means that the basis element associated with time \[(e.g. $\sigma_0$ of eq. (2.1)\] must anticommute with the other three mutually anticommuting basis vectors. In order to avoid confusion, we will introduce a new set of symbols: $\{\basis{\mu} \}$ as the basis vectors of space, with $\basis{4}$ (instead of $\basis{0}$) as the fourth basis element. Hence the coordinate [*four-vector*]{} will be expressed in boldface lower case (no overhead arrow), $${\bf x} = (x,y,z,t) = x^{\mu} \basis{\mu}=
x\basis{1}+y\basis{1}+z\basis{3}+t\basis{4}, \eqno(2.2)$$ where the Einstein summation notation is assumed on the repeated index: $\mu =1,2,3,4$.

The [*metric*]{} of time is empirically known to be of the opposite sign as that of the positional portion. Mathematically, we would say that the orthogonal space is either ${\bf R}^{3,1}$ or ${\bf R}^{1,3}$, corresponding to metric signatures $(+++-)$ or $(---+)$ respectively (physicists refer to these respectively as the [*east coast*]{} or [*west coast*]{} metrics). In the former case, the norm of the position four-vector would be, $$\|{\bf x}\|= x_{\mu}x^{\mu} = x^2+y^2+z^2-t^2, \eqno(2.3)$$ where $x_4=-x^4=t$, but $x_i=x^i$ for $i=1,2,3$. Physically, this hyperbolic nature is interpreted as a manifestation of the [*principle of causality*]{}. For an “event” at the origin to be the source or [*cause*]{} of another event at ${\bf 