Refreshed online version of Geometric Algebra book.
Posted by peeterjoot on August 6, 2009
My exploratory physics notes using Geometric Algebra have now been refreshed with the following recent updates:
- July 2, 2009 (ch73) Space time algebra solutions of the Maxwell equation for discrete frequencies
- July 17, 2009 (ch38) Stokes theorem applied to vector and bivector fields
- July 21, 2009 (ch39) Stokes theorem derivation without tensor expansion of the blade
- July 27, 2009 (ch107) Bivector form of quantum angular momentum operator
- July 30, 2009 (ch74) Transverse electric and magnetic fields
These were all first posted in this blog.
The first covers transverse waves in vacuum, and uncovers the solution utilizing a Fourier transform of the Maxwell equation.
The second of the Stokes articles above covers the N dimensional stokes theory proof usually expressed with differential forms. Here the integrated object is not a N-form, but instead just a multivector (blade) object, which has the required intrinsic antisymmetrical character that is often expressed using a form. The nice thing about this approach in my opinion is that could be considered less foreign to somebody used to the vector (i.e. the third term calculus course in an Engineering program). You don’t have to make the paradigm shift to start treating the integrated object (like an electric field) as a differential element. What this article is not, is a full treatment of Geometric Calculus as presented in Doran/Lasenby’s text for example. That’s a more complex topic and not one that I’ve tried to understand yet. That text gets Stokes Law “for free” as a special case, but if all you want is the generalization of Stokes to greater than three dimensions (like 4), then it is perhaps more complexity and abstraction than desirable.
The angular momentum article is a factorization of the spatial (or four vector space-time) gradient into and terms. This factorization is seen in the context of quantum mechanics treating the angular momentum operator in spherical polar coordinates, but only in coordinate form. I don’t think it is normal to see this outside of quantum mechanics, but there is nothing intrinsically quantum about this projection operation on the gradient or Laplacian operators. Nothing is done with this result here, and I’d like to revisit this later to play with the implications of this factorization in a classical context.
The last notes are on the relations between the propagation direction and transverse components of an e&m wave in media. This reproduces content available in Jackson (and Eli’s blog) in a somewhat elegant way.