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Geometric algebra: A small introduction to a powerful and general language of physics

Geometric algebra: A small introduction to a powerful and general language of physics. Michael R.R. Good Georgia Institute of Technology. Redundant Languages.

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Geometric algebra: A small introduction to a powerful and general language of physics

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  1. Geometric algebra: A small introduction to a powerful and general language of physics Michael R.R. Good Georgia Institute of Technology

  2. Redundant Languages • Synthetic Geometry Coordinate Geometry Complex Numbers Quaternions Vector Analysis Tensor Analysis Matrix Algebra Grassmann Algebra Clifford Algebra Spinor Algebra etc… • There are unnecessary consequences of so many languages. • Redundant learning • Complicates access to knowledge • Frequent translation • Lower concept density, i.e., theorems / definitions Geometric Concepts

  3. Geometric algebra • A unifying language for mathematics. • A revealing language for large areas of theoretical and applied physics. • Acts for both classical and quantum physics. • Applications in robotics, computer vision, image processing, signal processing and space dynamics.

  4. What has geometric algebra done for physics? • Maxwell’s electrodynamics has been formulated in one equation revealing a more simple physical relationship. • Relativistic quantum mechanics has been reformulated, replacing abstract complex inner space Dirac matrices by real space-time basis vectors. • General relativity has been improved by the construction of a new gauge theory using geometric calculus improving ease of calculations.

  5. `Physicists quickly become impatient with any discussion of elementary concepts'So why re-learn vectors? Geometric algebra: • allows the division by vectors. • introduces a more general concept than the cross product, which is only defined in three dimensions • this is needed so that full information about relative directions can be encoded in all dimensions. • gives the imaginary unit concrete and natural geometric interpretations. • is more intuitive than standard vector analysis. • is more efficient because it reduces the number of operations, and is coordinate free. • is well-defined for higher and lower dimensions. • handles reflections and rotations with ease and power.

  6. So what is geometric algebra? • A language for geometry. • The exploitation of the concept of a vector. • The use of higher dimensional vectors, called k-vectors. • The combination of different dimensional concepts, scalars, vectors, bi-vectors, tri-vectors, and finally k-vectors to form multi-vectors. Geometric Concepts Algebraic Language

  7. Geometric algebra makes use of dimensions called grades • Point  scalar grade 0 • Vector a directed line grade 1 • Bi-vector B directed plane grade 2 • Tri-vector T directed volume grade 3 They are all called k-vectors: • k-vector K directed objectgrade k

  8. So what is a bi-vector? • A bi-vector has the same magnitude as the familiar cross product. • The cross product is a vector, whereas a bi-vector is an area • The bi-vector is a directed area and its orientation lies in the plane that it rests. • The outer product a  b, or wedge product, defines a bi-vector and has magnitude: |a  b| = |a| |b| sin 

  9. The outer product is the natural partner of the inner product. • The inner product a · b, or dot product, is a scalar and has magnitude: |a · b| = |a| |b| cos  • The outer product ab, or wedge product is a bi-vector and has magnitude: |a  b| = |a| |b| sin  The outer product is more general than the cross product!

  10. Addition of different dimensions? • In complex analysis addition defines a relation: • z = x + i y • Clifford’s “geometric product” for vectors: • ab = a b + a  b • scalar bi-vector • (inner product) (outer product) • addition

  11. How can you add a scalar to a bi-vector? • A scalar added to a bi-vector is the most basic axiom of geometric algebra, it is called the geometric product, or Clifford product. ab = a b + a  b • Adding different quantities is exactly what we want an addition to do! The product has scalar and bi-vector parts, just like a complex number has real and imaginary parts.

  12. Sum of k-vectors are multi-vectors • A multi-vector M is the sum of k-vectors. ( M =  + a + B + T + … ) • A multi-vector has mixed grades. (grades are dimensions)

  13. The geometric product is basic We can define our inner product and outer product in terms of the basic geometric product. Define dot product in terms of geometric product: a · b = 1/2 (ab + ba) scalar Define wedge product in terms of geometric product: a  b = 1/2 (ab - ba) bi-vector Using the geometric product: a · b + a  b = ab

  14. What happened to the cross product? The cross product is translated from geometric algebra via a  b = -i a  b Or if you prefer via a  b = i a  b where i is a unit tri-vector, whose square is -1. i = 123 The basis for geometric algebra in 3 dimensional space is: {1, 1, 2, 3,12, 13, 23, 123 } scalar vectors bi-vectors tri-vector

  15. Why isn’t geometric algebra more widely known? • Clifford was a student of Maxwell, but died an early death allowing Gibbs’ and Heaviside’s vector analysis to dominate the 20th century. • It is hard to learn a new language, especially one that asks you to revisit elementary concepts. • Many physicists and teachers have not heard about geometric algebra’s advantages. W.K. Clifford 1845-1879

  16. The Future of Geometric Algebra Speculations and hope for: • An understanding of geometric algebra as a quantum algebra for a quantum theory of gravitation. • Complex numbers, as mystical un-interpreted scalars, to be proven unnecessary even in quantum mechanics • Unobserved higher dimensions to be proven unnecessary in the clarity created by geometric algebra. • Introducing geometric algebra: • High school: generalizing the cross product. • Undergraduate: complimenting rotation matrices. • Graduate: condensing the Maxwell equation’s into one equation.

  17. References and Sources Online Presentations: • Introduction to Geometric Algebra, Course #53 by Alyn Rockwood, David Hestenes, Leo Dorst, Stephen Mann, Joan Lasenby, Chris Doran, and Ambjørn Naeve. • GABLE: Geometric Algebra Learning Environment, Course #31 by Stephen Mann, and Leo Dorst. • Gull, S. Lasenby A. & Doran,C. 1993 ‘Imaginary Numbers Are Not Real – The Geometric Algebra Of Space-Time’ http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html • Lasenby A.N Doran C.J lecture notes 2000-2001 http://www.mrao.cam.ac.uk/~clifford/ptIIIcourse/ • Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics by David Hestenes Books: • Hestenes, D. 1966 ‘Spacetime algebra’ New York Gordon and Breach • Jancewicz, B. 1988 ‘Multivectors and Clifford Algebra in Electrodynamics’ World Scientific • Lounesto, P. 2001 ‘Clifford Algebras and Spinors’ Cambridge University Press Websites: • Hestenes’s Site: http://modelingnts.la.asu.edu/ • Lounesto’s Site: http://www.helsinki.fi/~lounesto/ • Cambridge Group: http://www.mrao.cam.ac.uk/~clifford/

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