## May 6, 2008

### Quaternionic Analysis

#### Posted by David Corfield

Nobody else has mentioned it, but perhaps a few extracts from a paper by one of the founding fathers of categorification, Igor Frenkel, might be of interest, even if not on our topic:

Quaternionic Analysis, Representation Theory and Physics, Igor Frenkel and Matvei Libine.

Abstract
We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.

They say

Unfortunately…the quaternionic analogues of the ring structure of holomorphic functions, local conformal invariance, Riemann mapping theorem and many other classical results of complex analysis have never been found. Such a failure has even led R. Penrose to say, “[Quaternions] do have some very significant roles to play, and in a slightly indirect sense their influence has been enormous, through various types of generalizations. But the original ‘pure quaternions’ still have not lived up to what must undoubtedly have initially seemed to be an extraordinary promise… The reason appears to be that there is no satisfactory quaternionic analogue of the notion of a holomorphic function.”

But,

The Minkowski space reformulation of quaternionic analysis brings us into a thorough study of Minkowski space realization of unitary representations of the conformal group SU(2, 2) by H. P. Jakobsen and M. Vergne who were motivated by the program of I. E. Segal on the foundational role of representation theory of SU(2, 2) in physics.

Irving Segal was John’s PhD. supervisor.

Furthermore,

Another unique feature of quaternionic analysis is its deep relation to physics, in particular, to the four-dimensional classical and quantum field theories for massless particles…We have already mentioned the implication of the Poisson formula…to the spectral decomposition of the Hamiltonian of the hydrogen atom. In general, the Minkowski formulation of various results of quaternionic analysis provides a link to the four-dimensional field theories. This is hardly surprising since the Minkowski space is the playground for these physical theories, but it is still quite remarkable that we encounter some of the most fundamental objects of these theories. It is certainly clear that the equations for the left- and right-regular functions (4) and (5) are nothing but the massless Dirac equation. But it comes as a surprise that the quaternionic analogue of the Cauchy formula for the second order pole (8) is precisely the Maxwell equations for the gauge potential. Moreover, the integral itself appears in the Feynman diagram for vacuum polarization and is responsible for the electric charge renormalization. Also, the quaternionic double pole formula in the separated form has a kernel (14) represented by the one-loop Feynman integral. There is no doubt for us that these relations are only a tip of the iceberg, and the other Feynman integrals also admit an interpretation via quaternionic analysis and representation theory of the conformal group. In fact, we make some explicit conjectures at the end of our paper. Thus we come to the conclusion that the quaternionic analysis is very much alive and well integrated with other areas of mathematics, since it might contain a great portion – if not the whole theory – of Feynman integrals. On the other hand, the latter theory – a vast and central subject of physics – might not seem so disconcerting and unmotivated anymore for mathematicians, and many of its beautiful results should be incorporated in an extended version of quaternionic analysis.

Posted at May 6, 2008 4:35 PM UTC

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### Re: Quaternionic Analysis

Thanks a lot for mentioning this.

I don’t have the time to look into this article at the moment, but from the quotes you give I get the impression that it ought to be true that the excitement that the authors are expressing is essentially about the same kind of phenomena that drives the excitement that practitioners in what is called “Geometric Algebra” are notorious for.

Lectures on “geometric algebra” typically emphasize a lot the naturalness with which Maxwell’s equations and Dirac’s equation arise when making use of quaternionic structures.

As one can probably already see from these links, practitioners in “geometric algebra” tend to have a stronger tendency to invest energy into pedagogics and physics than into heavy-duty research math, which may be the reason why some of the very good ideas developed there are having a hard time propagating meme-wise in the pure-math community.

Maybe with Frenkel this is changing now. But I haven’t even looked at his article yet, so don’t trust me.

Posted by: Urs Schreiber on May 6, 2008 5:56 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

I noted this paper when it came out. I would agree with Urs about the Geometric Algebra angle. By the way, I am a huge geometric algebra fan, and I’m in good company because Urs was also mightily impressed by it back in the sci.physics.research days.

