## Book review: The Order of Time

The Order of Time
by Carlo Rovelli (2018)
also his
Time in quantum gravity: an hypothesis, Phys Rev D 43, 442 (1991)
Statistical mechanics of gravity and the thermodynamical origin of time, Class Quantum Grav. 10 1549 (1993)
Relational Quantum Mechanics, Int. J. of Theor. Phys. 35 1637 (1996)

I saw Carlo Rovelli, inventor of loop quantum gravity, give a talk once.  I believe it was at GR22 in Warsaw.  It was my first exposure to his general philosophy of doing physics.  Rovelli thinks that questioning the core insights of quantum mechanics and general relativity is by this point an unpromising strategy for theoretical physics.  Our task is to extend and synthesize them.  Like his fellow Italian Thomas Aquinas, Rovelli is a synthesizer; by my count, in this book he synthesizes Anaximander, Aristotle, St. Augustine, Newton, Leibniz, Boltzmann, and Einstein.  All around his surprising claim that time, at the most fundamental physical level, does not exist.  Surprisingly for a work of science popularization, he ends up agreeing with phenomenologist philosophers who claim that the “lower-case t of physics” doesn’t capture the human reality of time.  Rovelli agrees that the essence of time is to be found in human subjectivity, and claims that physics itself leads to this conclusion by murdering the “lower-case t of physics”.

The Order of Time is written for a general but intelligent audience.  Rovelli is a widely read man, given to discursions into history and poetry, with a personal and sometimes eloquent prose style.  I find this distracting in science writing, so I’ve consulted some of his papers that he cites, which also bear the mark of an interesting personality but submit to the clarifying discipline of mathematics.  Below, I will occasionally mention things like Hamiltonian flows and gauge invariance to clarify for appropriate readers, but others should be assured that Rovelli, an excellent writer for lay audiences, avoids such terms in his book.

Dynamics without time

To judge the claim that time does not exist, one must first understand it properly.  On the one hand, Rovelli is not questioning that there are timelike-separated events, that proper time passes on worldlines connecting such events, that therefore there are clocks which can at least locally provide an order to events.  On the other, he means more than the commonplace observation that there is no unique absolute time variable or criterion for simultaneity for spacelike-separated events.  The claim is that there may not be a single chronological ordering for all the events in the universe.  There may be no variable $t$ in terms of which all other variables $q_i$ can be given as functions $q_i(t)$ defined for all the same values of $t$.  To put it in the most precise way possible, physics at the Planck or cosmological scale may not form a Hamiltonian system.

A universe timelike paths closed be an example of a case that would muck up attempts to define a global “absolute” time order.  Even an existing successful theory, general relativity (GR), Rovelli judges to be “timeless”.  This may sound surprising at first given that there is a space+time split version of GR, and numerical relativists use it routinely to evolve initial data for a spacetime “in time”.  However, GR is also a generally covariant theory, its Hamiltonian is weakly vanishing (i.e. =0 for allowed spacetimes), and time evolution cannot be disentangled from coordinate change.  This causes little trouble in classical GR (except for deciding computationally practical gauge choices), but in a quantum theory, observables should be gauge invariant, which here translates into the demand that they be stationary!  But if all observables are stationary, how can there be any dynamics?  Hence the famous “problem of time in quantum gravity”.

Rovelli’s solution has technical aspects, but he believes that the core problem to overcome is conceptual.  That the usual recipe for building a quantum dynamics in time doesn’t work is a good thing.  It’s telling us something.  We don’t need it.  Rovelli shows that one can do dynamics without time (defined by the flow of a Hamiltonian) using presymplectic mechanics and suggests how this can be carried over to quantum mechanics.  As for the problem of observables, it’s true that we cannot observe anything as a function of some global, external flowing time.  Nobody has ever done that anyway.  What we really do is measure change using clocks–how one variable (e.g. the location of a ball moving along a ruler) varies with another variable (e.g. the reading of a watch on the ball).  This can still be done in quantum gravity, although there may be no clock variable that can be extended to work as a Hamiltonian time over all spacetime.  These clock times correspond to observables such as “location – speed * (clock time – t)”.  This expression is indeed independent of clock time along the trajectory (because in the above the change in location and change in clock time cancel out) and singles out the location when clock time reads value “t”.  Rather than the observable changing, each choice of “t” is a different observable.  This is similar to the eternalist gambit of adding time indexicals to every property, although here ironically in the service of questioning the fundamentality of absolute time indexicals.  Another interesting point:  you must work out the whole trajectory of the ball (in this example) before you can even define observables!

To sum up, there is no time, but there are clocks.

Thermodynamics and the flow of time

Boltzmann famously speculated that what we experience as the flow of time is really the accumulation of entropy.  As Rovelli notes, it’s low entropy, not energy, which is the real resource.  An increase of energy of the Earth–global warming–is a bad thing.  What enables life and activity on Earth is ultimately that the fewer high-energy photons from the sun bring less entropy than the more numerous low-energy photons emitted by the Earth.  Rovelli then points out that entropy is entirely an artifact of coarse graining; it is a measure of the number of undetermined microstates corresponding to the given observed macroscopic state.  If one does not discard any information but regards each microstate as distinct, each is as improbable as any other and has multiplicity one; the idea of entropy disappears.  Entropy, and thus time “flow” itself, are thus an effect of a “blurred” vision of the world.  Fundamental physical laws do not pick a preferred axis of time or direction along it.  If a Hamiltonian is defined on the phase space, one can use (via Poisson brackets) it to construct a flow which we identify as evolution in time, but in its absence, another scalar function on phase space can be used the same way.  In particular, if one has only probabilistic knowledge of the state of the system (or an ensemble of systems), one can use a distribution function (basically, a function giving the probability at each point in phase space to be in that state) to define a “thermal time”.  For an equilibrium distribution of a Hamiltonian system, this distribution would just be $\rho\propto e^{-H/k_BT}$, so his thermal time reduces to Hamiltonian time rescaled by the temperature.  Rovelli proposes “thermal time” as a generalization valid for timeless systems.

