for a 2-diamond is an assignment across the body of the 2-diamond connecting path 1.) to path 2.)

If you draw this, it will look like a 2-morphism in a 2-category. Coincidence?!

As a result of this 1-orientation for 2-diamonds, we found that

$e^{ijl} = - e^{ikl}$ .

This was not put in by hand. It is Cartan-given and results simply from taking the Leibniz rule and $d^{2} = 0$ seriously (see the first paragraph on page 12 of our paper).

As you will have noticed, I’ve blabbered for several paragraphs and still have not made the connection to “extrusion”. This is a frustration of the limitations of “one dimensional language” that Baez has written about. This stuff is trivial in pictures! Too bad human language evolved in 1-d! Maybe in 100 years we will have corrected this, but I digress :)

Anyway…

The product of a 0-diamond $e^{i}$ and a 1-diamond $e^{ij}$ is

$e^{i} e^{ij} = e^{ij}$ .

This may be interpreted as “extruding” $e^{i}$ along $e^{ij}$ .

The product of a 1-diamond $e^{ij}$ and a 2-diamond $e^{jl}$ is the 2-diamond $e^{ijl}$ , i.e.

$e^{ij} e^{jl} = e^{ijl}$ .

This may again be interpreted as “extruding” $e^{ij}$ along $e^{jl}$ .

In an $n$ -diamond complex where each cell is an $n$ -diamond suitably connected, the product of an $n$ -diamond with a 1-diamond is an $(n + 1)$ -diamond essentially obtained by extruding the $n$ -diamond along the 1-diamond.

Re: Separability

If we did put a topology overlaying our stuff, it would be finitary and non-separable (otherwise it would be discrete!), which I think might be related to Urs’ question regarding separability (?)

Posted by: Eric on May 22, 2008 6:00 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Jenny and Eric,

sorry for not having participated more so far. One quick remark:

in everything that Eric said it is important to keep in mind that it is about differetial algebra on graphs. Possibly even finite graphs.

So this is not about handling and/or generalizing delta-distributions – it’s all about Kronecker deltas only! :-)

The starting point here, the questions one asks and is motivated by, is very different from the one in Jenny Harrison’s work. The main and crucial insight for the things Eric talked about is: the choice of a lift of the algebra of functions (no topology) on a finite set to a differential graded algebra (associative but not necessarily graded-commutative) is a choice of graph structure on that set. Essentially, for $δ_{i}$ the unit function supported on the $i$ th vertex, you have to choose whether the degree 1- element $δ_{j} (d δ_{i})$ is supposed to be non-vanishing in the DG-algebra lift or not. So it’s a choice of edges between vertices $i$ and $j$ .

The diamond concept comes in much later when one tries to extend a DG-algebra obtained from a graph this way into a spectral triple by first choosing an inner product on the DG-algebra, then taking the Dirac operator to be the differential plus its adjoint. It turns out that if the graph is cubical and $n$ -valent there is a “canonical” choice here, and that choice is pseudo-Riemannian and makes all edges light-like.

Posted by: Urs Schreiber on May 21, 2008 6:26 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Ah ha!

I am beginning to see the light. Thanks Jenny and thanks Urs. I am even more convinced there must be some relation and our stuff may in fact fit nicely into Jenny’s more general framework.

Urs and I never could agree on notation. He likes to write $δ_{i} (d δ_{j})$ to emphasize the distributional aspects (?), whereas I used to like to write $e (i) de (j)$ to emphasize the algebra aspects where $e (i)$ are basis elements. For some reason, I liked to multiply $e$ ’s rather than $δ$ ’s. It doesn’t matter, but the $δ$ ’s do help emphasize that we are multiplying and taking derivatives of distributions, but rather “finitary” derivatives of “finitary” distribitions.

A 0-form for us can be written as

$ϕ = \sum_{i} ϕ (i) e (i)$

(or as Urs might prefer $ϕ = \sum_{i} ϕ_{i} δ_{i}$ ). [Note: I am using inline equations because I am at the mercy of a corporate firewall and do not have the luxury of choosing a browser that handles MathML nicely.]

The product of $e$ ’s is about as obvious as it could be

$e (i) e (j) = δ_{ij} e (i)$ .

That is another reason I didn’t want to use $δ$ ’s since sometimes we really do want to use the traditional Kronecker delta.

The dual space is spanned by elements $e^{*} (i)$ and integration is defined by

$\int_{e^{*} (j)} e (i) = δ_{i, j}$ .

