Pictures Paths Particles Processes

142 November 7, 2013
under Lorentz transformations. It is somewhat surprising to see that the form of
the Lorentz transformation in Clifford space is quite simple. Since all spinorial
dyads ξη are Clifford elements, we find from the above that the transformation
rules are
ξ → Σ ξ , ξ → ξ Σ . (5.87)
Let us now select the spinor of a particle in its rest frame, and consider rotations
of the space axes. By x µ , y µ and z µ we shall mean the four-dimensional
extensions of the spatial unit vectors in the x-, y- and z-directions, respectively.
A rotation Σ z over an infintesimal angle θ from x towards y around the z axis 25
is then determined by choosing
p µ = x µ , q µ = cos(θ)x µ + sin(θ)y µ ≈ x µ + θy µ , (5.88)
if we restrict ourselves to first order in θ. To this order, we find that |C| = 1/2,
and so
Σ z ≈ 1 2 (1 − (/x + θ/y)/x) = 1 + θ /x/y . (5.89)
2
(realize that x 2 = y 2 = z 2 = −1). The generators of the rotation group must
therefore be 26 T x = β/y/z , T y = β/z/x , T z = β/x/y , (5.90)
where we have used cyclicity, but not specified the constant β. This constant
can be determined from the rotation group algebra requirement:
which for the Dirac system is seen to read
[T x , T y ] = T x T y − T y T x = i¯hT z , (5.91)
[T x , T y ] = β 2 ( /y/z/z/x − /z/x/y/z ) = 2β 2 /x/y = 2β T z , (5.92)
from which we see that β = i¯h/2. Noticing also that 27
T z 2 = β 2 /x/y/x/y = −β 2 x 2 y 2 = ¯h2
4 = T x 2 = T y 2 , (5.93)
we conclude that the total-spin operator comes to
⃗T 2 = T x 2 + T y 2 + T z 2 = 3 4¯h2 . (5.94)
The spinors are, therefore, representatives of a spin-1/2 system.
25 Here the confusing active-passive distinction rears its ugly head. We shall not worry about
it since the rotation algebra is the same in each case.
26 By inserting the Pauli representation of the Dirac matrices, one may figure out that these
generators are nothing but the Pauli matrices in disguise. The present treatment aims at a
more relativistic description.
27 The fact that the square of any of the generators is proportional to the unit matrix is
more or less a coincidence ; for systems with higher spins it no longer holds.