N-D geometric algebras are essentially the algebras of N anticommuting scalar-squared generators. That a multivector is a linear combination of our particular generators choice is what makes it a 1-vector. Grades are "geometry dependant" and arise from a particular choice of N anticommuting generators deemed to be 1-vectors. This choice determines the restricted products and conjugations and also the signatures and associated metric.  

    Let (e₁,e₂,..,e_N) be an orthogonal basis for geometric algebra Â_p,q,r for p³1 with e₁²=1. Then {e₁,e₁₂,e₁₃,..,e_1N} provides an anticommuting <1;2>-grade generating basis for Â_N having all but the first signature negated so Â_p,q,r @ Â_q+1,p-1,r for p³1 .
    For even N this has the disconcerting effect of making former (N-1)-blade e_234..N the new "N-blade" pseudoscalar i', having opposite signature to the previous pseudoscalar i but (anti)commuting with the new "1-vector" basis in like manner to i=e_12..N 's (anti)commutation with the original e_j.
    For any N, e₁'ºe₁ and e₂'ºe₁₂ must now both be regarded as orthogonal 1-blades with their geometric product e₁₂'=e₂ now regarded as a 2-blade arising from a new outter product Ù' with e₁'e₂'=e₁(e₁₂)=e₁Ù'e₁₂=-e₁₂Ù'e₁=e₂. Former 1-blade e₂ thus takes the role of a 2-blade e₁₂' while 2-blade e₁₂ assumes the role of a 1-blade e₂'. The grade of a blade is thus ultimately a matter of interpretation. e₁+e₁₂ is mixed grade in Â_p,q,r but a pure 1-vector in Â_q+1,p-1,r.
    As well as new restricted products Ù', ¿' etc. we aquire new conjugations ^§', ^#' etc. with e₁₂^§'=e₂'^§' = e₂'=e₁₂ and so forth, but since ^§# acts identically on 1 and 2 blades we have (^§#)'=^§# and can regard ^§# as geometrically invarient under the Â_p,q,r @ Â_q+1,p-1,r equivalence.

    Adoption of such alternate geometric basies can be considered as differing "geometric interpretations" of the underlying multivector algebra.

Odd N
   

     When N=p+q and ½N(N-1)+q are both odd (eg. Â_4k+3 or Â_4k,1) so that the pseudoscalar is central and negative squared then we can replace any even number of 1-vector generators whose signature we wish to change with their even (N-1)-vector duals. These have  opposite square and anticommutes with every other e_i and e_i^*. [  If we dual an odd number we no longer have a generating basis (eg. { e₂₃,e₂,e₃}  cannot generate e₁) ]
    For odd N>3, when an odd number of e₁,e₂, and e₃ have negative signature so that e₁₂₃²=1 then the anticommuting <1;4> generaters { e₁,e₂,e₃,e₁₂₃₄,e₁₂₃₅,..,e_123N} negates the signatures all but the first three elements with the same full product for odd N, giving Â_p,q @ Â_q+1,p-1 when p³2, q³1 and Â_p,q @ Â_q-3,p+3 when q³3 , with ^§ invariant.
    When N=4k+1 then starting from Â_4k+1 we can judiciously replace pairs of 1-vectors with 4k-vectors to obtain any Â_N-2m,2m with ^§ geometrically invarient. When N=4k+3 we can change 1-vectors to (4k+2)-vectors in pairs and it is Clifford conjugation ^#§ that is invarient.

    When p³4 then {e₁e₁₂₃₄,e₂e₁₂₃₄,e₃e₁₂₃₄,e₄e₁₂₃₄,e₅,e₆,..,e_N} provides anticommuting <1;3> generators with the first four positive signatures negated so Â_p,q,r @ Â_p-4,q+4,r for p³4. This works for odd and even N and keeps ^#§ invariant.

