Talk:Spacetime algebra

message: The only division algebras it could be, abstractly, are the real numbers, complex numbers, or quaternions.
Charles Matthews 22:04, 27 March 2007 (UTC)
replies: text: Fixed.  Thanks.  Jheald 12:14, 30 March 2007 (UTC)
thread_title:  Division algebra ? 

message: It would be nice to discuss how the Lorentz transformations can be settled in the space-time algebra (i.e. by rotations through rotors). I think that a Lorentz transformation between two frames R and R', with axes aligned, but with R' moving w.r.t. R in a given direction determined by the velocity vector v could be easier explained with geometric algebra than it is usually done without it. 
Best Regards. Physicist137 (talk) 15:25, 1 May 2020 (UTC)
replies: text: I know it's a little bit late, but there's at least a start to this section now Tjf801 (talk) 16:53, 8 May 2023 (UTC)
thread_title:  Discussion about Lorentz transformation 

message: This is is a proposed article revision: User:TMM53/Spacetime_algebra-2024-01-15. I will make this revision in about 2 weeks.  If you disagree with the revised content, please provide an alternative suggestion with secondary or tertiary sources.  I excluded a small amount of the current article that was not obvious to a reader or unsourced despite a comprehensive literature search.  The improvements achieved by this revision include:
* References increased from 14 to 26, citations increased from 12 to 44.
* Non-functioning links repaired or removed.
* The revision places citations in a Citation section, and references in a Reference section making it easier for the reader.
* Needs additional citations to secondary or tertiary sources: All primary sources were replaced by secondary sources. 
* This section needs expansion: I provided additional content, changed section titles, and merged sections when a comprehensive literature review failed to find additional content.
* This section needs additional citations for verification: Unsourced content is now sourced with secondary sources.
* "According to David Hestenes ...": I added an actual quote from Hestenes, a direct quote is better than an author's interpretation of what Hestenes thinks.
*Lead section: I added content, comparing STA to similar algebras i.e. APS and Dirac algebra.
* Structure section: I added needed essential content on vector, inner and outer products.
* "corresponding to the fact that under parity transformations ...":  This is unclear to the reader why this is correct, there is no source provided, and I cannot find any source, so this was deleted.
* Mathematical statements are more explicit, easier for a reader to understand.  For example, \{ \gamma_{\mu}\}  is replaced by \{ \gamma_{0}, \gamma_{1}, \gamma_{2}, \gamma_{3} \} . Also, sentences do not begin with a math symbol.  The revision follows Wikipedia guidelines for mathematical articles.    
* "This generates a basis of one ...":  I edited this section to distinguish the vector basis from the tensor basis.
* "The spacetime algebra also contains a non-trivial sub-algebra ..."  I expanded this section into a separate section to better explain the relationship to complex numbers, the exact relationship of the even-subalgebra to the APS (Pauli) algebra and the basic features of the Pauli subalgebra.  This is essential as this is the foundation for the space-time split and subsequent equations that rely on the subalgebra.
* "The spacetime algebra is not a division algebra ... ":  This content is unsourced. It is unclear to a reader why STA is a non-division algebra and how this is related to idempotents.  Null vectors are common in STA, proper zero divisors, related to the lightcone, but not discussed.  The revision clarifies these issues.
* Lorentz transformations section title changed to Transformations and additional content added as requested.
* Lorentz Force section and Physical observables sections merged to the preceding sections, as I could not find seconary source content to augment these sections.  If you disagree, provide sufficient additional seconary sourced content needed to maintain these separate sections.
* The Pauli equation section:  I added the explicit method for transforming the matrix representation of a spinor to the STA representation.  This is essential as it is unclear to the reader how this transformation occurs.
* U(1) gauge symmetry section: Title was changed and rewritten because there were no sources for the content or the equations.  I maintained similar content including equations and references from a secondary sources.  I added additional content. 
* The revision uses consistent symbols.  The I always refers to a pseudoscalar of a Clifford algebra, and the i always refers to the imaginary number of a complex number i.e. the pseudoscalar in the algebra of complex numbers only.
Let us work together to improve this article, thanks.TMM53 (talk) 19:56, 15 January 2024 (UTC)
thread_title:  Article revision