Feynman Path Integral Formulation

6.4 Rotations, Parallel Transports and Voronoi Loops 175
l μ ij (s′ )=R μ ν(s ′ ,s) l ν ij(s) . (6.16)
Under individual Lorentz rotations in s and s ′ one has of course a corresponding
change in R, namely R → Λ(s ′ )R(s ′ ,s)Λ(s). In the Euclidean d-dimensional case
R is an orthogonal matrix, element of the group SO(d).
In the absence of torsion, one can use the matrix R(s ′ ,s) to describes the parallel
transport of any vector φ μ from simplex s to a neighboring simplex s ′ ,
φ μ (s ′ )=R μ ν(s ′ ,s)φ ν (s) . (6.17)
R therefore describes a lattice version of the connection (Lee, 1983). Indeed in the
continuum such a rotation would be described by the matrix
R μ ν =
(e Γ ·dx) μ
ν , (6.18)
with Γ λ
μν the affine connection. The coordinate increment dx is interpreted as joining
the center of s to the center of s ′ , thereby intersecting the face f (s,s ′ ). On the other
hand, in terms of the Lorentz frames Σ(s) and Σ(s ′ ) defined within the two adjacent
simplices, the rotation matrix is given instead by
R a b (s′ ,s) =e a μ(s ′ )e ν b (s) Rμ ν(s ′ ,s) , (6.19)
(this last matrix reduces to the identity if the two orthonormal bases Σ(s) and
Σ(s ′ ) are chosen to be the same, in which case the connection is simply given by
R(s ′ ,s) μ
ν = eμ a e ν a). Note that it is possible to choose coordinates so that R(s,s ′ ) is
the unit matrix for one pair of simplices, but it will not then be unity for all other
pairs if curvature is present.
This last set of results will be useful later when discussing lattice Fermions. Let
us consider here briefly the problem of how to introduce lattice spin rotations.Given
in d dimensions the above rotation matrix R(s ′ ,s), the spin connection S(s,s ′ ) between
two neighboring simplices s and s ′ is defined as follows. Consider S to be
an element of the 2 ν -dimensional representation of the covering group of SO(d),
Spin(d), with d = 2ν or d = 2ν + 1, and for which S is a matrix of dimension
2 ν × 2 ν . Then R can be written in general as
[ ]
R = exp 12
σ αβ θ αβ , (6.20)
where θ αβ is an antisymmetric matrix The σ’s are 1 2d(d − 1) d × d matrices, generators
of the Lorentz group (SO(d) in the Euclidean case, and SO(d − 1,1) in the
Lorentzian case), whose explicit form is
[ ] γ σαβ
δ = δ γ α η βδ − δ γ β η αδ , (6.21)
so that, for example,