I don’t understand the quote by Penrose:

…The reason appears to be that there is no satisfactory quaternionic analogue of the notion of a holomorphic function.

As far as I understand, there is precisely such a notion - it is one of the major achievements of the geometric algebra way of looking at things.

A holomorphic function on $\mathbb{R}^n$ is a even-grade multivector field $G(x)$ (a linear combination of scalars and even wedge-products of vectors) whose geometric derivative vanishes:

(1)$\nabla G(x) = 0$

Here the geometric derivative is the differential operator

(2)$\nabla = \sum_i e_i \nabla _i$

So… it’s basically the Dirac operator. In two-dimensions it reduces to the ordinary notion of a holomorphic function, because there $G(x)$ has a scalar and a bivector part,

(3)$G(x) = f(x) + g(x) e_1 \wedge e_2,$

and the vanishing of the geometric derivative is just the Cauchy-Riemann equations.

So… there is a beautiful notion of a holomorphic function in higher dimensions. And it satisfies Cauchy integral formulas like

(4)$G(x_0) = \frac{1}{IS_n} \oint_{\partial V} \frac{x-x_0}{|x-x_0|^n} dS G$

where the integral is a geometric integral. By the way, I’m getting all this stuff from the “7. Geometric Calculus” part of these notes.

I don’t know why geometric algebra hasn’t taken off in the mathematical community. Urs is probably right - his theory is simply that they concentrated too much on pedagogics and too little on cracking outstanding problems with it.

By the way, the name “geometric algebra” is not something made up by David Hestenes in order to look snazzy - it’s the name and the original approach used by Clifford himself!

Posted by: Bruce Bartlett on May 6, 2008 6:55 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

Bruce wrote in part:

I don’t understand the quote by Penrose:

…The reason appears to be that there is no satisfactory quaternionic analogue of the notion of a holomorphic function.

As far as I understand, there is precisely such a notion - it is one of the major achievements of the geometric algebra way of looking at things.

But this gives a notion of holomorphic function from R3 to H, not from H to H. (Even in the complex case, it gives directly holomorphic functions from R2 to C, but this can be fixed: any orthonormal basis on R2 tells us how to interpret these as functions on C, and the question of which such functions are holomorphic is independent of the choice of basis.)

Geometric algebra is cool, but does it answer the question at hand?

(Sorry this comment is so late; I was just searching through the posts for my name to see if I should reply to anything, and I came across this.)

Posted by: Toby Bartels on August 24, 2008 5:27 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

Heh, David, thanks for the link! We hadn’t spotted this paper, but I’m sure it’s interesting.

Posted by: Kea on May 6, 2008 11:27 PM | Permalink | Reply to this

### Tim Poston comment; Re: Quaternionic Analysis

Professor Tim Poston emailed me from Bangalore to comment on this passage:

“Unfortunately – the quaternionic analogues of the ring structure of holomorphic functions, local conformal invariance, Riemann mapping theorem and many other classical results of complex analysis have never been found.”

His comment is as follows:

For a very basic reason.

If a function from the quaternions to the quaternions is quaternionically differentiable – that is, if it has
an approximation that is linear over the quaternions,
d( [number] [function]) / dx
= [number] d( [function]) / dx

and additive, then it actually is linear, in the first place.

You can see this by writing down the analogue of the Cauchy-Riemann equations: with all the signs involved, they end up implying that all the second partial derivatives vanish. (Alternatively, just look at differentiation over any non-commutative ‘field’.)

So there is a theory of holomorphic quaternionic functions, in a way … but it is identical to the theory of linear quaternionic functions, and not separately interesting.

Similarly, you can fill a library with books on complex-differentiable mappings,
but not find one on piece-wise complex-linear mappings (though piece-wise real-linear mappings are richly studied). Piece-wise complex-linear mappings are globally complex-linear.