    The distinct odd dimensioned geometric algebras of nonnull pseudoscalar for N£7 are as follows, with positive signature pseudoscalared listed first and geometrically invarient conjugations indicated in []..
    Â₁     ;; Â_0,1 @ C_1×1 .
    Â_2,1 ; Â_0,3 @ C_0,2 ;; Â₃ @ Â_1,2 @ C₂ @ C_2×2 [^§#]
    Â₅ @ Â_1,4  ; Â_3,2   ;; Â_4,1 @ Â_2,3 @ Â_0,5 @ C₄ @ C_4×4 [^§] ;
    Â_0,7 @ Â_4,3 ; Â_6,1 @ Â_2,5 @ C_0,8  ;; Â₇ @ Â_1,6 @ Â_5,2 @ Â_3,4 @ C₈ @ C_8×8 [^§#]

Even N
    When N and ½N(N-1)+q are both even (eg Â_4k-2l,2l) so that i has positive signature and anticommutes with 1-vectors we cannot selectivly dual the basis 1-vectors but we can dual them all to obtain Â_4k-2l,2l @ Â_2l,4k-2l and in particular Â_4k @ Â_0,4k.
    Hence the distinct even dimensioned geometric algebras of nonnull pseudoscalar for N£8 are as follows, with invarient conjugations indicated in []. These do not divide according to pseudoscalar signature, which for even N is not geometrically invariant.
    Â_0,2 @ Q_1×1 ; Â₂ @ Â_1,1 @ Â_2×2 [^§#]    
    Â₄ @ Â_0,4 @ Â_1,3 @ Q_2×2 ; Â_3,1 @ Â_2,2 @ Â_4×4 [^§#].
    Â₆ @ Â_5,1 @ Â_2,4 @ Â_1,5 @  Q_4×4     ; Â_4,2 @ Â_3,3 @   Â_0,6 @ Â_8×8[^§#].
    Â₈ @ Â_0,8 @ Â_4,4 @ Â_1,7 @ Â_5,3 @ Â_16×16 ; Â_7,1 @ Â_3,5 @ Â_2,6 @ Â_6,2 .

    Thus we have Minksowski spacetime Â_3,1 and timespace Â_1,3 fundamentally distinct .

    Lounesto [16.4] shows that Â_p+8,q @ Â_p,q+8 @ (Â_p,q)_16×16 , ie. the space of 16×16 matrices of Â_p,q multivectors.

Algebraic Product Formulations
    By considering generating sets some of which commute we can reduce Â_p,q,r into a "product algebra" of two smaller geometric algebras. Suppose geometric algebra U_N has a basis e₁,e₂,..e_N for N > 2 with e_N and e_N-1 nonnull. We can express any a in U_N not only as a real weighted sum of products of the e_i (the e_i "generate" U_N) but alternatively as a real-wieghted sum of products of the N-2 3-blades e_N-1e_Ne₁ , e_N-1e_Ne₂ , ... , e_N-1e_Ne_N-2 and the two 1-vectors e_N-1 and e_N which commute with all the 3-blade generators but anticommute with eachother. The 3-blade generators all anticommute with eachother so we have U_N @ U_N-2 × U₂   where U₂ is a 2-D geometric algebra having signatures e_N-1² and e_N-2²    and U_N-2 is an (N-2)-D geometric algebra with signatures (e_N-1e_Ne_i)² = -e_N-1²e_N²e_i²   .
    Symbol × here indicates the "product" algebra obtained by allowing the generators of seperate algebras to multiply commutatively.  
     If N³ 4 we can repeat the trick, "picking out" a pair of the nonnull 3-blades to produce an alternate set of N generators for U_N consitisting of two 1-vectors, two 3-blades, and N-4 5-blades; all generators commuting with those of different grade and anticommuting with those of the same grade. If N³6 , we can replace all but two of the 5-blades with 7-blades, and so forth.
    By such means, we can express any N-D multivector in terms of N odd-graded blade generators that each commute with all but at most one other generator (the unique generator sharing the same grade). The signatures of these "multigrade generators" depend on the signatures of the original 1-vector basis. We can express these results most succinctly as
    Â_p+2,q+0,r @ Â_2,0 × Â_q,p,r    @ Â_2×2 × Â_q,p,r   
    Â_p+1,q+1,r @ Â_1,1 × Â_p,q,r    @ Â_2×2 × Â_p,q,r   
    Â_p+0,q+2,r @ Â_0,2 × Â_q,p,r    @ Q × Â_q,p,r   

    For p³q we thus have Â_p,q @ (Â_1,1)^q × Â_p-q @ (Â_1,1)^q × (Â₂)^2l × Â_m,0,r where p-q = 4l+m with mÎ {0,1,2,3} .

