Feynman Path Integral Formulation

174 6 Lattice Regularized Quantum Gravity
δ
Fig. 6.4 Illustration of the deficit angle δ in two dimensions, where several flat triangles meet at a
vertex.
Fig. 6.5 Deficit angle in three dimensions where several flat tetrahedra meet at an edge, and in
four dimensions where several flat four-simplices meet at a triangle.
φ(s)=(φ 0 ...φ d−1 ). Under a Lorentz transformation of Σ(s), described by the d ×d
matrix Λ(s) satisfying the usual relation for Lorentz transformation matrices
the vector φ(s) will rotate to
Λ T ηΛ = η , (6.14)
φ ′ (s) =Λ(s)φ(s) . (6.15)
The base edge vectors e μ i
= l μ 0i
(s) themselves are of course an example of such a
vector.
Next consider two d-simplices, individually labeled by s and s ′ , sharing a common
face f (s,s ′ ) of dimensionality d − 1. It will be convenient to label the d edges
residing in the common face f by indices i, j = 1...d. Within the first simplex s one
can then assign a Lorentz frame Σ(s), and similarly within the second s ′ one can assign
the frame Σ(s ′ ).The 1 2 d(d − 1) edge vectors on the common interface f (s,s′ )
(corresponding physically to the same edges, viewed from two different coordinate
systems) are expected to be related to each other by a Lorentz rotation R,