These rigidifications of the theory as we go to preserving more structure seem to belong with the way everywhere-once-complex-differentiable mappings are automatically analytic (everywhere-∞fold-complex-differentiable,
plus convergence of the Taylor series) and thus far ‘stiffer’ than everywhere-∞fold-real-differentiable mappings, where one neighbourhood’s values cannot pin down the whole mapping.

Posted by: Jonathan Vos Post on May 8, 2008 4:21 PM | Permalink | Reply to this

### Re: Tim Poston comment; Re: Quaternionic Analysis

Jonathan wrote:

Tim Poston wrote:

So there is a theory of holomorphic quaternionic functions, in a way … but it is identical to the theory of linear quaternionic functions, and not separately interesting.

That’s true for one definition of ‘holomorphic quaternionic function’, but not the Fueter definition. That’s why people prefer the Fueter definition! The first three pages of Sudbery’s paper explain the situation quite clearly. Maybe you can show Poston this paper.

Posted by: John Baez on May 8, 2008 6:35 PM | Permalink | Reply to this

### Re: Tim Poston comment; Re: Quaternionic Analysis

There is tradition to consider linear map $f(x)$ as product of number over variable $x$. Since product in quaternion algebra is noncommutative this view creates problem. But why not to consider that linear map has form

(1)$f(x)=axb$

where $a$ and $b$ are quaternions. In such case we extend the set of linear maps and can consider derivatives of polynomial or exponent of quaternion argument.

Posted by: Aleks Kleyn on January 17, 2016 1:53 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

Tony Sudbery wrote a nice review of different attempts to define holomorphic functions from the quaternions to the quaternions:

I’ve been wanting to learn this stuff and write about it in This Week’s Finds, but I’d lost track of this paper for a long time. I just found it now, on Citeseer!

Sudbery says that Fueter’s notion of holomorphic functions is the best. Is this what Frenkel uses? Is this the same one the geometric algebra people use, Bruce?

It would be great if people have converged on some good notions and are starting to do really interesting things with them.

Posted by: John Baez on May 8, 2008 5:28 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

I now think Frenkel and Sudbery are using the same notion of ‘quaternionic holomorphic function’ as the one the geometric algebra people are using — it’s basically a solution of a certain Dirac equation, a 4d analogue of the Cauchy-Riemann equation.

Posted by: John Baez on May 8, 2008 7:09 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

I now think Frenkel and Sudbery are using the same notion of ‘quaternionic holomorphic function’ as the one the geometric algebra people are using

Yes it seems so, at least from a distance. I recall for for the benefit of others that in geometric algebra the quaternions are thought of geometrically as bivectors (blades) in 3d space,

(1)$i = e_x \wedge e_y, j = e_x \wedge e_z, k = e_x \wedge e_y.$

These blades are genuine geometric objects - they are oriented units of area. A quaternion is an even-grade multivector, so it has scalar and bivector components,

(2)$q = \alpha + \beta_x e_x \wedge e_y + \beta_y e_x \wedge e_z + \beta_z e_x \wedge e_y.$

A quaternion field $q(x,y,z)$ is holomorphic if the geometric derivative vanishes,

(3)$\nabla q = \sum_i e_i \partial_i q = 0.$

From Lecture 7 of the Part III course on Geometric Algebra I gather that if one writes this out, one finds that the solutions are the spin harmonics from relativistic quantum mechanics.

So… if Frenkel and Sudbury’s definitions also end up giving the spin harmonics, then they’re the same.

Posted by: Bruce Bartlett on May 8, 2008 7:49 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

I should add that if they are the same, then I would prefer the geometric algebra formulation since it gives the quaternions concrete geometric meaning by embedding them in a broader context; i.e. stressing that they are bivectors in 3d space. This is important because the geometric derivative operator

(1)$\nabla = \sum_i e_i \partial_i$

sends a quaternion (a scalar + bivector) into an odd multivector (a vector plus a pseudoscalar). In other words, if one refuses to envisage the quaternions as really sitting inside the algebra of multivectors, it is very difficult to even conceive the geometric derivative.

