Quaternionic Analysis

May 6, 2008

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

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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 $ℝ^{n}$ 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} \land 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

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: Quaternionic Analysis

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

Anthony Sudbery, Quaternionic analysis.

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

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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

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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} \land e_{y}, j = e_{x} \land e_{z}, k = e_{x} \land 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 = α + β_{x} e_{x} \land e_{y} + β_{y} e_{x} \land e_{z} + β_{z} e_{x} \land 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

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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

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