Feynman Path Integral Formulation

246 7 Analytical Lattice Expansion MethodsOne then observes the following: the first two terms describe the local fluctuation ofδA on a given hinge; the third and fourth terms describe correlations between δAterms on different hinges. But because the action is local, the only non-vanishingcontribution to the last two terms comes from edges and hinges which are in the immediatevicinity of the hinge in question. For hinges located further apart (indicatedbelow by “not nn”) one has that their fluctuations remain uncorrelated, leading to avanishing variance∫( ∫dμ(l 2 )δ A ∑′ δ h A h dμ(l 2 )δ A) ( )∫dμ(l 2 ) ∑′ δ h A hhnotnnhnotnn∫−( ∫dμ(l 2 2= 0 ,)dμ(l )) 2(7.80)since for uncorrelated random variables X n ’s, < X n X m > − < X n >< X m >= 0.Therefore the only non-vanishing contributions in the last two terms in Eq. (7.79)come from hinges which are close to each other.The above discussion makes it clear that a key quantity is the correlation betweendifferent plaquettes,∫dμ(l 2 )(δ A) h (δ A) h ′ e k ∑ h δ h A h< (δ A) h (δ A) h ′ > = ∫, (7.81)dμ(l 2 )e k ∑ h δ h A hor, better, its connected part (denoted here by C )< (δ A) h (δ A) h ′ > C ≡ < (δ A) h (δ A) h ′ > − < (δ A) h >, (7.82)which subtracts out the trivial part of the correlation. Here again the exponentials inthe numerator and denominator can be expanded out in powers of k, as in Eq. (7.69).The lowest order term in k will involve the correlation∫dμ(l 2 )(δ A) h (δ A) h ′ . (7.83)But unless the two hinges are close to each other, they will fluctuate in an uncorrelatedmanner, with < (δ A) h (δ A) h ′ > − < (δ A) h >< (δ A) h ′ >= 0. In order toachieve a non-trivial correlation, the path between the two hinges h and h ′ needs tobe tiled by at least as many terms from the product (∑ h δ h A h ) n in( )∫ndμ(l 2 )(δ A) h (δ A) h ′ ∑δ h A h , (7.84)has are needed to cover the distance l between the two hinges. One then has< (δ A) h (δ A) h ′ > C ∼ k l ∼ e −l/ξ , (7.85)