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Why does dilation invariance often imply proper conformal invariance?

What exactly is the difference between the two? Can someone give an example of a theory which is scale invariant but not conformally invariant?

  • $\begingroup$ Two counterexamples are e.g. given here arxiv.org/abs/hep-th/0006037 $\endgroup$ – Qmechanic May 3 '12 at 13:53
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    $\begingroup$ This question is answered here and hence essentially a duplicate. $\endgroup$ – Qmechanic May 3 '12 at 13:55
  • $\begingroup$ @Qmechanic: the answers to that previous question are now outdated due to some progress over the last two years. $\endgroup$ – Vibert Sep 20 '13 at 6:30