Is a CFT quantized on flat space trivial? My understanding of CFT (as taught in an introductory class for example) is that we work in Euclidean signature and quantize on a sphere of radius $R$. The spectrum is given by the conformal weights $\Delta$ of the primary operators (via the state-operator mapping) and their descendants via 
$$E_n = \frac{\Delta_n}{R}.$$
If we were to instead quantize on a slice of flat space, we would get a trivial spectrum $E_n = 0$ since there is no energy scale. This can also be seen from taking $R\to \infty$ above. So the spectrum on flat space is trivial. 
Since, in almost all examples I've encountered, a quantum theory is specified by it's spectrum, is it correct to say that CFT quantized on flat space is trivial? Are there any subtleties?
Edit: There is a misstatement in this question, that the theory on flat space has a trivial spectrum, as per the comment. (The nonexistence of a mass scale only prohibits a gap, not a spectrum altogether.) This is a point I'd like to understand better.
 A: If you take the limit of $\frac{\Delta_n}{R}$ for $R \to \infty$ with fixed $n$ you do get zero, but keep in mind that you also have infinitely many $\Delta_n$. Thus it might be possible (and is in fact true) that by taking the limit $n \to \infty$ at the same time in appropriate way you end up with a nonzero result for infinite $R$. Then you can immediately conclude that the spectrum is $[0, \infty)$, because by scaling symmetry for every state of energy $E$ there is a state of energy $\lambda E$ for any given $\lambda >0$.
Let me give you a simple example: consider the Laplace equation on a circle of length $L$. Its eigenvalues are $\lambda_n(L):= \left( \frac{n}{L} \right)^2$ where $n$ is an integer. For any given $n$ the limit of $\lambda_n(L)$ for $L \to \infty$ is zero. On the other hand, for every finite $L$ the spectrum is not bounded above. It becomes more and more dense in the half-line $[0, \infty)$ as $L$ is increased. In this sense you do recover correct spectrum of the Laplace operator on $\mathbb R$ in the limit $L \to \infty$.
