Algebra of Physical Space (APS)
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Direct sum of subspaces are also linear subspaces of the algebra.
The sum of scalar and vector subspace is know as the paravector space.  
Dimention is 1 + 3.
 
definition of paravectors as       p = p0 + pv           =  grade(p,0) + grade(p,1)
ex : 3e0 + 5e1 - e3
p0 = 3e0                         scalar part
pv = 5e1 - e3                  vector part
 
Paravectors constitue a four-dimentional linear subspace with the metric of spacetime.
the interpretation is both in space-time (paravector) and spacial (vector) terms.
 
Every elements of Cl3 reduces to a complex paravector :
with i = e123    the unit pseudoscalar.     i² = -1  
 
scalar                          e0                            grade 0
vectors                     e1,e2,e3                       grade 1
bivectors                 ie1,ie2,ie3                      grade 2
trivectors                       i                             grade 3
 
we may write a multivector (a Cliffor)  :
 
  A = a0*e0 + a1*e1 + a2*e2 + a3*e3 + a12*ie3 + a23*ie1 + a13*ie2 + a123*i
  A = (a0 + a123 i)*e0 + (a1 + a23 i)*e1 + (a2 + a13 i)*e2 + (a3 + a12 i)*e3
  A = p = z0*e0 + z1*e1 + z2*e2 + z3*e3    with    z0, z1, z2, z3  complex numbers.
   
  p0 = z0*e0                                         scalar part
  pv = z1*e1 + z2*e2 + z3*e3             vector part
  p = p0 + pv
ex :
a= 3e0 + 5e1 - e3 + 2e12 + 5e123
p = (3+5i)e0 + 5e1 - (1-2i)e3
p0 = 3+5i                             scalar part
pv = 5e1 - (1-2i)e3              vector part
 
Convertion functions :  
CtoP(u)  (Cliffor to Paravector)  and  PtoC(p)    (Paravector to Cliffor)
ex :
CtoP(u) return p                    grade(0,1)
PtoC(p) return u    only if p = grade(0,1)
 
Paravector metric  :     pseudo-Euclidian metric of Minkowski space-time :
 p p generaly is not a scalar. We will use the quadratic form Q(p) = p cj(p) wich is a always scalar where                      p" = cj(p) = p0 - pv
 
p² = p p" = p cj(p) = p0² - p
 
note : the square length in spacetime is not necessarily positive :
p² > 0  -> p is timelike        
p² = 0  -> p is ligth like or null          
p² < 0  -> p is spacelike
 
ex :
r   =  ct*e0 + x*e1 +y*e2 + z*e3
r² =  gp(r,cj(r)) = (c²t² - x² - y² - z²)e0
r² =  (ct)² - r²          r  paravector,  r  space vector = (x,y,z)
 
geometric product :      p cj(q) = p p"              
scalar product  p.q      <p cj(q)>0,3        =  1/2 (p cj(q) + q cj(p))  =  1/2(p q" + q p")
biparavector    p^q     <p cj(q)>1,2        =  1/2 (p cj(q)  - q cj(p))  =  1/2(p q"  - q p")
 
pabs(p)=sqrt(p cj(p))          = sqrt(p p")
punit(p)= p/pabs(p)             = p/|p|
cinv(p) = cj(p)/gp(p,cj(p))   = p"/(p p")
 
ex :  p = e0 + e1 + (1+i)e3
punit(p) = (0.5+0.5i)e0 + (0.5+0.5i)e1 + i*e3
|unit(p)| = pabs(punit(p)) = 1
1/p    = cinv(p) = i/2 e0 - i/2 e1 + (0.5+0.5i)e3
p 1/p = gp(p,cinv(p)) = e0
 
