Talk:Paravector

Expert Review

After (brief) discussion on the Mathematics project page, I have added the expert review banner after receiving a handful of comments which I interpreted to be consent. The author's engagement would be appreciated. Sojourner001 (talk) 18:53, 24 February 2010 (UTC)

That discussion appears to have vanished. Andrewgdotcom (talk) 10:11, 12 August 2010 (UTC)

At the time of this writing, the discussion can be found here Gary2468 (talk) 10:58, 1 August 2012 (UTC)

Error in ψ-equation in Matrix Representation Section

Matrix Representation Section:

  Equation for a general Clifford number in 3D: 

There should be a plus sign in front of coefficient ψ12 because the 2x2 matrix P3e1 has positive one in row 1 column 2 and zeros elsewhere.

The same error is in Wikipedia "Dirac equation in the algebra of physical space" Jro8163 (talk) 21:02, 3 July 2012 (UTC)

Error in section "Closed Subspaces respect to the product"

Identification of bivectors by quaternions. In my oppinion, these equations should be

$-\mathbf {e} _{23}=i$

$-\mathbf {e} _{31}=j$

$-\mathbf {e} _{12}=k$

Prove: These equations imply:

$ij=\mathbf {e} _{23}\mathbf {e} _{31}=\mathbf {e} _{2}\mathbf {e} _{3}\mathbf {e} _{3}\mathbf {e} _{1}=\mathbf {e} _{2}\mathbf {e} _{1}=-\mathbf {e} _{1}\mathbf {e} _{2}=-\mathbf {e} _{12}=k$.

On the other hand, the equations in the text imply $ij=-k$.

Can somebody confirm this? I will change the article if this issue is accepted. — Preceding unsigned comment added by Michaelis (talk • contribs) 20:12, 27 July 2014 (UTC)

Yup, you're right. I've made this same mistake myself before elsewhere (and had someone correct it). Go ahead with the fix. —Quondum 21:57, 27 July 2014 (UTC)

Thanks for your work and your reply!

--Michaelis (talk) 11:43, 28 July 2014 (UTC)

Thanks for this article

I just wanted to chime in that this article is amazing. It uses some different language from other clifford algebra articles, but it's thorough and has helped me multiple times to fill in the gaps left by the other articles. This is the only place on Wikipedia with a clear explanation of what exactly the anti-automorphisms are. Other articles assume a very high up view. This one teaches you the basics, like offering a tree to climb up so that you can get that higher view. — Preceding unsigned comment added by 75.72.215.209 (talk) 10:04, 18 February 2016 (UTC)

Not-so-fundamental axiom?

It's pretty obvious why: $(\mathbf {u} +\mathbf {w} )^{2}=\mathbf {u} \mathbf {u} +\mathbf {u} \mathbf {w} +\mathbf {w} \mathbf {u} +\mathbf {w} \mathbf {w} ,$ in conjunction with "the fundamental axiom": $\mathbf {v} \mathbf {v} =\mathbf {v} \cdot \mathbf {v}$ would imply the right hand side of: $\mathbf {u} \cdot \mathbf {u} +2\mathbf {u} \cdot \mathbf {w} +\mathbf {w} \cdot \mathbf {w} =\mathbf {u} \cdot \mathbf {u} +\mathbf {u} \mathbf {w} +\mathbf {w} \mathbf {u} +\mathbf {w} \cdot \mathbf {w}$. However, the left hand side -- specifically $2\mathbf {u} \cdot \mathbf {w}$ -- is not implied by appeal to that "axiom" because the two factors are not the same.

Moreover, if one appeals to some other "axiom" to justify $2\mathbf {u} \cdot \mathbf {w} =\mathbf {u} \mathbf {w} +\mathbf {w} \mathbf {u}$ the "fundamental" axiom becomes a mere theorem, correct? Jim Bowery (talk) 23:03, 26 January 2019 (UTC)

Missing closed subspaces with respect to the product?

The article states that there are only two closed sub-spaces with respect to the product. I think there are at least six more. I believe what I called plane-like subspaces are isomorphic to 2D clifford algebras, which are isomorphic to the 2x2 real matrices. Additionally I think what I called screw-like subspaces are isomorphic to the tessarines. Were these spaces simply overlooked, or is there a more fundamental reason they are not included? JasonHise (talk) 00:09, 21 September 2019 (UTC)