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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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**Geometric algebra: A small introduction to a powerful and**general language of physics Michael R.R. Good Georgia Institute of Technology**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**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.**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.**`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.**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**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**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 **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 ab, 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!**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**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.**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)**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**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 = 123 The basis for geometric algebra in 3 dimensional space is: {1, 1, 2, 3,12, 13, 23, 123 } scalar vectors bi-vectors tri-vector**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**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.**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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