    "Factoring" geometric algebras in this way does not make for compacter storage since we still have N elements generating 2^N distinct products amd so requiring 2^N real coordinates for the genreral real-weighted sum. We merely obtain more commutative multiplication tables for the "multigrade basis" than for the 1-vectors one.

Complexification
    For odd N, if the central pseudoscalar i has negative square (ie. if ½(N-1)N+q is odd) we can express any mutlivector a as b+ic for even b and c giving Â_p,q @ (C_p,q)₊ @ ((Â_0,1)_p,q)₊ , associating i with i. The complex conjugation b-ic is provided by the main involuation ^#.
    If ½(N-1)N+q is even we have Â_p,q @ ((Â_1,0)_p,q)₊ .

Minimal Geometric Algebras

    Our geometric vocabulary is somewhat broad with various product symbols ¿,.,Ù,_°...; operators +,/,^↑,...;  conjugations ^§,^#,... ; and so forth. Which symbols are fundamental in that they cannot be semantically defined using other fundamental symbols?
    Let us first suppose just + and ¨ ; precedence indication brackets ( and ) ;  orthonormal N-frame symbols {e₁,e₂,...e_N} ; and some multivector "variable" symbols a, b, ... together with an assigment statement = .
    Unity symbols 1 and (-1) (and hence the left and right negation conjugation operators - and ^- ) are available from {e₁e₁ , e₂e₂, e₁e₂e₁e₂} depending on the signatures and we obtain, semantically at least, all the integer-coordinated multivectors in the given frame.
    Exponmentiation ^↑ can be defined using + and ¨ given some form of infinitite summation å symbolism and if we then suppose logarithm ^↓ as a particular inverse of ^↑ we obtain a^-1 as (-(a^↓))^↑ which we can write as ^-1 = ^↓-↑ whence a/b º a(b^-1)   =   a(b^↓-↑) and we attain the general real-coordinate multivector space. [under construction]

Bivectors

    Left multiplication by a bivector ( a ® b₂a ) casts scalars into b₂. If b₂ is a 2-blade It rotates directions within b₂ (by b₂), while casting directions perpendicular to b₂ into trivectors. For N=3, it casts bivectors in b₂ to scalars while bivectors normal to b₂ are rotated by b₂ and the pseudoscalar is cast to 1-vector b₂^*.
    Right multiplication by a bivector has similar effects. Indeed a 2-blade commutes with all multivectors in its dual space , which is not the case for a 1-vector. Consequently, it is occasionally advantageous to represent 3D 1-vectors by their bivector duals.
    Let b be multivector having only bivector and scalar components. a¿b = a.b while b¿a = b.a + b₀a
    b₂×a sends ^(a,b₂) to 0 and rotates ¯(a,b₂) in b₂ by ½p, scaling it by |b₂|.
    The operation a ® (ba).a for nonnull b is interesting, casting ¯(a,b) to 0 and ^(a,b) into
|^(a,b)|²b = |aÙb|²(b^-2)b = |aÙb^~|²b .

Expressed as sum of commuting 2-blades
    Any Euclidean     N-D 2-vector b₂ can be represented as the scalar-weighted sum of at most ½N orthogonal (by which we mean geometrically commuting) unit 2-blades. Such a decomposition is unique except in cases where two or more wieghts are identical. "Decomposing" a given bivector b₂ in this way is nontrivial but a computational algorithm exists.
[ We seek 2-blades a₁,A₂..,a_m with a_ia_j=a_iÙa_j=a_ja_i and b₂=a₁+A₂+..+a_m .
    Now (b₂^k)_<2k> = b₂Ù...Ùb₂ = k!S_r<s<..<va_rÙa_s..Ùa_v = k!S_r<s<..<v a_ra_s..a_v     where there are k terms in each product and k suffices r,s,..v .
    Hence (b₂^k-1)_<2k-2> ¿ (b₂^k)_<2k> = k!(k-1)! å_i=1^m  a_i (    S_{r<s<..<u ¹ i} a_r²a_s²..a_u² )     for 1£k<m which, if the scalar a_i² are known and distinct, provides m linear ^NC₂-dimensional equations , solvable for a_i by conventional numerical methods.
    The scalar a_i² can shown to be the m roots of the m^th order scalar polynomial å_k=0^m  (-l)^m-k((b₂^k)_<2k> ¿ (b₂^k)_<2k>)=0 . See Hestenes & Sobczyk (3-4) for a fuller treatment. ]