Involutions    
p~  = prev(p)                  reversion    
p'   = pinvol(p)                 grade involution
p"  = cj(p)                       Clifford conjugation
 
grades :
<p>s   = <...>0,3        scalarlike part of p               <p>v   = <...>1,2       vectorlike  part of p
<p>re  = <...>0,1        real                                    <p>im  = <...>2,3       imaginary
<p>+   = <...>0,2       even                                   <p>-    = <...>1,3       odd
 
invariant Cl3 subspaces :
scalar subspace                     invariant under Clifford conjugation
vector subspace                    reverses sign under Clifford conjugation
real subspace                        invariant under reversion conjugation
imaginary subspace               reverses sign under reversion conjugation
         
Biparavector :
from the quadratic form of the sum p+q of paravectors, we obtain the scalar product of paravectors :
<p cj(q)>s = 1/2 (p cj(q) + q cj(p))
p and q are orthogonal iff the scalar product = 0.
 
the vectorlike (vector + pseudovector) part of  p cj(q) is :
<p cj(q)>v = 1/2 (p cj(q) - q cj(p))  = <p cj(q)> - <p cj(q)>s
and represents a spacetime plane containing paravectors p and q.
the biparavectors form a six dimensional linear subspace equal to the direct sum of vector and bivector spaces.
Since bivectors of Cl3 are also pseudovectors, any biparavector is also a complex vector.
 
ex :
p = (3+5i)e0 + 5e1 - (1-2i)e3
q = e0 + e1 + e2 + (1+i)e3
pq = gp(p, cj(q)) = (1+4i)e0 + (2-5i)e1 - (3+5i)e2 + (1-6i)e3 - 5e12 - (1-2i)e23 - (6+3i)e13
<pq>v = (2-5i)e1 - (3+5i)e2 + (1-6i)e3 - 5e12 - (1-2i)e23 - (6+3i)e13   is a biparavector
(pq)² = gp(pq,pq) = -(121+34i)e0  is a scalar.
 
In analogy with bivectors, biparavectors generate rotation in paravector space.
 
example of biparavector in physics :
electomagnetic field  F = E + iB
where both electric and magnetic fields are real vectors
E~ = E
B~ = B
 
Lorentz transformation :
Rotations and reflections in paravector space preserve the scalar product <p cj(q)>s of any two paravectors. Paravector rotations have the same form as vector rotations,
 
   p -> L p L~      odd multiparavector grade
   F -> L F L"       even multiparavector grade
                   
 
where L is a unimodular element (L L" = 1) known as a Lorentz rotor.  
Lorentz rotations are the physical Lorentz transformations of relativity : boosts, spacial rotations, and their product.
Lorentz rotors for spacial rotations are just the same rotors as for vectors, and those for boosts are similar except that the rotation plane in paravector space include the time axis e0.
 

EVA internal representation :
 
scalar                p  = <p>0                         (p0,0,0,0,0,0,0,0)                     3e0
                           = CtoP(p)                      (p0,0,0,0,0,0,0,0)                     3e0  
 
paravector        p  = <p>0 + <p>1            (p0,p1,p2,p3,0,0,0,0)                3e0+e1-2e2+e3  
                           = CtoP(p)                      (p0,p1,p2,p3,0,0,0,0)                3e0+e1-2e2+e3
 
biparavector      p  = <p>1 + <p>2            (0,p1,p2,p3,p4,p5,p6,0)            e1+e2-3e23+e13
                           = CtoP(p)                      (0,z1,z2,z3,0,0,0,0)                  (1-2i)e1+(1-i)e2
 
triparavector     p  = <p>2 + <p>3            (0,0,0,0,p4,p5,p6,p7)                3e23+e13-5e123
                           = CtoP(p)                      (z0,z1,z2,z3,0,0,0,0)                -5ie0+2ie1-ie2
 
pseudoscalar     p  = <p>3                         (0,0,0,0,0,0,0,p7)                      5e123
                           = CtoP(p)                      (z0,0,0,0,0,0,0,0)                      5i                                                          
 
where z = pi +ipj                  
 
Differential operators :
 
pnabla(u,x)
dalembertian(u,x) = pnabla²
 

                               
 
 

                           
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