the Majorana generators: $\sigma_j = \basis{4}\basis{j}$ for $j=1,2,3$.

Automorphisms and Conservation Laws
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Fundamental to physical theories are [*conservation laws*]{}, e.g.conservation of energy. These are intimately related via Noether’s Theorem[@Drell65] to the physical principles being invariant under certain transformation (e.g. rotations). The “allowed” symmetries should correspond mathematically to those generated by the automorphsim group of the geometric algebra.

Of particular interest are the orthogonal transformations associated with hyperbolic rotations between space and time. These have the physical interpretation of connecting two reference systems, one at rest and the other moving at velocity $\vector{V}={d\vector{X} \over dt}$. A particle of mass $m$ at rest in the “moving” frame, will be perceived as moving at velocity $\vector{V}$ in the “rest” frame, with [*four-vector momentum*]{} [**p**]{}, $${\bf p}=p^{\mu}\basis{\mu}=
{\cal L} \ m\basis{4}\ {\cal L}^{-1}, \eqno(2.5a)$$ $${\cal L}=\expf{ {r \hat{\beta}} \over 2} =
{ {-{\bf p}\basis{4} + m} \over \sqrt{2m(E+m)}}=
{\momentum + E + m \over \sqrt{2m(E+m)}},\eqno(2.5b)$$ $$E=p^4 = -\basis{4}\cdot{\bf p}, \eqno(2.5c)$$ $$\vector{P}=E\vector{V}=\basis{4}\wedge{\bf p}, \eqno(2.5d)$$ $$\tanh(r) = \nnn{V} = E^{-1} \nnn{P}, \eqno(2.5e)$$ $$\hat{\beta} = {\vector{V}\over \nnn{V}}=
{\vector{P}\over \nnn{P}}. \eqno(2.5f)$$ The term ${\cal L}$ is called the (half-angle) [*Lorentz Boost operator*]{}, where $r$ is the [*rapidity*]{} associated with the velocity. The unit bivector $\hat{\beta}$ points in 