Posted by: Bruce Bartlett on May 8, 2008 7:58 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

Hello:

Twice over the years I have had math people doing reviews of quaternion calculus email me to say they thought my definition of a quaternion derivative looked quite useful. It might even classify as “cute”. I will share it here with people with better math training than myself.

The idea comes from L’Hospital’s rule: use a two limit process. On the quaternion manifold H1, let the 3-vector of the quaternion go to zero first. In the next limit process, the scalar goes to zero. A scalar will commute with anything, so it makes no difference if the differential is written on the left or the right. In effect, the two limit process is a way to write a directional derivative along the real axis. This definition of a derivative should work fine for real and complex analysis.

What happens if the limits are reversed? Then the most we can define is the norm of a derivative.

What I do is think of all of these derivatives applying to events in spacetime. My differential element is thus (c dt, dx1, dx2, dx3). What the first order of limits says is it applies to c dt > |dR|, or for events that have a light like separation. These events are ordered in time, like a movie. This is the domain of classical physics. The second ordering of limit processes is for spacelike separated events, c dt < |dR|. Events cannot be linked by cause because nothing travels faster than the speed of light. Yet we can say something about a set of spacelike events governed by a quaternion function, specifically what the norm of the change of a function happens to be. This may be the domain of quantum mechanics.

It certainly would be neat if the division between classical and quantum physics was due entirely to getting the most useful definition of quaternion derivative written down right.

Posted by: Douglas Sweetser on May 17, 2008 9:46 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

Hello:

I thought I’d share the story of where this proposal for a two limit definition of a quaternion derivative came from because it is a nice story with fun implications for physics, specifically the magic of the factors of 1/3 and 2/3 that pop up for holomorphic functions.

In 1999, I went to the only meeting devoted to quaternions and physics I had ever heard about (and have not heard of any since) in Rome. I traveled with Prof. Guido Sandri of BU. In the second talk of the conference by David Joyce of Oxford, he explained the problem of quaternion analysis well. He then described his own approach to hypercomplex geometry. I got angry at his proposal, thinking it was far too complicated and clucky (I know, clucky isn’t a technical word in math, but it accurately reflects my feelings of yucko). I decided that this was the problem to focus on during the conference: come up with a good definition of a quaternion derivative driven by the needs of physics that does analytical functions gracefully.

I was fortunate that Guido was happy to be my personal tutor in complex analysis. He taught me the importance of the manifold C1 in the topic (and I apologize ahead of time for any poor phrasing I use since I have not continued my education since then, consumed mostly by the activities of daily living outside academia). The pair of variables z and z* on C1 can do everything that x and yi can do on R2. Yet they can also do more because they have the properties of the complex division algebra built in.

For quaternions, it was clear that q and q* was not enough, there being 4 degrees of freedom. Had I been a diligent student of the literature, I would have picked up Feuter’s approach. Yet in the outdoor cafes in Rome, I came up with a different idea. What the conjugate does is keep the sign of the scalar positive, while flipping the signs of the other three. Why not make a new widget, that would flip the sign of all but the first term in the 3-vector? I called this the first conjugate, q*1 = (e1 q e1)*. This could be classified as “cute”, not too tricky, yet does something unexpected, taking a quaternion such as G = (f, g, u, v) to (-f, g, -u, -v). There is also a second conjugate, q*2 = (e2 q e2)* that flips the sign of all but the e2 term. They can be isolated by different combinations of these conjugates:

(f, 0, 0, 0) = (G + G*)/2
(g, 0, 0, 0) = -e1(G + G*1)/2
(u, 0, 0, 0) = -e2(G + G*2)/2
(v, 0, 0, 0) = -e3(G - (((G*)*1)*2))/2

Back in my SPR days, I recall one comment from Toby Bartels that this little widget should be acknowledge in a paper he and John Baez were writing. What their paper was, its outcome, any one-to-one emails on the topic didn’t happen. Why don’t we just say whatever slim relationship between John and I had was tense. I was hoping someone would ask why I had developed the first and second conjugates: for a new approach to quaternion analysis!

