Physical interpretation of the generators of the conformal symmetries

Question

The Poincare group has ten generators, which have the physical interpretation of energy, momentum, angular momentum, and the system center of mass, and which are of course conserved in any Poincare invariant system. Adding five more generators (of dilitation and the four special conformal transformations) extends the Poincare group to the conformal group. Do these five new quantities, which are conserved in any conformally invariant system, have any natural physical interpretation (something I can picture in my head)?

Edit: To reiterate, I am familiar with the physical interpretation of the comformal symmetries. I am looking for a physical interpretation for the generators of the conformal symmetries. I'm not looking for the analog of "momentum is the generator of spatial translations," I'm looking for the analog of "conservation of momentum tells you that something moving in a straight line will continue to move in a straight line."

Answer 1

There is a physical interpretation for each of the transformations. The conformal transformations, as you have noted, consist of translations $x'^\mu = x^\mu + a^\mu$ which has the momentum $P_\mu = -i\partial_\mu$ as the generator. The other is dilations, $x'^\mu=\alpha x^\mu$ with generator $D = -ix^\mu\partial_\mu$, and a dilation is nothing more than a rescaling. We also have rotations, $x'^\mu = M^\mu_\nu x^\nu$ with $L_{\mu\nu} = i(x_\mu\partial_\nu - x_\nu \partial_\mu)$.

The interesting transformation which is not immediately physically obvious is the special conformal transformation which has the finite representation,

$$x'^\mu = \frac{x^\mu - b^\mu x^2}{1-2b\cdot x + b^2 x^2}.$$

By inspection it is not obvious, but it in fact corresponds to an inversion followed by a translation and another inversion. That is,

$$x^\mu \to x'^\mu=\frac{x^\mu}{x^2}, \quad x'^\mu \to x''^\mu = x'^\mu - b^\mu, \quad x''^\mu \to x'''^\mu = \frac{x''^\mu}{x''^2}.$$

I encourage you to check this yields the finite representation of the SCT. The book by Blumenhagen has a neat illustration of this:

Answer 2

The very interesting paper "Electric–magnetic symmetry and Noether's theorem" gives the expressions for the generators of the conformal symmetry in classical E&M (note that these quantities are not conserved in QED, because the conformal symmetry is anomalous and quantum effects spontaneously break it). If we denote the Maxwell stress-energy tensor by $T^{\mu \nu}$, then dilitation symmetry is generated by $D^\mu := x_\nu T^{\mu \nu}$ and the special conformal transformation for $x^\mu$ is generated by $I^{\mu \nu} := 2 x^\mu D^\nu - x^2 T^{\mu \nu}$. The paper goes on to say

... Bessel-Hagan commented that 'the future will show if they have any physical significance.' It appears that their physical significance is still not understood ... The independence of [$I^{\mu \nu}$] from the others has been questioned ... For a single plane wave, the conservation [of $D^\mu$] can be interpreted as a statement of the familiar dispersion relation $\omega = c |k|$.

Answer 3

Besides the Poincare group the other five generators are:

-Dilations: these are the most obvious, they uniformly re-scale the coordinates. Basically scale transformations

-Special Conformal Transformations: these are less obvious, they generate translations of the inverted coordinates, so $X^\mu$/$X^2$ -----> $X^\mu$/$X^2$ + $A^\mu$ @JamalS shows a figure with the geometrical interpretation of these inverted translations

See also for instance the math properties of the conformal group at http://bolvan.ph.utexas.edu/~vadim/classes/13f/SCA.pdf besides the more basic wiki article

The conformal transformations are important for a number of reasons, among them

1) in 4D Lorentzian spacetime that symmetry group is a representation of SO(4,2),

From an answer in PSE at, this I part of the reason there is an AdS-CFT correspondence:

From Commutation relations of the generators of the conformal group:

"One very interesting thing about all this: you might ask, what is a spacetime where SO(4,2) really is just generalized rotations (as opposed to rotations + SCTs + dilatations)? Well, $AdS_5$ is one! This might be your first clue towards the existence of the AdS-CFT correspondence! A CFT in 3+1-dimensional spacetime obeys the same algebra as the isometries of $AdS_5$. See "ANTI-DE SITTER SPACE" by Ingemar Bengtsson - pages 1-5 give a nice concise introduction to AdS spacetime and it's isometries."

2)null cones in Minkowski spacetime transform to null cones under a conformal transformation. And these symmetries exist at null infinity in the horizon of a black hole, which Hawking and his colleagues used to conclude that there are other conserved hair in black holes, specifically soft hair, and that those may (not proven, but hinted at) carry the previously missing information that was understood to be lost as particles fall into the blac holes (or at least into the horizon).

See the paper at https://arxiv.org/abs/1601.00921

3) the whole area of conformal field theory.