Statistical Approach to Quantum Field Theory by Andreas Wipf
Author:Andreas Wipf
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg
(9.71)
where x is any site on the lattice. In the limit N→0 all terms containing coefficients vanish. This means that graphs with tvertices that have more han two lines attached, or equivalently with several lines between nearest neighbors, do not contribute. All graphs must be connected since disconnected graphs contain at least one full contraction of indices and hence are suppressed by at least one power of N. We conclude that only non-intersecting connected graphs contribute which connect the lattice points 0 and x. Two examples are shown in Fig. 9.19. Since 〈s a s b 〉∝δ ab the internal index is preserved along the line connecting 0 and x and hence all spins on the line have the same index as the spins at the endpoints. The contribution of a graph is equal to , where 0 and x are connected by a line with length ℓ. Thus we find
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