Feynman Path Integral Formulation

172 6 Lattice Regularized Quantum Gravitywith 1 ≤ i, j ≤ n, and which in the Euclidean case is positive definite. In componentsone has g ij = η ab e a i eb j . In terms of the edge lengths l ij = |e i −e j |, the metric is givenbyg ij (s) = 1 (2 l20i + l0 2 j − lij2 ). (6.3)Comparison with the standard expression for the invariant interval ds 2 = g μν dx μ dx νconfirms that for the metric in Eq. (6.3) coordinates have been chosen along the ne idirections.The volume of a general n-simplex is given by the n-dimensional generalizationof the well-known formula for a tetrahedron, namelyV n (s) = 1 n!√detg ij (s) . (6.4)An equivalent, but more symmetric, form for the volume of an n-simplex can begiven in terms of the bordered determinant of an (n + 2) × (n + 2) matrix (Wheeler,1964)1/2∣V n (s) =∣ ∣∣∣∣∣∣∣∣∣∣ 0 1 1 ...1 0 l01 2 ...1 l 2 10 0 ...n!2 n/2 1 l20 2 l21 2 ...... ... ... ...1 ln,0 2 ln,1 2 ∣...n+1(−1) 2. (6.5)It is possible to associate p-forms with lower dimensional objects within a simplex,which will become useful later (Hartle, 1984). With each face f of an n-simplex (inthe shape of a tetrahedron in four dimensions) one can associate a vector perpendicularto the faceω( f ) α = ε αβ1 ...β n−1e β 1(1) ...eβ n−1(n−1) , (6.6)where e (1) ...e (n−1) are a set of oriented edges belonging to the face f , and ε α1 ...α nis the sign of the permutation (α 1 ...α n ).The volume of the face f is then given byV n−1 ( f )=( n∑ 1/2ωα( 2 f )). (6.7)α=1Similarly, one can consider a hinge (a triangle in four dimensions) spanned by edgese (1) ,..., e (n−2) . One defines the (un-normalized) hinge bivectorω(h) αβ = ε αβγ1 ...γ n−2e γ 1(1) ...eγ n−2(n−2) , (6.8)with the area of the hinge then given by( ) 1/21V n−2 (h) =(n − 2)!∑ ω 2 αβ (h) . (6.9)α