How to understand Carlo Rovelli's notion that time "does not exist"? Media coverage of Carlo Rovelli's book The Order of Time has had headlines like "There is no such thing as past or future", or "Carlo Rovelli: 'Time does not exist'." Is there a way to explain what he means that is more concrete than "the dance of nature does not develop to the rhythm kept by the baton of a single orchestral conductor"? (Rovelli's words from the second linked reference.) To be fair, he precedes that by "elementary processes cannot be ordered along a common succession of instants", which is fairly concrete but leaves a lot of questions open. In particular, what is an elementary process?
Can anybody who is familiar with Rovelli's work clarify what he's getting at? I would also like to know just how speculative the ideas in these papers are seen to be within the physics community. 
There are a lot of other questions about whether time is real, but I think this one is different because it regards a specific author's claims about time.
 A: I believe Carlo sometimes monitors the loop-quantum-gravity tag here, so hopefully he will answer this question. In the mean time, I will attempt to answer.
What he refers to is known in the quantum gravity literature under the name of problem of time.
Naively, if you attempt to convert General Relativity to the Hamiltonian description, you will find that the Hamiltonian is identically zero when the equations of motion are imposed. This can lead you to believe that time is somehow absent from the Hamiltonian description, and that time is not a part of quantum gravity.
The modern understanding is that this is indeed true, but only for coordinate time. Physical time is still present, it hides in the relations between partial observables.
There's also a separate but related concept of thermal time by Rovelli and Connes (the one with the non-commutative Standard Model), which is basically a conjecture that the macroscopic physical arrow of time is of thermodynamic origin. More precisely, to each quantum gravity density matrix $\rho$ we can associate the "thermal Hamiltonian"
$$ H = - \ln \rho, $$
for which by definition $\rho$ is stationary. However, fluctuations around $\rho$ can be shown to live in the thermal time associated with the Hamiltonian flow generated by the thermal Hamiltonian.
