Tuesday, 20 December 2005

Geometric Algebra

The algebra of complex numbers is related to geometry by the Argand plane. Using it, we see that the operation of multiplying by i is equivalent to a 90 degrees rotation in the counterclockwise direction. A little more advanced concept is that of quaternions, that as complex numbers, are a set of numbers that can represent rotations in 3D space. In both these cases, there is a beautiful connection between algebric structures and geometry that can be used to express physical laws in a concise way.

The notorious way to use geometry in physics is by means of Gibbs' vector calculus, which became widespread in physical sciences and engineering. In 1878 Clifford created a structure with the name geometric algebra uniting the dot and the cross products of two vectors into a single entity named the geometric product, which for two vectors a and b is written as
\[ab=a\cdot b +a \wedge b,\]

where the first term is the dot (scalar) product and the second the wedge or exterior product, which generalize the cross product that turns out to be a particular case in 3 dimensions.

Although it has a lot of applications in physics, it was eclipsed by Gibbs' vector calculus and was forgotten untill 1960 when David Hestenes, trying to recover the geometric meaning of the Clifford algebra related to spin discovered that geometric algebra is a "universal language for mathematics, physics and engineering."

There are a complete introductory course as Lecture Notes in the site of the Department of Physics of the University of Cambridge.

The interesting fact, that my former PhD advisor pointed me, is that there is a hope that this structure can lead to a geometric interpretation of the misterious use of complex numbers in Quantum Mechanics. However, I need to read more the lecture notes to talk about that.

Papers over my desk (or in my desktop):

  • Vegetation's Red Edge: A Possible Spectroscopic Biosignature of Extraterrestrial Plants - Seager et al. (astro-ph/0503302)
  • Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School) - Sorkin (gr-qc/0309009)
  • The General Quantum Interference Principle and the Duality Computer - Long (quant-ph/0512120)
  • Entropic Priors - Caticha and Preuss (physics/0312131)
  • On Math, Matter and Mind - Hut et al. (physics/0510188)
Picture: Quantum Notions - Gerard von Harpe

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