Could you go through a demonstration of how half-derivatives lead to spinors?
Let's have 3-dimensional Euclidean space. Let's define 3 scalar fields
x,
y and
z that are linear in one dimension and constant in other dimensions.
We can define 3 "spinor differentials" that are anticommutative. They are elements of Grassman algebra. Let's call them
sx,
sy and
sz.
Let
s be the half-differential operator. We can postulate its linearity, anticommutativity and the differential product rule:
s(a F) = a sF
s(F + G) = sF + sG
s(F G) = sF G + f sG
sF sG = -sG sF
F and
G are fields,
a is a scalar. Note that this is noncommutative algebra, so order of multiplication is important, especially in the product rule. The unit spinors
sx,
sy and
sz are half-differentials of the scalar fields
x,
y and
z respectively.
Also note this algebra is not quaternions. In quaternions you have
i2 = -1, here we have
dx2 = 0. We have derived geometric algebra.
A general spinor is a field of a form:
S = a sx + b sy + c sz
Of course
a,
b and
c may depend on
x,
y and
z. Let's name their half-differentials too.
sa = a
x sx + a
y sy + a
z sz
sb = b
x sx + b
y sy + b
z sz
sc = c
x sx + c
y sy + c
z sz
Now let's take the half-differential once again.
sS = s(a sx) + s(b sy) + s(c sz) = sa sx + a ssx + sb sy + b ssy + sc sz + c ssz =
= (a
x sx + a
y sy + a
z sz) sx +
+ a ssx +
+ (b
x sx + b
y sy + b
z sz) sy +
+ b ssy +
+ (c
x sx + c
y sy + c
z sz) sz +
+ c ssz =
= a ssx + b ssy + c ssz +
+ 0 + a
y sy sx + a
z sz sx +
+ b
x sx sy + 0 + b
z sz sy +
+ c
x sx sz + c
y sy sz + 0 =
= a ssx + b ssy + c ssz +
+ (b
x - a
y) sx sy + (c
y - b
z) sy sz + (a
z - c
x) sz sx
Note that we have not said anything about the product of two spinors. The only thing we know that a quare of a spinor is always zero and multiplication is antisymmetrical. In particular, it is not true that
sx sy = sz . So this is not a quaternion algebra. In fact, the product of two spinors is another object we may call a pseudovector.
So the double half-derivative of a scalar field gave us another object. The first part, the sum
V = a ssx + b ssy + c ssz
is a familar covariant vector.
The second part
P = (b
x - a
y) sx sy + (c
y - b
z) sy sz + (a
z - c
x) sz sx
we may call a general pseudovector.
We may now define the differential operator
dF = s2F = s(sF).
The differentials
dx = ssx,
dy = ssy,
dz = ssz are unit vectors. They are commutative and they behave as vectors should do.
This construction can be generalized to any number of dimensions, except pseudovectors have 3 components only in 3d space. Number of components of pseudovectors is always
(d2 - d)/2. It is an important coincidence that pseudovectors and covariant vectors have the same number of components in 3d. This is also the reason why quaternion algebra describes some properties of 3d space.
A different coincidence occurs in 7d space, where pseudovectors have
21 = 7 * 3 components. The related algebra describing some aspects of rotations in 7d space is octonions.
I personally believe, but this is only my gut feeling, that this vector plus pseudovector result has something to do with the V-A weak interaction theory and the CP symmetry breaking. But this is only my personal opinion.