The trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of nth order. The coefficients of this expansion (the nonlocal form factors) are calculated to third order in the curvature inclusive. The earlytime and late-time asymptotic behaviours of the trace of the heat kernel are presented with this accuracy. The late-time behaviour gives the criterion of analyticity of the effective action in quantum field theory. The latter point is exemplified by deriving the effective action in two dimensions.
The one-loop effective action for a generic set of quantum fields is
calculared as a nonlocal expansion in powers of the curvatures (field
strengths). This expansion is obtained to third order in the curvature. It is
stressed that the covariant vertices are finite. The trace anomaly in four
dimensions is obtained directly by varying the effective action. The nonlocal
terms in the action, producing the anomaly, contain non-trivial functions of
three operator arguments. The trace anomaly is derived also by making the
conformal transformation in the heat kernel.Comment: 26 pages, REVTeX, Alberta Thy 15-9
A complete basis of nonlocal invariants in quantum gravity theory is built to third order in spacetime curvature and matter-field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality 2ω < 6. The present results are used in heat-kernel theory, theory of gauge fields and serve as a basis for the model-independent approach to quantum gravity and, in particular, for the study of nonlocal vacuum effects in the gravitational collapse problem.
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