I am trying to show that the Polyakov action of a massless particle in a D dimensional Minkowski spacetime S=∫dτ(e−1˙xμ˙xνημν)
is invariant under the conformal transformation generated by a vector field
ξ in the Minkowski spacetime, i.e.
Lξη=λη, where
Lξ is the Lie derivative.
It turns out that in order to have conformal invariance, the auxiliary field
e has two transform in the following way:
Lξe=2De(∂μξμ)
But the above transformation is counter-intuitive to me. I cannot understand the physical or geometrical meaning of that transformation. Why should the field
e transform in that way under conformal transformation?