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In this paper, authors propose taking into account the variation
of the metric to study the relation between scale and conformal
invariance.
Since the proposed transformation is not an isometry
of the flat metric, the transformed theory describes a field
theory in the presence of background metric such as the dilaton
background. By requiring that two field theories with and without
the background are equivalent, we can define the scale or
conformal invariant field theories. In this way, we can discuss
whether or not the scale invariance implies the conformal
invariance.
This paper would have been more interesting if the program is
performed in the curved space time. In this paper, however,
authors stick to the flat space time, and no new result is
obtained, except to pint out there scale and conformal invariance
is not equivalent, in principle.
Since calculation is correct, this paper may be published even
though it is not so interesting.
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