Apfeldorf K. M., Ordonez C.
Field Redefinition Invariance in Quantum Field Theory
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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.