Therefore, we see that letting

$S = \sum_{i} e^{*} (i)$

we have

$\int_{S} e (i) ϕ = ϕ (i)$

so that the $e (i)$ ’s can certainly be thought of as distributions and we have essentially assumed multiplication exists from the beginning.

Jenny said:

Chainlets are the study of distributions without the test functions.

They rely on a geometric/algebraic definition of Dirac deltas and its derivatives which are used as the basic elements, the building blocks, of the calculus. They rigorize the atoms of the Greeks, the monads of Leibniz, and fluxions of Newton. They encode algebraic structures at a single point which enrich the smooth continuum.

This sounds quite similar to what we did, but in a finitary setting, as Urs mentioned. A difference appears to be that our derivatives of ditributions span two points introducing some nonlocality. Distributions form the basis for 0-forms and derivatives of distributions form the basis for 1-forms. Higher forms are generated from these.

If you are talking about derivatives of Dirac delta’s at points, then that makes me think there may be some relation to synthetic differential geometry. It would be quite fascinating to find that a chainlet is the continuum limit of a diamond. Then we can start a jewelry store and sell diamond chainlets :)

All this must be tied together somehow. Fun stuff!

Posted by: Eric on May 21, 2008 7:20 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

PS: I’m not sure if I ever sent you these earlier papers. They are written in the “engineering” dialect, but I think also help convey the basic ideas:

Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance
May 2002

Abstract:

The present report contains an introduction to some elementary concepts in noncommutative differential geometry. The material extends upon ideas first presented by Dimakis and Meuller-Hoissen. In particular, stochastic calculus and the Ito formula are shown to arise naturally from introducing noncommutativity of functions (0-forms) and differentials (1-forms). The abstract construction allows for the straightforward generalization to lattice theories for the direct implementation of numerical models. As an elementary demonstration of the formalism, the standard Black-Scholes model for option pricing is reformulated.

Financial Modelling Using Discrete Stochastic Calculus
October 2004

Abstract:

In the present report, a review of discrete calculus on directed graphs is presented. It is found that the binary tree is a special directed graph that contains both the exterior calculus and stochastic calculus as different continuum limits are taken. In the latter case, we arrive at something that may be referred to as “discrete stochastic calculus.” The resulting discrete stochastic calculus may be applied to any stochastic financial model and is guaranteed to produce results that converge in the continuum limit. Discrete stochastic calculus is applied to the Black-Scholes model for an illustration. The resulting algorithm generated by discrete stochastic calculus agrees with that of the Cox-Ross-Rubinstein model, as it should. The results presented here are preliminary and are intended to encourage others to learn discrete stochastic calculus and apply it to more complicated financial models.

Posted by: Eric on May 21, 2008 7:46 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Hi Jenny,

two quick questions:

a) When you say “quantum exterior calculus”, is the word “quantum” just supposed to be suggestive? And how suggestive is it really? I might not see clearly on your objectives here, yet.

b) In this comment you mention the “tripoles that show up in the standard model”. I am not sure I know what you mean by that. Is it more than saying that 1-particles in 3+1 dimensions have “order three” delta distributions in your formalism (sorry, I won’t have said this correctly, but I guess you see what I am asking).

Posted by: Urs Schreiber on May 26, 2008 6:33 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Hi Urs,

When you say “quantum exterior calculus”, is the word “quantum” just supposed to be suggestive? And how suggestive is it really?

There are features of quantum mechanics and quantum field theory emerging from this viewpoint, but it is not yet clear how far it will go. This is a work in progress. The change in variance effects everything (commutation relations, Fock spaces, axioms of quantum mechanics, etc) so everything has to be carefully checked as if it were brand new. There is many a slip twixt cup and lip!

What we do have is a new theory of calculus with a new integral and derivative that unifies the Cartan theory and various discrete theories. It has a new discrete theory within it that may deserve the name `quantum exterior calculus’. This should be clear soon, once the new topology is woven in and actually does what it is supposed to do.

you mention the “tripoles that show up in the standard model”. I am not sure I know what you mean by that. Is it more than saying that 1-particles in 3+1 dimensions have “order three” delta distributions in your formalism

This is what we are talking about, but we have taken it a little further, another work in progress.

Posted by: Jenny Harrison on May 27, 2008 5:21 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

The prederivative operator takes monopoles to dipoles, dipoles to quadrupoles, etc. This is continuous on chainlets and thus extends to a continuous operator on differential chains such as surfaces and fractals. There is also a tripole operator on chainlets that takes monopoles to tripoles. This also seems to be continuous. (If it is, then there should be a dual tripole operator on forms.) When we combine these with projection operators in Hodge theory, also continuous in chainlets, we get interesting fields.

