First, I think you may be thinking about a global conformal transformation. In the context of (relativistic) Bohmian mechanics and scalar theories of gravity, the conformal transformation is local: the magnitude of the conformal rescaling varies from point to point, which is what leads to dynamical behavior.

A conformal spacetime transformation amounts to a change in the gravitational constant. A local conformal transformation, therefore, amounts to a variable gravitational constant. Conversely, a variable gravitational constant can be absorbed into a conformal transformation; this is how the variable gravitational constant of Jordan-Brans-Dicke theory can be eliminated by a conformal transformation from the Jordan frame to the Einstein frame. Of course, since the transformation is local, it is described by a scalar field, and as a result of that transformation, the scalar field will couple to other fields. The phenomenological interpretation is that a particle trajectory that is a geodesic in the Jordan frame is the trajectory of a particle that responds to a nonmetric scalar "fifth force" in the Einstein frame, and thus not a geodesic in that frame.

Conformal transformations are of course angle preserving. In the context of relativistic spacetime, this has the special significance that they leave light cones (in the case of a local conformal transformation, infinitesimal light cones) invariant. More generally, Maxwell's laws for electromagnetism are invariant under a conformal transformation (in four dimensions). Furthermore, the Weyl tensor (which is also known as the conformal tensor), which describes, among other things, free gravitational waves in empty space, is also invariant under conformal transformations.

The relationship between conformal transformations and Bohmian mechanics is well desrcibed in this paper: http://arxiv.org/abs/0810.2786. The essence of the argument is that we first introduce a scalar theory of gravity, and observe that the resulting metric is always proportional to the Minkowski metric, therefore the theory amounts to introducing a local conformal transformation; and then, using a suitably chosen stress-energy tensor representing matter yields equations that are the defining equations behind the de Broglie-Bohm interpretation.

By the way, the full conformal group in 4 dimensions has 15 degrees of freedom; in addition to the 10 degrees of freedom of the Lorentz-Poincare group, five additional degrees of freedom are introduced by a) a conformal rescaling, and b) an inversion, followed by a translation, and then a second inversion. This latter bit is referred to as the special conformal transformation. However, the metric is invariant up to a rescaling under the conformal group, which is why we are only interested in that one degree of freedom. (Matter, on the other hand, may transform differently.)

• A conformal field theory (CFT) is a field theory which is invariant under some transformations. This means that the physics of the theory looks the same at all length scales. Conformal field theories cares about angles, but not about distances.
• A transformation of CFT has a different interpretation depending on whether we are considering a fixed background metric gαβ, or a dynamical background metric.
• When the metric is dynamical, the transformation is a diffeomorphism; this is a gauge symmetry.
• When the background is fixed, the transformation should be thought of as

an honest, physical symmetry, taking the point to other. This is now a global

symmetry with the corresponding conserved currents.

Dear Rahmani, for a brief expression for the first question, due to the Noether's theorem, any external or internal symmetries in physics corresponds to a conserved charge, these charges of the Poincare's group are well determined as energy-momentum angular momentum etc. Consequently the trend toward investigating of all possible governing symmetry resulted in the diffeomorphism groups and more applicable Dilation (scaling) and special conformal symmetry. these transformation are subsets of conformal transformation where just the angles remain  preserved. Now adding these Lie algebra generators to aforementioned Poincare's algebra leads to conformal group and its generators commutations. On the other hand conformal group transformation,sometimes refer to the transformation of metric tensor as you pointed out in second question. For your second question please review this article from Golshani et.al : 1) A Quasi-Newtonian Approach to Bohmian Mechanics I: Quantum Potential Annales de la Foundation Louis de Broglie 34 (2009), 67-82   2) Shojai,F.,Shojai,A., Golshani,M., “Conformal Transformations and Quantum Gravity,” Modern Phys. Letters A. 13, No. 34, 1998, P. 2725 ,  I think these papers contains the Bohmian viewpoints in quantum gravity and quantum potential approaches.

Can some one explain the special confromal symmetry or at least can give its generator?