Is Geometric Algebra inconsistent/circular?

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SUMMARY

The discussion centers on the potential circularity in defining the geometric product of vectors in Geometric Algebra, specifically referencing the textbook by Doran and Lasenby. The geometric product is defined using axioms of associativity, distributivity, and the square of a vector being a real scalar. Participants express concerns about the lack of clarity in computing the geometric product, particularly when considering the inner and outer products. They emphasize the need for a rigorous definition that accommodates multivectors and suggest alternative resources for better understanding.

PREREQUISITES
  • Understanding of Geometric Algebra principles
  • Familiarity with vector operations, including inner and outer products
  • Knowledge of axiomatic definitions in mathematics
  • Basic concepts of multivectors and their properties
NEXT STEPS
  • Study the axiomatic approach to Geometric Algebra as outlined in the paper from arXiv
  • Learn about the definitions and properties of multivectors in Geometric Algebra
  • Explore the book "Geometric Algebra for Physicists" by Dorst et al. for intuitive understanding
  • Research the differences between scalar, vector, and bivector products in Geometric Algebra
USEFUL FOR

This discussion is beneficial for students and researchers in mathematics and physics, particularly those interested in advanced algebraic structures and their applications in theoretical physics.

  • #31
micromass said:
I'm actually planning to do an insight series about it too!

Wonderful idea! I am really looking forward to see it published soon.
 
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  • #32
Although I've postponed learning geometric algebra (GA) [putting my efforts into differential forms instead],
I have found myself having to learn to it in order to understand/decode some calculations.

In writing an Insight,
it might be useful to include translations between statements and calculations in GA
and statements and calculations in (say) more-standard tensor-calculus and/or vector-calculus,
accompanied by some diagrams.
To me, the notation found in various sources in quite dense, rather abstract for a beginner, and not quite standardized,
forcing the reader to juggle the various objects and operations.

My $0.03.
 
  • #33
I guess the original idea for the insight was not to focus only on the details of GA, but more generally on "Non-mainstream but useful math to know". As micromass said:
micromass said:
There are many other things in mathematics that I feel should be basic knowledge for undergrads, but that is somehow something that is only known to a select few.
 

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