Spinors

Recall that for puresquare a,     a↑ º ea =	cos(|a|) + a~ sin(|a|)	if a2 < 0 ;
	cosh(|a|) + a~ sinh(|a|)	if a2 > 0 ;
	1+a	if a2 = 0.

    Authors differ in precise meaning of the terms "spinor" and "rotor". [ Hestenes & Sobczyk define a spinor as a multivector a satisfying (aba^§)_<1>= aba^§     " 1-vector b, which is equivalent to a being an even versor except when N=p+q divides 4, in which case we have a=(a+bi)v where v is an even versor and a and b are scalars ]
    Here we use k-rotor to indicate a nonnull 2k-versor and pure/impure k-spinor to indicate an exponentiated unit or null pure/impure k-vector. e^b_kf for scalar f. Postive scalars are thus pure 0-spinors. If b is a nonnull 2-versor, then so is e^b [   cos(f)+ sin(f)e₁₂ = e₁( cos(f)e₁+ sin(f)e₂) ] . Since any 2-blade is a 2-versor, any 2-spinor is a 1-rotor.

    Under the geometric product, the pure spinors generate Â_N⁺ which for N=3 is isomorphic to quaternion space (the 2-spinor (inf)^↑ corresponding to normalised quaternion { cos(f/2), sin(f/2)n} ).

Spinor Adjustment Rule
    Whenever i squares to -1 and commutes with a we have the spinor adjustment rule
    (1+a)e^qi = (1+a)e^qai = (1+a)e^-qa* (when i=i)     provided a²=1.
    (1-a)e^qi = (1-a)e^-qai = (1+a)e^qa*     provided a²=-1 .
[ Proof : (1+a)(e^qi - e^qia) = (1+a) sinq(i - ia) = (1+a)(1-a)i sinq = 0 for a²=1  .]
    In Â₃ we have (1+a)e^qi=(1+a)e^qai which means that under a (1+a) mutiplier we can replace ai exponentiations by i exponentiations, which tend to be more commutative.

Alternate representations of spinors
    Physicists tend to represent spinors in convoluted and confusing ways that obsfucate more than they reveal. In Quantum Mechanics , a "spinor" is typically defined via two complex numbers a₀+b₀i = r₀e^q₀i and a₁+b₁i = r₁e^q₁i = re^fi r₀e^q₁i when r₀¹0 as the singular (zero determinant) 2×2 complex matrix

b=

æ

(a0+ib0)(a1+ib1)

-(a0+ib0)2

ö

=

æ

r0r1e(q0+q1)i

-r02 e2q0i

ö

=

r02e2q0i

æ

refi

-1

ö

è

(a1+ib1)2

-(a0+ib0)(a1+ib1)

ø

è

r12e2q1i

-r0r1 e(q0+q1)i

ø

è

r2e2fi

-r efi

ø

    The spinor is considered normalised if the real scalar part ½(a₀²+b₀²+a₁²+b₁²) =½(r₀²+r₁²) of the singular matrix

b(b§)=	æ	(a0+ib0)(a0-ib0)	(a0+ib0)(a1-ib1)	ö	is equal to ½.
	è	(a0-ib0)(a1+ib1)	(a1+ib1)(a1-ib1)	ø

This gives r₀Î[0,1] , r₁=(1-r₀²)^½ , r₀=(1+r²)^-½ , r₀r₁=r₀²r=r(1+r²)^-1 .

