The HyperDiamond Lattice
The random walk did not reach all points in space time:
only those points 
where n+and n-are non-negative integers and 
if we draw the random
walk in the complex plane with the imaginary axis being the time axis.
Alternatively, if we use a real axis for time then the a± would be the 2-vectors:
where the top row is
the time coordinate.
In either case, the a± are the
basis vectors of a two-dimensional diamond lattice.(Feynman checkerboard) and
we can generate all possible random walks by combining integer multiples of
these two basis vectors.
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In four dimensions generate all possible random walks by combining integer multiples of the following hyper-diamond basis vectors.(see appropriate entries in Tony Smith’s web page )
(1)
The column vectors are relative to an ordinary Cartesian
system:
.
The vectors projected on three space look like this:
Concerning the
normalization: In the two dimensional case, each basis vector covers one
unit of spatial distance in one unit of time. The speed of light is taken to be
one (c=1). In the four dimensional case, each basis vector covers a distance of
units of spatial
distance. If we want to set all
coordinates to ½., then we must set
: 
,
In order to study the rotation properties of these vectors we represent them as
quaternions. We use the Clifford algebra notation.
(Note that all four vectors point forward in time; There is also a set of conjugate vectors that point in the opposite spatial direction.
(A random walk in space time will consist of a sequence of steps along any combination of the vectors in (2) or (3) but not both. Certain sequences are excluded. For example a 180 degree reversal from qR to qRb is not allowed. However a particle can reverse direction thru a sequence of three direction changes along the unit vectors. For example one step along the qR vector brings the particle to the space time point {t=1,x=1,y=1,z=1}. Three successive steps along {qB,qG,qY} brings the particle to the space time point {t=3,x= -1,y= -1, z= -1}. The reason for this restriction is that we consider each change in direction from one unit vector to another as having equal probability weight. The 180 degree reversal is of lower probability and should not be weighed the same as the 90 degree change of direction.
The diamond lattice vectors of (2) are space like unit vectors (|qX|=-1). And they are orthogonal with respect to the four-vector dot product:, for example:
(4)
(Note that the 3D
projected vectors in the picture are at a tetrahedral 109° ).
We can verify the following:
(Get the remaining 9 relations by reversing the rotation, and cyclic permutation.
From HyperDiamond Coordinates to Cartesian and Back
. Suppose an event location E is described in terms of the hyper-diamond basis:
(6)
The same event expressed in terms of a Cartesian basis would be:
(7)
Likewise, to go from Cartesian coordinates to hyperdiamond:
(8)