1. generates 3-dimensional rotations
and with the basis:
serve as the angular
momentum operators!
And of course we already had
and with the simpler basis:

2. U s3U = †
U s1U = †
U s2U = †

3. U†1UU†2U- U†2UU†1U = iU†3UJx
Since U†U = UU† = I
Jx Jy- Jy Jx = iJz

4. , a b c x y z This 6×6 matrix also satisfies the same algebra:a b c x y z ,
3-dimensional transformations (like rotations) are
not limited to 3-dimensional “representations”

5. Besides the infinite number of similarity transformations that could
produce other 3×3 matrix representations of this algebra

6. R ( ) = e- iJ·/ħ in the orthonormal basis “DEFINING” representation^
can take many forms
The 3-dimensional representation
in the orthonormal basis
that diagonalizes z is the
“DEFINING” representation
of vector rotations

7. ℓ = 2
mℓ = -2, -1, 0, 1, 2
ℓ = 1
mℓ = -1, 0, 1 2 1 1
L2 = 1(2) = 2
|L| = 2 =
L2 = 2(3) = 6
|L| = 6 =
Note the always odd number
of possible orientations:
A “degeneracy” in otherwise identical states!

8. Spectra of the alkali metals (here Sodium) all show lots of doublets
1924: Pauli suggested electrons
posses some new, previously
un-recognized & non-classical
2-valued property

9. Perhaps our working definition of angular momentum was too literal
…too classical
perhaps the operator relations
Such “Commutation Rules”
are recognized by
mathematicians as
the “defining algebra”
of a non-abelian
(non-commuting) group
may be the more fundamental definition
[ Group Theory; Matrix Theory ]
Reserving L to represent orbital angular momentum, introducing the
more generic operator J to represent any or all angular momentum
study this as an
algebraic group
Uhlenbeck & Goudsmit find actually J=0, ½, 1, 3/2, 2, … are all allowed!

10. ( ) ( ) ( ) ( ) spin : p, n, e, , , e ,  ,  , u, d, c, s, t, b
quarks
leptons
spin : 1 2
p, n, e, , , e ,  ,  , u, d, c, s, t, b
the fundamental constituents of all matter!
spin “up”
spin “down”
s = ħ = ħ 3 2
ms = ± 1 2
sz = ħ 1 2
( ) 1
| n l m > | > = nlm 1 2 1 2
“spinor”
( )
( )
( )
the most general state is
a linear expansion in this
2-dimensional basis set  
=  
with a 2 + b 2 = 1

11. s- s- s- s+ obviously: How about the operators sx, sy ? = 1ħ ħ = 1ħ ħ
eigenvalues of each are also ħ/2 but
their matrices are not diagonal
in this basis
How about the operators sx, sy ? s- s-
= 1ħ ħ s- s+
= 1ħ ħ
obviously work on
the basis we’ve defined

12. You already know these as the Pauli matrices
obeying the same commutation rule:
but 2-dimensionally!
What if we used THESE as generators?

13. x , y , z Should still describe “rotations”.
Its 3-components will still require
3 continuously variable
independent parameters:
x , y , z
But this is not the defining representation
and can not act on 3-dimensional space vectors.
These are operators that obviously act on the SPINORS
(the SPIN space, not the 3-dimensional wave functions.

14. Spinors are 2-component objects
intermediate between scalars (1-component) and vectors (3-component)
When we rotate the “coordinate system” scalars are unchanged.
Vector components are mixed by the prescriptions we’ve outlined.
What happens to SPINORS?
actually can only act on the spinor part

15. The rotations on 3-dim vector space involved ORTHOGONAL operators
Rt = R-1 i.e. RtR = RRt = 1
These carry complex elements,
cannot be unitary!
As we will see later this
is a UNITARY MATRIX
of determinate = 1
Let’s stay with the simplified case of rotation  =  z
(about the z-axis) ^

16. Notice: zz =
and obviously: zzz = z ( ) ( )
+ iz

17. ( ) ( ) = -1 + 0 an operator analog to: ei = cos + isin 1 0 1 0
( )
( )
1 0
0 -1
1 0
0 1
Let’s look at a rotation of 2 (360o) =
This means
A 360o rotation does not bring a spinor “full circle”.
Its phase is changed by the rotation.

18. Limitations of Schrödinger’s Equation
1-particle
equation
2-particle
equation:
mutual interaction

19. np+e++e n+p  n+p+3 e-+ p  e-+ p + 6 + 3g
But in many high energy reactions
the number of particles is not conserved!
np+e++e
n+p  n+p+3
e-+ p  e-+ p + 6 + 3g

20.  then and as we’ve argued before,
Let’s expand the DEL operator from 3- to 4-dimensions
i.e. 
lowered
since operates on x
and as we’ve argued before,
the starting point for a relativistic
QM equation:
then
The Klein-Gordon
Equation
or if you prefer:

21. - = 0 *( ) ( ) Look at Schrödinger’s Equation for a free particle
But this equation has a drawback:
Look at Schrödinger’s Equation for a free particle
*( )
( ) -
= 0
The Continuity
Equation
probability
density
probability
current density

22. *( ) -  ( )* = 0 starting from the Klein-Gordon Equation
*( ) -
 ( )* = 0
not positive definite
The Klein-Gordon Equation is 2nd order in t!
much more
complicated
time evolution
Need to know initial (t=0) state as well as (0)