FIG. 17 The projection of the FCHC lattice into two dimensions.

In two dimensions, mi=4 along cube edges, while mi=1 along face diagonals. In the twodimensional case, four of the FCHC velocities project to a “rest” direction for which the velocity vanishes.

If one uses the notation ciα to denote the component of ci along the α direction, in a Cartesian coordinate system aligned with its axes parallel to the edges of the cubic lattice, one can derive the following identities:

∑imiciα=0 (149)

∑imiciαciβ=12δαβ (150)

∑imiciαciβciγ=0 (151)

∑imiciαciβciγciδ=4δαβδγδ+4δαγδβδ+4δαδδβγ. (152)

The above identities are valid both for three-dimensional and two-dimensional projections of the FCHC lattice.