## Clifford Algebra, 3 and 4-Dimensional Analytic Functions with Applications. Manuscripts of the Last Century.

**Authors:** Giuliano Bettini

The greatest revolution in the number from the days of Pythagoras.
The similarity between quantum mechanics and electromagnetism. Are we a three-dimensional television show?
These and other fascinating topics are addressed by the author in this paper at once popular and
mathematical, which leads us to a world still largely unexplored. Are we facing with what is (up to now) the true language of Physics? "Clifford's algebra - he called it 'geometric algebra' - is now well recognized as the natural
algebra for describing physics in 3-space, but it hasn't yet caught on in engineering, or even in
standard treatments of electricity and magnetism or fluid dynamics, where vector analysis with its
ugly cross product still holds sway" (Mark Buchanan, Nature Physics 7, 442, 2011).
But can physics laws be derived from Clifford algebra and analytic functions? And why?
From simple postulates of geometrical nature (or, I mean, which simply precisely define our
language) it seems that we arrive at equations of relativistic dynamics, electromagnetism, fluid
dynamics and quantum mechanics.
Issues covered more or less in depth in this paper are: numbers and algebra, the analysis and the δ*
operator, analytic functions in 3 and 4 dimensions, Maxwell's and Dirac equations, analytic
functions in circular waveguides, analytic functions in four dimensions, i.e. spherical cavity,
Physical Optics and heuristic derivation of the Hydrogen spectral lines.
Many disciplines are then influenced by this approach in a way that the paper often only suggests,
so as it suggests several areas of future development.
I mulled over these topics for more than 40 years, and I then summarized in an unpublished
manuscript dated March 2000, which is almost entirely reported here in his complete, also if naïve,
form.

**Comments:** 63 pages, v2 in Italian, v3 in English

**Download:** **PDF**

### Submission history

[v1] 30 Jul 2011

[v2] 13 Aug 2011

[v3] 17 Aug 2011

**Unique-IP document downloads:** 350 times

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