# Is a universe without massive particles scale-invariant?

In a popular talk by Roger Penrose about spacetime geometry, when introducing his conformal cyclic cosmology starting at 17:15 I think he says that as soon as there are no massive particles left in the universe, the scaling of the spacetime metric loses its meaning, and we are left with a conformal geometry, which is scale invariant.

Did I interpret this correctly? If so, is it that all physics not involving massive particles is scale invariant? Is that so for all known physics, or for all possible physics (I suspect the latter, because right at 17:15 he says that scaling is essentially equivalent to how clocks measure time, and that without massive particles, there are no clocks).

• This isn't a proper answer because I don't know a lot, but I believe that this is true in the classical theory but not in the quantum theory. I've heard that the Yang-Mills Lagrangian is scale invariant but the quantum theory is not. – Javier Jun 18 '17 at 23:49
• @Javier Indeed, a classical field theory is scale invariant if there is no dimension-full (with dimension of mass) parameter. In a quantum theory however it is also necessary that the beta function vanishes. – Diracology Jul 15 '17 at 0:10