Feynman Path Integral Formulation

6.14 Lattice Fermions, Tetrads and Spin Rotations 211
Without loss of generality we can fix the average edge length to be equal to one,
〈l〉 = 1, which then requires λ 0 = 5 2
+ σ. In order for the model to be meaningful,
the measure parameter is constrained by σ > −5/2, i.e. the measure over the edges
cannot be too singular.
6.14 Lattice Fermions, Tetrads and Spin Rotations
On a simplicial manifold spinor fields ψ s and ¯ψ s are most naturally placed at the center
of each d-simplex s. In the following we will restrict our discussion for simplicity
to the four-dimensional case, and largely follow the original discussion given in
(Drummond, 1986). As in the continuum (see for example Veltman, 1975), the construction
of a suitable lattice action requires the introduction of the Lorentz group
and its associated tetrad fields e a μ(s) within each simplex labeled by s.
Within each simplex one can choose a representation of the Dirac gamma matrices,
denoted here by γ μ (s), such that in the local coordinate basis
{γ μ (s),γ ν (s)} = 2g μν (s) . (6.161)
These in turn are related to the ordinary Dirac gamma matrices γ a , which obey
{
γ a ,γ b} = 2η ab (6.162)
by
γ μ (s) =e μ a (s)γ a , (6.163)
so that within each simplex the tetrads e a μ(s) satisfy the usual relation
e μ a (s) e ν b (s) ηab = g μν (s) . (6.164)
In general the tetrads are not fixed uniquely within a simplex, being invariant under
the local Lorentz transformations discussed earlier in Sect. 6.4.
In the continuum the action for a massless spinor field is given by
∫
I = dx √ g ¯ψ(x)γ μ D μ ψ(x) , (6.165)
where D μ = ∂ μ + 1 2 ω μabσ ab is the spinorial covariant derivative containing the spin
connection ω μab . It will be convenient to first consider only two neighboring simplices
s 1 and s 2 , covered by a common coordinate system x μ . When the two tetrads
in s 1 and s 2 are made to coincide, one can then use a common set of gamma matrices
γ μ within both simplices. Then in the absence of torsion the covariant derivative D μ
in Eq. (6.165) reduces to just an ordinary derivative. The fermion field ψ(x) within
the two simplices can then be suitably interpolated, by writing for example