    Such 2×2 complex matrices are often represented via the s_i and 1 "Pauli" matrix representors for 1,e₁,e₂,e₃ defined above using

æ	a	b	ö	= ½(a+d)1 + ½(b+c) s1   + ½i(b-c) s2 + ½(a-d) s3
è	c	d	ø

    We can associate i with i=e₁₂₃ and regard b as the null (b²=0) Â₃ multivector
    b= ½r₀²e^2q₀i (e₁(r²e^2fi-1) -e₂i(r²e^2fi+1) + e₃2re^fi )
    = ½r₀²e^2q₀i(r²+1)(u+vi)     where
    u = (e₁(r² cos2f-1) + e₂r² sin2f + e₃2r cosf)(r²+1)^-1     ; v = (e₁r² sin2f - e₂(r² cos2f+1) +e₃2r sinf)(r²+1)^-1     are orthonormal 1-vectors (u²=v²=1) ; uv = (r²+1)^-1(e₂₃ 2r cosf + e₃₁ 2r sinf + e₁₂ ( 1-r²) ) .
    Letting w=-u^-1v i= e₁ 2r cosf + e₂ 2r sinf + e₃ (1-r²) = (r²+1)R_iem(re^fi) we have
    b= ½r₀²e^2q₀i(r²+1)u(1-w)       where w²=1 .

Riemann Sphere Representation

    In Riemann sphere representation of C + ¥ we map complex "point" z=a+bi=re^fi to the real 3D unit 1-vector corresponding to the intersection of the line joining ³ point ae₁+be₂ to the "south pole" -e₃ with the unit sphere centred at 0. This point is given in spherical polar coordinates as
    R_iem(z) º [2 tan^-1(r),f,1] = e^fe₁₂) e^{-2( tan-1(r))e₃₁} e₃ = (1+r²)^-1( e₃(1-r²) + 2r( cosfe₁ + sinfe₂) ) .
     We further define R_iem(¥)=-e₃ .
    We have R_iem(z^-1) = [ p-2 tan^-1(r),-f,1] ; R_iem(-z) = [2 tan^-1(r),f+p,1] ; R_iem(e^fi) = e₁e^fe₁₂ .
    [ If we site the argand plane tangent at the "north pole" e₃  rather that at the equator we have R_iem(z)º[ 2 tan^-1(½r),f,1] ]    
[ Proof : u² = (r² cos2f-1)² + r⁴ sin2f² + 4r² cosf² = r⁴ + 1 - 2r² cos2f+ 4r² cosf² = r⁴ - 1 - 2r²( cosf²- sinf²)+ 4r² cosf² = r⁴ - 1 + 2r²( cosf²+ sinf²) = (r² + 1)²
v ² = r⁴ sin2f² + (r² cos2f+1)² + 4r² sinf² = r⁴ + 2r² cos2f +1 + 4r² sinf² = 1+r⁴ + 2r²
u¿v= (r² cos2f-1)r² sin2f -r² sin2f(r² cos2f+1) +2r² cosf sinf = (r² cos2f-1) sin2f - sin2f(r² cos2f+1) + sin2f
uv = (e₁(r² cos2f-1) + e₂r² sin2f + e₃2r cosf) (e₁r² sin2f - e₂(r² cos2f+1) +e₃2r sinf)
    = e₂₃ (2r₃ sin2f sinf+2(r² cos2f+1)r cosf) + e₃₁ (2r₃ sin2f cosf-2(r² cos2f-1)r sinf) + e₁₂ ( -(r² cos2f-1)(r² cos2f+1) - r⁴ sin2f²)
    = e₂₃ 2(r₃( sin2f sinf+ cos2f cosf)+r cosf) + e₃₁ (2r₃( sin2f cosf- cos2f sinf)+r sinf) + e₁₂ ( 1-r⁴)
    = e₂₃ 2r(r²+1) cosf + e₃₁ 2r(r²+1) sinf + e₁₂ ( 1-r⁴)
    = (r²+1)(e₂₃ 2r cosf + e₃₁ 2r sinf + e₁₂ (1-r²) )  .]
    u = e₁(r² cos2f-1) + e₂r² sin2f + e₃2r cosf     ;
    v = e₁r² sin2f - e₂(r² cos2f+1) +e₃2r sinf
    w = e₁ 2r cosf + e₂ 2r sinf + e₃ (1-r²)

