Re: Analytic Functions in 3D?--

I post this a second time because of a frustrating typing error and because I forgot to remove _no spam_.
I typed "Well, there is on 3D field over the reals." instead of "Well, there is no 3D field over the reals."


Jeff wrote:

>Hello
>
>I've been studying the remarkable properties of analytic functions in the
>complex plane and their relationship to conformal mapping.
>
>I was wondering if there was any extension of this to 3 dimensions - though
>I suspect not.
>And what about higher dimensions? Will the quaternions do for 4D what
>complex numbers do for 2?
>
>Any pointers or keywords for searching would be appreciated.
>
>Many Thanks
>Jeff.
>

I am afraid you are right to guess that there is no extension to 3D -

The ingredients for differentiability are the four principal arithmetical operations and the limit concept,
in other words: one needs a topological field, preferably commutative, the topology preferably derived from a metric, the field being complete as a metric space.

Well, there is no 3D field over the reals.

As regards quaternions: the norm in the quaternion skew field yields the well-known Euclidean metric with its accompanying topology. Multiplication is not commutative, so one needs to distinguish between left-division ( p^(-1).q ) and right-division ( q.p^(-1) ), and consequently, between left- and right-difference and differential quotients.
I am not aware of a fully developed quaternion differential calculus. Google and Wikipedia might be helpful.

Concerning conformal mappings of 4D regions:
First look at the 2D case. In the plane we have at every point a combined translation, rotation and magnification, plus something infinitesimal.

Something similar happens in 4D: 4D translation, 4D rotation, magnification, plus something infinitesimal.

The snag is in the 4D rotations. They come in several different kinds. Read more about 4D rotations in Wikipedia > SO(4) and in my arXiv articles at http://arxiv.org/abs/math.GM/0501249 and http://arxiv.org/abs/math.GM/0701759 ,
all of this also reachable from my web page on 4D rotations at http://www.xs4all.nl/~jemebius/4drot.htm .

Also read about hyper-Kähler mainfolds, a class of 4D Riemann manifolds with special, quaternion-based porperties.

Cheers: Johan E. Mebius
.

Relevant Pages

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