Feynman Path Integral Formulation

212 6 Lattice Regularized Quantum Gravity
ψ(x) =θ(n · x)ψ(s 1 )+[1 − θ(n · x)]ψ(s 2 ) , (6.166)
where n μ is the common normal to the face f (s 1 ,s 2 ) shared by the simplices s 1 and
s 2 , and chosen to point into s 1 . Inserting the expression for ψ(x) from Eq. (6.166)
into Eq. (6.165) and applying the divergence theorem (or equivalently using the fact
that the derivative of a step function only has support at the origin) one obtains
I = 1 2 V (d−1) ( f )(¯ψ 1 + ¯ψ 2 )γ μ n μ (ψ 1 − ψ 2 ) , (6.167)
where V (d−1) ( f ) represents the volume of the (d − 1)-dimensional common interface
f , a tetrahedron in four dimensions. But the contributions from the diagonal
terms containing ¯ψ 1 ψ 1 and ¯ψ 2 ψ 2 vanish when summed over the faces of an n-
simplex, by virtue of the useful identity
n+1
∑ V ( f (p) )n (p)
μ = 0 , (6.168)
p=1
where V ( f (p) ) are the volumes of the p faces of a given simplex, and n (p)
μ the inward
pointing unit normals to those faces.
So far the above partial expression for the lattice spinor action was obtained by
assuming that the tetrads ea μ (s 1 ) and ea μ (s 2 ) in the two simplices coincide. If they do
not, then they will be related by a matrix R(s 2 ,s 1 ) such that
e μ a (s 2 )=R μ ν(s 2 ,s 1 ) e ν a (s 1 ) , (6.169)
and whose spinorial representation S was given previously for example in Eq. (6.24).
Such a matrix S(s 2 ,s 1 ) is now needed to additionally parallel transport the spinor ψ
from a simplex s 1 to the neighboring simplex s 2 .
The invariant lattice action for a massless spinor takes therefore the form
I = 1 2
∑
faces f(ss ′ )
V [ f (s,s ′ )] ¯ψ s S[R(s,s ′ )]γ μ (s ′ )n μ (s,s ′ )ψ s ′ , (6.170)
where the sum extends over all interfaces f (s,s ′ ) connecting one simplex s to a
neighboring simplex s ′ . As shown in (Drummond, 1986) it can be further extended
to include a dynamical torsion field.
The above spinorial action can be considered analogous to the lattice Fermion
action proposed originally in (Wilson, 1973) for non-Abelian gauge theories. It is
possible that it still suffers from the Fermion doubling problem, although the situation
is less clear for a dynamical lattice (Lehto, Nielsen and Ninomiya, 1987; Christ
and Lee, 1982).