An obvious objection is that a definition of time that relies on “blurring”, on lack of information, would be entirely subjective.  I could get different times by considering different information.  Rovelli replies that entropy is not necessarily subjective if we acknowledge that it is observer-dependent–what information gets ignored depends on what can’t be extracted from the observer’s physical interactions with his world.  He even suggests that perhaps the universe did not start out in a low entropy state but only in a low entropy state for us, we being somehow physical systems relating to the universe in a peculiar way (although he doesn’t guess, and I can’t imagine, how).  In this sense, the flow of time would, indeed, be different for every observing system.

Relational quantum mechanics

Rovelli carries his relationalism even farther in his interpretation of quantum mechanics.  His interpretation is based on a meditation on the Wigner’s friend paradox.  System S is in some superposition of states.  Observer O measures S and finds it to be in a particular state, say 1 rather than 2.  Observer P then observes the S-O combined system (say, by talking to O), but before doing so uses standard quantum mechanics to predict that S-O is in a superposition of states |O in 1>|S measures 1> and |O in 2>|S measures 2>.  After P’s measurement, both will agree, but before, there seems to be a contradiction between O and  P.  Which is right?  Either choice, O or P, is unpalatable in various ways, violating either quantum mechanics or everyday experience.  Then Rovelli suggests the way out.  A system has no one unique state, but rather a different state relative to each observer.  Thus, “S is in state 1 for observer O” and “O-S is in a 1-2 superposition for observer P” can both be true statements with observational consequences.  Relative statements become absolute once all the indexicals are supplied.  (To use Rovelli’s favored historical analogy, in special relativity, the energy of a particle is frame-dependent, but the energy of a particle as measured in a given frame is absolute, something all frames can agree upon.)  I am afraid in interpreting Rovelli this way, I am doing violence to his metaphysical ambitions to put relations before all and all on an equal footing.  If those two observer-dependent states are both facts of the world, then clearly O in the state measured by S is in some sense preferred to the state S does not measure, even if P can’t know that prior to measurement and even if the superposition of (an ensemble of) O-S for P has measurable interference effects.  I fear I am collapsing relational QM into a modal interpretation against Rovelli’s wishes, but I can think of no other way to make sense of it.

Rovelli’s explanation for why unitary evolution breaks down when one does an observation (“wavefunction collapse”) is that by interacting, the observer and the system become one system.  Why doesn’t a Schroedinger evolution apply to a system of which one is a part?  His argument on this point is short and not entirely clear, but I think here is to be found the most profound insight of the relational interpretation.  The perspective on system O-S is necessarily an outsider’s perspective; the problem of measurement is the same as the problem of self-reference.  An utterly remarkable revelation.

Concluding remarks

There is more in The Order of Time.  Rovelli is sympathetic to event ontology.  I was surprised at how quickly he dismisses the idea that fields are the fundamental beings in physics; this seems to me a somewhat natural reading of modern physics.  He sets up philosophical questions of space and time well by comparing the beliefs of Aristotle and Newton.  As a relationist, he is more partial to Aristotle, but he argues that Einstein has reconciled the two by making space and time dynamical fields like any other.  I’m not sure I buy that in detail.  Christoffel symbols are a close analog to the potentials of other fields, but other fields have no analog of the spacetime metric, which carries information about causal structure that all fields respect.  I would say that spacetime remains special.  Rovelli professes to be unafraid of death, a psychological peculiarity that seems very common these days if men are to be taken at their word.

I am wary of adjusting my worldview to accommodate quantum gravity.  As I see it, a theory that doesn’t exist yet is in no position to make demands.  Rather, we are in no position to know what those demands will ultimately be.  Even so, it is intriguing that one can get by in known physics without an absolute time variable.  If such a variable does no work for physics or other aspects of human experience, then perhaps we should get rid of it.  In fact, once one makes the idea of time sufficiently precise, with claims about Hamiltonians and phase space topology and whatnot, one begins to wonder if anyone except mathematical physicists ever believed in time to begin with.

### 4 Responses

1. […] be meaningless without matter filling space, and time would not pass without something changing. Carlo Rovelli praises Aristotle highly for his relationism, for relationism is still a strong contender in the […]

2. Hi Bonald, was just reading through this again when something rather opaque stuck out to me: what do you mean by “the problem of self-reference”?

3. Philosophers usually mean things like Russell’s paradoxes that arise from sets having inclusion criteria that can involve reference to themselves so that the set must be a member of itself and also must not be. My meaning is more general. I mean the whole class of more general epistemological problems having to do with the subject-object split, that the subject can never really include itself in its picture of the world. There are analogous issues in the theory of computation with Turing machines modeling themselves, which in turn end up being closely connected to Goedel incompleteness. Interestingly, the Stanford Encyclopedia entry on relational quantum mechanics has made me aware that the connection of all of these philosophy of mathematics issues with the measurement problem in quantum mechanics had already been made by philosopher Marisa Dalla Chiara in a paper that I am eager to read.

4. […] be meaningless without matter filling space, and time would not pass without something changing. Carlo Rovelli praises Aristotle highly for his relationism, for relationism is still a strong contender in the […]