User:Jheald/sandbox/GA

Current targets:

• The mixed multivector ((sin(θ) + cos(θ) e₁), created by the "flattening" the e₃ direction on right-multiplying by the primitive idempotent (1+ e₃). What does this mixed multivector mean? Why might projective space be a clue?
• Conformal geometric algebra -- as the place to look, so see what projective transformations come up
• Weyl & Brauer's decomposition of rotations -- what exactly were their steps, & how did their spinors get used?
• "Isotropic space" -- i.e. (most often) one with mixed signature. Why are Cartan's spinors considered so embedded in such a space? Is there a relation with Cl⁺_(p,q) ≃ Cl_(p-2,q+1).
• The non-columnlike reappropriation of the word "spinor" by several GA books: document,
• "Geometric algebra" vs. "Clifford algebra" -- write up results of reading survey for Talk:Geometric algebra.

• Wigner D matrix -- put "change of basis" into the intro. Non-integer-valued spherical harmonics?
  • Citizendium, webpage on SH.

Scratchpad for pages where adding a GA perspective might be useful, or where existing GA material might usefully be tweaked.

• Spinor -- pretty horrible at the moment -- the part that motivates in 2D and 3D is anyway, thanks in no small amount to myself; much of the later technical detailed stuff is pretty good.
  • could use more physics though: above all, more interpretation/application.
  • a thread: [1] Desperately seeking spinors. Another thread [2]
  • Tony Smith: "What ARE Clifford algebras and spinors?" old newer 1newer 2
  • Review Doran et al mapping them to the bivectors -- User:Jheald/sandbox/GA/Spinors -- sui generis, or more general?
  • Leigh "Spinors in flat spacetimes" [3] covers 2d, 3d, 3+1d
  • User:Jheald/sandbox/GA/Some groups found in GA -- see table in Lundholm, Svensson notes at p.58
• Spinors in two dimensions -- some notes at User:Jheald/sandbox/GA/Spinors in two dimensions
• Spinors in three dimensions -- corresponding formulas in GA notation -- User:Jheald/sandbox/GA/3D spinors workpage; spinors for Cl₃(C) might also be useful; as may Cl_4,0(R)
• Spinors in 4-dimensional Euclidean space -- bits and pieces at User:Jheald/sandbox/GA/Spinors in 4-dimensional Euclidean space

• Bott periodicity -- off-puttingly mathematical at the moment. Some more general remarks/conclusions would be good to float higher up, to give people an idea of what it means. (Is the pseudo-scalar the key to this? eg: its square, square of its reverse, etc; also chirality).

• Module (mathematics) -- User:Jheald/sandbox/GA/squeak

• Conformal geometric algebra -- User:Jheald/sandbox/GA/Conformal geometric algebra
  • Update screw axis and screw theory to reflect CGA methods, as well as Plucker co-ordinate methods -- how to resolve T₂R₂ + T₁R₁ and T₂R₂T₁R₁
  • Update inversive geometry, discussing how stereographic projection which maps spheres to planes, along with homogenisation for dealing with points at infinity, can be used to map R3 to R5, with GA operators then able to represent transformations.
  • Update Möbius transformation for higher dimensional case.
• User:Jheald/sandbox/GA/Affine geometric algebra

• Dirac operator -- per expansion suggestion on talk page
• Cauchy-Riemann equations -- scratchpad at User:Jheald/sandbox/GA/Cauchy-Riemann add -- though might work better in an expanded Dirac operator ?
• Clifford analysis -- needs further wiki-formatting at the least. Why the use of Cl_0,1 as the basic signature? Says FT is a problem for real-valued CAs. But is complex everything the right way to go? Or do we need to introduce another (real) dimension, or set of dimensions, giving imaginaries with more closely controlled meanings? Probably leave any more here till after Dirac operator.

• Rotation representation (mathematics) -- could use Quaternion <-> Bivector -- User:Jheald/sandbox/GA/Quaternions to Bivectors
• Rotations in 4-dimensional Euclidean space -- add GA perspective; Rotation_(mathematics)#Four_dimensions gives a lead
  • Failure of bivector exponentiation in this and higher dimensions?
• Quaternions and spatial rotation -- might use a look.

• Electromagnetic tensor -- Muphrid15 (talk · contribs) has a draft add at User:Muphrid15/Electromagnetic tensor draft
• Versor -- check GA definition
• Curl (mathematics) -- at the moment goes into huge length about differential forms; perhaps all we need to say is: it's a bivector ?

• Pseudosphere

• Pauli matrices -- add more physics?
• Clifford module -- what is this?
  • We used to have an explicit construction somewhere for representations of Clifford algebras / spin matrices -- where did it go?

Found it: [4]; was rm'd from the page as "nonstandard". Not all the representations are faithful (a.k.a. "universal" in the language of the page)

Instead there are Higher-dimensional gamma matrices and Weyl–Brauer matrices.

Woit [5] introduces a "Mathematical finds" by Baez [6]

• Spherical harmonics
• Vector spherical harmonics -- pics for all of these would be nice. From Mathematica demonstrations ?
• Spin spherical harmonics
• Spin-weighted spherical harmonics
• paper on double-valued spherical harmonics: [7]

Some good GA pages

• Bivector

Websites

• [8] Czech page with good bibliography / links

GA contributors

• JohnBlackburne (talk · contribs)
• Brews ohare (talk · contribs) -- banned (fell out with JohnBlackburne and Headbomb)
• Chessfan (talk · contribs)
• Muphrid15 (talk · contribs)
• Nbarth (talk · contribs)
• Peeter.joot (talk · contribs) -- blog
• Prof McCarthy (talk · contribs)
• Quondum (talk · contribs)
• Rschwieb (talk · contribs) -- subpage User talk:Rschwieb/GA Discussion
• Selfstudier (talk · contribs)
• Chrisjldoran (talk · contribs) -- Chris Doran
• Ajohnryan (talk · contribs) -- John Ryan

also try

• RDBury (talk · contribs)
• Rgdboer (talk · contribs)