Feynman Path Integral Formulation

7.3 Lattice Diffeomorphism Invariance 239l 2 i = l 2 0i + q i + δl 2 i , (7.49)where q i describes an arbitrary but small deviation from a regular lattice, and δl 2 i isa gauge fluctuation, whose form needs to be determined. We shall keep terms O(q 2 )and O(q δl 2 ), but neglect terms O(δl 4 ).The squared volumes V 2 n (σ) of n-dimensional simplices σ are given by homogeneouspolynomials of order (l 2 ) n . In particular for the area of a triangle A Δ witharbitrary edges l 1 ,l 2 ,l 3 one hasδA 2 Δ = 1 8 (−l2 1 + l 2 2 + l 2 3)δl 2 1 + 1 8 (l2 1 − l 2 2 + l 2 3)δl 2 2 + 1 8 (l2 1 + l 2 2 − l 2 3)δl 2 3 , (7.50)and similarly for the other quantities which are needed in order to construct theaction. For our notation in two dimensions we refer to Fig. (7.5). The subsequentFigs. 7.6 and 7.7 illustrate the difference between a gauge deformation of the surface,which leaves the area and curvature at the point labeled by 0 invariant, and aphysical deformation which corresponds to a re-assignment of edge lengths meetingat the vertex 0 such that it alters the area and curvature at 0. In the following we willcharacterize unambiguously what we mean by the two different operations.Consider therefore an expansion about a deformed equilateral lattice, for whichl 0i = 1 to start with. A motivation for this choice is provided by the fact that inthe numerical studies of two-dimensional gravity the averages of the squared edgelengths in the three principal directions turn out to be equal, 〈l1 2〉 = 〈l2 2 〉 = 〈l2 3 〉.Thebaricentric area associated with vertex 0 is then given byA = A 0 (q)+ 1 [2 · 3 5/2 δl01 2 (3 + q 06 − 4q 01 + q 02 + q 16 + q 12 )+ δl02 2 (3 + q 01 − 4q 02 + q 03 + q 12 + q 23 )+ δl03 2 (3 + q 02 − 4q 03 + q 04 + q 23 + q 34 )+ δl04 2 (3 + q 03 − 4q 04 + q 05 + q 34 + q 45 )+ δl05 2 (3 + q 04 − 4q 05 + q 06 + q 45 + q 56 )]+ δl06 2 (3 + q 05 − 4q 06 + q 01 + q 56 + q 16 )+ O(δl 4 ) .(7.51)The normalization here is such that A 0 =in more compact notation, at the vertex 0√32 for q i = 0. Equivalently one can write,A = A 0 (q)+ 1 3 v A(q) · δl 2 + O(δl 4 ) , (7.52)with δl 2 =(δl01 2 ,...,δl2 06). After adding the contributions from the neighboringvertices one obtains∑ A = ∑ A 0 (q)+v A (q) · δl 2 + O(δl 4 ) . (7.53)P 0 ...P 6 P 0 ...P 6