spinors
A Clifford algebra CLn upon En always has a ring structure. The structure of vectorial space imply an additive Abelian group, and there is an exterior product (generally non commutative) wich is distributive in relation to addition. Ideals exists for these rings and we call spinor every element of an ideal (left or right) wich elements are obtained from one of them, well specified, by left (or right) multiplication with any element from Cln.
 
Pauli spinors :
 
Let Psi an element of Cl3. We say that Psi is a positive spinor if                
                                             Psi e3 = Psi                  
and Psi is a negative spinor if
                                             Psi e3 = -Psi.
 
If Psi is an element of Cl3, we can associate a principal left ideal of positive spinors if we take as particular element
                                             Psi+ = Psi f   with f = 1/2(1 + e3)     -->                      
                                             Psi+ e3 = 1/2 Psi (1 + e3) e3 = Psi+
and a left ideal of negative spinor
                                             Psi- =  Psi f'  with f' = 1/2(1 + e3)    -->  
                                             Psi- e3 = 1/2 Psi (1 - e3) e3 = -Psi-
 
                                       As   Psi = 1/2 Psi (1 + e3) + 1/2 Psi (1 - e3)  -->  
                                             Psi = Psi+ + Psi-
 
we see that Psi is the sum of a positive spinor and a negative spinor.
These two ideals are independant :
 let   l Psi+ +  m Psi- = 0  
 if we multiply this equality by e3, we get
        l Psi+ -  m Psi- = 0
 that imply   l Psi+ = 0    and  m Psi- = 0      ==>     l = m = 0
 
We may determinate a basis for Psi+ and Psi- :
 
let Psi = a0 + a1*e1 + a2*e2 + a3*e3  with ak complex.  
 
Psi+ = 1/2 Psi (1 + e3) = 1/2 gp(Psi, e0 + e3) = 1/2 (a0 + a3, a1, a2, a0 + a3, 0, a2, a1, 0)
 
if we write Psi+ as a paravector :    Psi+ = 1/2 CtoP(a0 + a3, a1, a2, a0 + a3, 0, a2, a1, 0)
                                                       = 1/2 (a0 + a3, a1 + i*a2, a2 - i*a1, a0 + a3)
                                                       = 1/2 (a0 + a3, a1 + i*a2, i*(a1 + i*a2), a0 + a3)
 
                                                Psi+ = 1/2 (a0 + a3)(e0 + e3) +
                                                         1/2 (a1 + i*a2)(e1 + e13)   or
 
                                                Psi+ = 1/2 (a0 + a3)(e0 + e3) +
                                                         1/2 (a1 + i*a2)(e0 - e3) e1
 
same for Psi- with result                 Psi- = 1/2 (a0 - a3)(e0 - e3) +
                                                         1/2 (a1 - i*a2)(e0 + e3) e1
 
with
g1 = 1/2 (e0 + e3)  and   g2 = 1/2 (e0 - e3) e1    form a basis for Psi+    and
h1 = 1/2 (e0 - e3)   and   h2 = 1/2 (e0 + e3) e1   form a basis for Psi-
 
Psi+ = Psi1*g1 + Psi2*g2
Psi- = Psi3*h1 + Psi4*h2
 
In matricial form, with e1 = ((0, 1), (1, 0), e2 = ((0, -i), (i, 0), e3 = ((1, 0), (0, -1)) and e0 = ((1, 0), (0, 1))
 
g1 = 1/2 (e0 + e3)      =  ((1, 0), (0, 0))    = |1 0|
                                                              |0 0|
 
g2 = 1/2 (e0 - e3) e1  =  ((0, 0), (1, 0))    = |0 0|
                                                              |1 0|
and
h1 = 1/2 (e0 - e3)      =  ((0, 0), (0, 1))    = |0 0|
                                                              |0 1|
 
h2 = 1/2 (e0 + e3) e1  =  ((0, 1), (0, 0))    = |0 1|
                                                               |0 0|
 
Psi+ = Psi1*g1 + Psi2*g2 = |Psi1  0|          with  Psi1, Psi2 <- C
                                      |Psi2  0|
 
Psi- = Psi3*h1 + Psi4*h2 = |0  Psi4|          
                                      |0  Psi3|
 


f = 1/2 (e0 + e3) = |1  0|             is idempotent, that is, f² = f
                           |0  0|
Psi+ = Psi f corresponds, in term of Clifford algebra, to an ideal spinor in Cl3 f
 
f' = 1/2 (e0 - e3) = |0  0|             is idempotent
                           |0  1|
Psi- = Psi f' corresponds, in term of Clifford algebra, to an ideal spinor in Cl3 f'
 

  Operator spinors in the even subalgebra Cl3+
 
Let Psi = Psi + Psi'  <- Cl3+        the even part of Psi       (Psi <- Cl3 f,  Psi' = invol(Psi) )
This even spinor Psi <- Cl3+  carries the same amount of information as the original spinor Psi : no information was lost in projecting out the even part of Psi.
 
The spin vector s = s1*e1 + s2*e2 + s3*e3 can be computed as
 
                                          s = Psi e3 Psi~      (Psi~ = rev(Psi) )
 
The spin vector is obtained without using any projection operators : this form of s is applicable in further computations. Since the even spinor Psi acts like an operator, it is also referred as a spinor operator.
 

                           
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