Operator averages, centroid density

From the cyclic invariance of the trace, the operator can be evaluated at any point along the cyclic imaginary-time path (t). This is depicted schematically in Fig. 2, where the centroid variable, which is off the thermal loop, is also shown. It is nontrivial, therefore, to reformulate the rules for calculating an operator average in order that the centroid density be used as the statistical distribution [34, 45]. To illustrate this principle, the case of an operator average is described here in some detail. As the first step, Eq. (2.43) is rewritten as... 

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