    For small r, u » -e₁ + e₃2r cosf     ; v » - e₂ +e₃2r sinf     ; w » 2r(e₁ cosf + e₂ sinf + e₃ (1-r²)

    For large r,
    u » r²(e₁ cos2f + e₂ sin2f) v » r²(e₁ sin2f - e₂ cos2f) w » -r²e₃ . Thus with r=0 we have u=e₁, v=-e₂, w=e₃ . As r increases u aquires a cosf e₃ component,
    When r₀=0 we have b= ½ (e₁r₁²e^2q₁i - e₂i(r₁²e^2q₁i) ) = ½ e^2q₁i r₁² (e₁ - e₃₁ ) = ½ r₁² e^2q₁i e₁ (1 + e₃ ) .

    Let z = re^fi =r₁(r₀^-1)e^q₁-q₀ .

• z=1 Þ r=1, f=0 Þ b=½ e^2q₀ie₃(1-e₁)
  [ Proof : ½ ½ e^2q₀i(-2e₃₁ + e₃2 )    = ½ e^2q₀i(-e₃₁ + e₃) = ½ e^2q₀ie₃(1-e₁)  .]
• z=-1 Þ r=1, f=p Þ b= ½ e^(2q₀+p)i e₃(1+e₁)
  [ Proof : (e₁(e^2fi-1) - e₃₁(e^2fi+1) + e₃2e^fi )   = ( - e₃₁(2) + e₃(-2) )  = -2e₃(1+e₁)  .]
• z=i Þ r=1, f=½p Þ b= ½ e^(2q₀+p)i e₁(1-e₂)
  [ Proof : (e₁(e^2fi-1) - e₃₁(e^2fi+1) + e₃2e^fi )   =(e₁(-2) + e₃2i )   =2(-e₁ + e₁₂ )   =-2e₁(1-e₂ )    .]
• z=-i Þ r=1, f=-½p Þ b= ½ e^2q₀i e₁(1+e₂)
  [ Proof : (e₁(e^2fi-1) - e₃₁(e^2fi+1) + e₃2e^fi )   =(e₁(-2) - e₃2i )   =2(-e₁ - e₁₂ )   =2e₁(1+e₂ )    .]
• z=0 Þ r₀=1,r₁=0 Þ b= =½ e^2q₀+pi   e₁(1-e₃)     
  [ Proof : e^2q₀i (e₁(-1) - e₃₁(+1)  )    =½ e^2q₀i   e₁(-1+e₃)      =½ e^(2q₀+p)i e₁(1-e₃)       .]
• z=¥ Þ r₀=0,r₁=1 Þ b= ½ e^2q₁i e₁ (1 + e₃ ) .

    A' = W.A where W º 2R'R^§ is a pure bivector known as the angular veclocity bivector.
[ Proof :   RR^§=1 Þ R'R^§ + R(R^§)' = 0 Þ R'R^§ = -R(R^§)' = -(R'R^§)^§ Þ W pure.
    A' = (RA₀R^§)' = (R'A₀R^§ + RA₀(R^§)') = (R'R^§)A - A(R'R^§) = (R'R^§)×A = (2R'R^§).A = W.A  .]
    For N=3 letting w=W^* gives     A' = W.A = (w^*).A = (wÙA)i = w×A     so w is the conventional 3D angular velocity 1-vector.

    2R'R^§ = W Þ R solves R' = ½WR .
    Writing W_L = R^§WR = 2R^§R' for the local angular-velocity bivector expressed in the frame of the object, we have R' = ½RW_L .
    These rotor equations are generally easier to solve than their matrix counterparts.
    An aesthetic advantage in defining angular velocities and momenta as bivectors rather than vectors is the avoidance of invoking a third dimension to "hold" them in the case of otherwise planar kinematics.  

    Next : Conclusion

Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 18 May 2007.