Despite owning quaternions.com, my “quaternions [accounting] rule the universe” approach is considerably more subtle than such a label implies. If one uses standard quaternions in the standard way, there is zero new physics in such an exercise. Zero, none, nada - most of my site is devoted to nothing new, but it is great practice. New physics is exclusively the result of new math. This is why the majority of theoretical physics research money can and should go to work on strings. This is why we should encourage the new math being developed in loop quantum gravity and the spin foams. This is why we should be skeptical that someone can find anything actually new to do with quaternions since they have been around since Gauss’ time (in an overlooked notebook). Yet I will continue with this post since this definition of a quaternion derivative - nine years old - still looks new and unknown.

The flaw with Feuter’s approach to the quaternion derivative (and Sudbery’s technical update) is that a function as simple as the square of a quaternion cannot be shown to be analytical in q. If you don’t have the polynomials, you don’t have anything. Guido told me there were five equivalent ways to show a function was analytic. One was to use the limit definition, another the Cauchy-Riemann equations, a third was to apply the chain rule. I forget what the fourth one was, my bad. I was able to show q2 was analytic in q, (q*)2 was analytic in q*, (q*1)2 was analytic in q*1, and (q*2)2 was analytic in q*2. It was fun to work through all the details. It was the fifth one that was a stumper. I had to show that a derivative of a regular function vanishes. They do that in geometric algebra by a geometric derivative as Bruce Bartlett described, an approach that does not work with quaternions. I wanted this “dot derivative” to vanish:

scalar((d/dx0, d/dx1, d/dx2, d/dx3)(f, g, u, v))
= d f/dx0 - d g/dx1 - d u/dx2 - d w/dx3 = 0

The problem was this didn’t want to vanish. There was too much from the 3-vector. The idea I have seen adopted by Feuter and everyone else I have read looks at the 4 degrees of freedom as equals. If I used that approach, I was able to get four of five tests for an analytic function to work, and that is not good enough. My derivative definition does not look at the four as equals. Instead, it puts the scalar equal on equal footing with the 3-vector. So I decided to put in a factor of 1/3 for the 3-vector, and if done just right - the details get confusing - I was able to get the dot derivative to vanish. Five for five is good :-)

Now that I had my extra factor of 1/3 for the basis of the 3-vector, I had to go back to my other 4 tests for analytic functions, and redo them with this factor. It was amazing to see all these factors of +2/3 and -2/3 in the chain rule calculation work out. The only other place where I have heard of factors of 2/3 showing up in calculations has to do with quarks. Accident? I don’t know, but it was exciting. On the Air France flight back, I worked out the details for the cube of q being analytic in q. If you ever want to be left alone on an international flight, do a page of algebra. People will leave you alone and assume you are brilliant.

It is clear folks who chat here are absurdly busy, citing papers yet to be fully read and digested. There is no way someone like myself could prove this quaternion derivative definition works unless people do it themselves. I have met my responsibility by posting it here since the topic is spot on. My email box is open to discuss the new quaternion math. The other two new math bits I have developed is a way to animate any quaternion expression what-so-ever, and a way to write a 4D division algebra that commutes, a variation of Clyde Daver’s hypercomplex numbers.

Off to Brazil for the 8th International Conference on Clifford Algebras to talk about the new quaternion math and its implications for new physics.

Doug
sweetser@alum.mit.edu

Posted by: Douglas Sweetser on May 22, 2008 12:33 PM | Permalink | Reply to this

### Re: Quaternionic Analysis

Doug Sweetser wrote in part:

Back in my SPR days, I recall one comment from Toby Bartels that this little widget should be acknowledge in a paper he and John Baez were writing.

That paper was never published, but you can find what was written at http://toby.bartels.name/notes/#quaternions. Holomorphic functions never came into that, but I certainly intend to credit you for inspiring the discussion if the paper is ever published (unlikely but possible).

(Sorry this comment is so late; I was just searching through the posts for my name to see if I should reply to anything, and I came across this.)

Posted by: Toby Bartels on August 24, 2008 5:33 PM | Permalink | Reply to this

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