Talk:Spacetime algebra

Division algebra ?

The only division algebras it could be, abstractly, are the real numbers, complex numbers, or quaternions.

Charles Matthews 22:04, 27 March 2007 (UTC)

Fixed. Thanks. Jheald 12:14, 30 March 2007 (UTC)

Spinors

Conventionally in physics, spinors are things that can be represented by complex column-vectors that transform under the one-sided action of spin matrices. Translated directly into geometric algebra, that one-sided action of spin matrices turns into the one-sided action of rotors or similar objects, and the complex column vectors turn into the product of a general multivector with an idempotent projector like ½(1+e₃), where (e₃)² = 1.

However, in publications by Hestenes and many others (eg Dorst et al), the word "spinor" is redefined as "a (normalised) element that preserves grade under a sandwiching product in a Clifford algebra" [i.e. S x S⁻¹] (Dorst et al (2007), p. 195 (Google books).

Lasenby, Doran and Gull (1998) give as justification that "[R]otation of a multivector is performed by the double-sided application of a rotor. The elements of a linear space which is closed under single-sided action of a representation of the rotor group are called spinors. In conventional developments a matrix representation for the Clifford algebra of spacetime is introduced and the space of column vectors on which these matrices act defnes the spin-space. But there is no need to adopt such a construction. For example the even subalgebra of the STA forms a vector space which is closed under single-sided application of the rotor group. The even subalgebra is also an eight-dimensional vector space, the same number of real dimensions as a Dirac spinor, and so it is not surprising that a one-to-one map between Dirac spinors and the even subalgebra can be constructed." (this pdf, p. 20; the one-to-one map is later presented in Appendix A).

However, as far as I can see this is only true of the STA. In other geometric algebras, the two types of object have algebraic dimensionalities, so presumably are not placeable in one-to-one correspondence.

It seems to me that the attempts of the likes of Dorst et al to redefine the meaning of the word spinor, on the fly without any warning flags or discussion that this is something different, has the potential to be extremely confusing.

The present welcome and commendable active development by Chris Howard (talk · contribs) of this article on STA looks to be moving in the right direction with material most recently added relating a column-vector Pauli equation to a multivector form one. No doubt a similar treatment of the Dirac equation may be next on the to-do list. But at the moment, simply saying ψ is a spinor field as the next section baldy does, without further explanation, when ψ appears not to be what is understood (at least conventionally) by the word "spinor" appears insufficient.

A more general discussion (either here or at spinor) of the relation between what the GA crowd call a spinor vis-a-vis what is conventionally called a spinor might be useful. Jheald (talk) 18:54, 25 February 2012 (UTC)

When I had seen the mention of the spinor field in the article, I had also found it came as a bit of a surprise, and it had not been sourced so I had added the "citation needed" template to it.

Regarding spinors and the way they can be represented in geometric algebra I find the following reference helpful: Chris J. L. Doran, Anthony N. Lasenby: Geometric algebra for physicists, Cambridge University Press, 2003, ISBN 0-521-48022-1, p. 268–271. It discusses the spinor when seen in analogy to a rotor, and several examples of a “one-to-one map between conventional quantum mechanics and the multivector equivalent” – the one-to-one map is given for spin-up and spin-down states, for Pauli operators, and for the unit imaginary of quantum theory. --Chris Howard (talk) 10:40, 26 February 2012 (UTC)

I've come upon another article on spinors in relation to geometric algebra: Matthew R. Francis, Arthur Kosowsky: The Construction of Spinors in Geometric Algebra, arXiv:math-ph/0403040v2 (submitted 20 March 2004, version of 18 October 2004). Rather tough reading, but might be of interest here. --Chris Howard (talk) 20:47, 1 April 2012 (UTC)

Discussion about Lorentz transformation

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)

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)

Article revision

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, ${\textstyle \{\gamma _{\mu }\}}$ is replaced by ${\textstyle \{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}}$. 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)