Posted by: Jenny Harrison on May 29, 2008 12:23 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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If In understand correctly, the “tripoles” that we are talking about are essentially like duals to 3-forms which are supported on a single point. And you want to think of them as descibing, for instance, the charge distribution of some charaged particle sitting at that point.

If we were talking about ordinary 3-forms, these would be called the electric/magentic current 3-forms of the corresponding electrically/magnetically charged particle distribution.

Posted by: Urs Schreiber on May 29, 2008 1:10 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

The tripoles I am thinking about are different. To get the idea, take the three vertices of an equilateral triangle and place at each of them a 3-vector with mass 2/3, 2/3, -1/3, respectively, to form a 3-chainlet. We take a limit of such chainlets (with mass adjusted) as the side length of the triangle tends to zero. If we can convert this into a continuous operator on chainlets (still have to check), then the dual operator on forms is well defined and would be a kind of three difference derivative. This should generalize to n-poles. (The tripole operator on forms is only hours old, and I hope I have sufficient caveats in place to gracefully withdraw if it should it not work out :-). ) What gives me optimism is that the prederivative operator can be defined in a similar fashion if you replace the triangle with a 1-cell to get dipoles. The dual to prederivative of chains is the Lie derivative of forms.

Posted by: Jenny Harrison on May 29, 2008 5:11 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

I should have said that the 3-vectors have mass 2/3, 2/3, 1/3, with the first two given positive orientation and the last one negative.

Posted by: Jenny Harrison on May 29, 2008 5:20 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

Is it possible that the masses should be 1/3, 1/3, and 2/3 with the last one having negative orientation?

If so, this “three difference derivative” would have some relation to a “discrete diffusion operator” a.k.a. “discrete heat equation”. I could ramble on a bit about why this would be interesting, but will wait to verify the masses.

“How can I go forward if I don’t know which way I’m facing?”
- John Lennon

Posted by: Eric on May 30, 2008 4:24 PM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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I just lost a fairly long comment because the server failed to respond, and I have now run out of time.

In quick summary, this idea seems to extend to a new type of k-pole differential of j-forms, a kind of Lie derivative wrt integrable k-vector fields replacing the 2-vector of the triangle in the example above, and j-elements replacing the 3-elements at the vertices. (In the example, there was a typo and the sum of the masses should be zero. Thanks for pointing this out. The example, as it is written, was developed for proton representations.) The integer k can vary on iteration making all sorts of new PDE’s possible. In my chainlets undergraduate class a couple of years ago, we solved the Dirichlet problem for nonsmooth surfaces in n-space using chainlet methods that may be extendable.

I need to slow down, though, because the rigorous mathematics needed to make this all into a bona fide new theory has to be developed carefully. There are numerous new tools available, but there are also some nontrivial questions that have to be dealt with.

But he who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in Divinity. Bishop Berkeley, The Analyst, 1734

Posted by: Jenny Harrison on May 31, 2008 1:04 AM | Permalink | Reply to this

Re: Harrison’s Geometric Calculus

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Interesting.

So now that we verified the oriented sum of masses is zero, I’ll outline the relation to the discrete heat equation. I’m not sure if this will be of interest to you.

Let our chainlet be denoted

and (linear) functional evaluation denoted simply

f(c)

which could just as well have been denoted by $〈 f, c 〉$ or $\int_{c} f$ . In any case, we have

f(c) = 1/3 f(-1,0) + 1/3 f(1,0) - 2/3 f(0,1).

Now introduce a “vacuum chainlet” or a “virtual chainlet”

() = 2/3 (0,0) - 2/3 (0,0),

which has the obvious property

Therefore,

$f (c) = 2 / 3 {1 / 2 [f (- 1,0) - 2 f (0,0) + f (1,0)] - [f (0,1) - f (0,0)]}$

The condition

$f (c) = 0$

leads to the (time-reversed) heat equation

$(1 / 2 \nabla^{2} - \partial_{t}) f = 0$

in an appropriate continuum limit where the first coordinate is labelled “x” and the second coordinate is labelled “t”.

If that is even remotely interesting, I’m happy to continue the line of thought. There is more that could be said, but I’m afraid I might be talking to myself already.

Cheers

Posted by: Eric on May 31, 2008 5:21 AM | Permalink |

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