There appears to be a cult following of geometric algebra/geometric calculus (GA/GC) as developed by David Hestenes. Many questions on the web regarding this but also many contradictions appear. I wanted to make this question different than others, this question hopes to bring together many theoretical physicists/mathematicians/mathematical physicists whom can contribute to the discussion of the validity of the claims and soundness of GA/GC. I want to ask the experts here on physics overflow what they know and what they think about GA/GC.
The claims are always as follows, GA/GC provides a unified formalism for theoretical physics and mathematics. It provides a framework which unifies many of the concepts in fields of mathematics such as differential geometry, algebra and many more.... Why are theoretical physicists/mathematicians not rushing into this field? If it was so universal and apparently simpler to learn (also as claimed), then why is it not as widespread? If anyone has made use of GA/GC for their research into our best theory, quantum field theory, it would be much more appealing I think. Anyone who has made use of GA/GC for quantum field theory or the standard model please share your opinions.
Now most textbooks and articles on this subject give the answer that Clifford developed this formalism concurrently with Gibbs standard formalism for vector calculus. Gibbs was a famous physicists/mathematician and worked at Princeton and so it is the reason his approach was so widespread and Cliffords approach was swept under the rug. Not until David Hestenes revived it, and now there are many descendants/followers of his whom repeat what I just said in the introduction or preface to their book or article. A prolific follower and firm believer of GA/GC is Alan MacDonald a professor of mathematics at Luther College in Decorah, IA. He wrote the following article (there are MANY more by other authors).
https://www.astro.umd.edu/~jph/GAandGC.pdf
Just read the introduction paragraph and this is generally the type of introduction which comes from most articles and books on GA/GC. It is basically a rephrasing of what Hestenes said in his pioneering work and books on GA/GC, such as his book Space-Time Algebra.
I need honest opinions from reputable mathematicians/physicists all over. It is important for us to get feedback from mathematicians and physicists outside the field of GA/GC because it seems the only people HIGHLY RECOMMENDING using GA/GC are only those who are using it exclusively. It would be good to get many opinions from mathematicians/mathematical physicists in fields related to GA/GC, such as Differential Geometry, Mathematical Physicists who make extensive use of Differential Geometry, Modern Algebraic Geometry, abstract algebra and especially those studying General Relativity and Quantum Field Theory, our two most successful and far reaching theories.
I know it is difficult for someone to comment on other peoples work, but when something makes such big claims as Unifying much of mathematics and physics, it is important for the community to discuss this, so that opinions and ideas are shared and cult following is either justified or unjustly spreading.
I am not against GA/GC, I am curious and want to know what other professional mathematicians/physicists think of this field and their claims. Please spread this article around so that others can make their comments and we can either begin to embrace GA/GC or refute its claims.
Thank you.
Note : There are many textbooks on this subject now and here a few so you can see for yourself that it is being used by a small community people. amazon.com/Geometric-Algebra-Physicists-Chris-Doran-ebook/dp/… , springer.com/gp/book/9780817682828 , link.springer.com/book/10.1007/978-1-84628-997-2 –
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