94-273 Apfeldorf K. M., Ordonez C.

Field Redefinition Invariance in Quantum Field Theory (85K, LaTeX) Aug 18, 94

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Abstract. We investigate the consequences of field redefinition invariance in quantum field theory by carefully performing nonlinear transformations in the path integral. We first present a ``paradox'' whereby a $1+1$ free massless scalar theory on a Minkowskian cylinder is reduced to an effectively quantum mechanical theory. We perform field redefinitions both before and after reduction to suggest that one should not ignore operator ordering issues in quantum field theory. We next employ a discretized version of the path integral for a free massless scalar quantum field in $d$ dimensions to show that beyond the usual jacobian term, an infinite series of divergent ``extra'' terms arises in the action whenever a nonlinear field redefinition is made. The explicit forms for the first couple of these terms are derived. We evaluate Feynman diagrams to illustrate the importance of retaining the extra terms, and conjecture that these extra terms are the exact counterterms necessary to render physical quantities invariant under field redefinitions. We see explicitly how these extra terms are essential to understanding why the unphysical practice of dimensional regularization works at all. We indicate how the extra counterterms cancel out unwanted divergent contributions to physical quantities so that the result is {\em consistent} with simply setting divergences such as $\delta^{(d)}(0)$ equal to zero in evaluations of Feynman diagrams. An exciting possibility is that these extra terms could allow for the presence of anomalies of higher orders in $\hbar$ in quantum field theories with nonlinear symmetries.

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