Feynman Path Integral Formulation

7.3 Lattice Diffeomorphism Invariance 241A similar calculation can be done for the curvature associated with vertex 0. Onehas for the deficit angle at 0δ = δ 0 (q)+ 13 3/2 [δl 2 01 (3 − 2q 06 − q 01 − 2q 02 + q 16 + q 12 )+ δl02 2 (3 − 2q 01 − q 02 − 2q 03 + q 12 + q 23 )]+ ··· + O(δl 4 ) ,(7.56)and therefore for the variation of the sum of the deficit angles surrounding 0( )Δ ∑δ h = ∑ Δδ (7.57)hP 0 ...P 6and∑ δ = ∑ δ 0 (q)+v R (q) · δl 2 + O(δl 4 ) (7.58)P 0 ...P 6 P 0 ...P 6with in this case, as expected, v R (q) ≡ 0.Finally for the curvature squared associated with vertex 0 one computesδ 2A = δ 2 0A 0(q)+ 43 3/2 [δl 2 01(q01 +q 02 + q 03 + q 04 + q 05 + q 06− q 12 −q 23 − q 34 − q 45 − q 56 − q 16)+δl 2 02(q01 +q 02 + q 03 + q 04 + q 05 + q 06− q 12 −q 23 − q 34 − q 45 − q 56 − q 16)+ ···]+ O(δl 4 ) .(7.59)Adding up all seven contributions one gets( )Δ ∑δh 2 /A h = ∑ Δ(δ 2 /A) , (7.60)hP 0 ...P 6and therefore∑ δ 2 (/A = ∑ δ 2 /A ) 0 + v R 2(q) · δl2 + O(δl 4 ) . (7.61)P 0 ...P 6 P 0 ...P 6In this case the curvature squared associated with the vertex 0 will remain unchanged,provided the variations in the squared edge lengths meeting at 0 satisfythe constraintv R 2(q) · δl 2 = 0 , (7.62)