conformal invariance

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In differential geometry, a conformal transformation is diffeomorphism which leaves the metric unchanged except for an overall conformal factor. I.e., a diffeomorphism x^\mu \to x^{\mu'} = f^\mu(x)\,, under which

g_{\mu\nu}(x)\, \to g_{{\mu'}{\nu'}}(x')\, = g_{\mu\nu} \frac{\partial x^\mu}{\partial x^{\mu'}} \frac{\partial x^{\nu} }{\partial x^{\nu'}}(f^{-1}(x))\,,

such that g_{{\mu'}{\nu'}} = \Omega^2 g_{\mu\nu}\,, with \Omega^2\, being some conformal factor. Infinitesimally, we demand that f^\mu\, be a conformal Killing vector. We therefore require that the vector field that generates the flow satisfy the conformal Killing equation.

In conformal field theory, two metrics are considered equivalent if they are related by a conformal factor (i.e., they are related by a Weyl transformation). Therefore, a conformal transformation is a change of coordinates[1] of the above form, without the accompanying change in the functional form of the metric. Such transformations form a group, known as the conformal group, and the study of conformal field theory is partly the study of fields which form representations of this group.

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Conformal vs Weyl invariance

We have found that a theory with diffeomorphism invariance and Weyl invariance has a new conformal symmetry in which we do not alter the metric g_{\mu\nu}\,. We may use this theory to construct one in which the metric is fixed, and non-geometrical (i.e., not a true tensor), as is the case when g_{\mu\nu} = \eta_{\mu\nu}\,. This new theory now has conformal invariance, but obviously not Weyl invariance.

Conformal vs Scale invariance

A special case of conformal invariance is scale invariance, where x \to x' = \lambda x\, where \lambda\, is some scale factor. While scale invariance in general does not imply conformal invariance, it is however true in 2 dimensions[2] [3]

References

Further reading:[4] [5] [6] [7]

  1. ↑ We preserve the term diffeomorphism to its original meaning.
  2. ↑ A. B. Zamolodchikov, "Irreversibility" of the flux of the renormalization group in a 2d field theory, JETP Lett. 43 (1986) 730–732.
  3. ↑ J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B303 (1988) 226.
  4. ↑ Paul Ginsparg, Applied Conformal Field Theory. arXiv:hep-th/9108028.
  5. ↑ P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, 1997. ISBN 0-387-94785-X.
  6. ↑ A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333–380.
  7. ↑ Mack, G.; Salam, A. (1969). "Finite-component field representations of the conformal group". Annals of Physics 53: 174-202. DOI:10.1016/0003-4916(69)90278-4. 
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