# Can the Montevideo interpretation of quantum mechanics do what it claims?

Partly inspired by the great responses to a my previous physics.SE question about "reversing gravitational decoherence, today I was rereading the intriguing papers by Gambini, Pullin, Porto, et al., about what they call the "Montevideo interpretation" of quantum mechanics. They've written lots of papers on this subject with partly-overlapping content; see here for a list.

The overall goal here is to try to identify, within the (more-or-less) known laws of physics, a source of decoherence that would be irreversible for fundamental physics reasons, rather than just staggeringly hard to reverse technologically. One can argue philosophically about whether anyone should care about that, whether such a decoherence source is either necessary or sufficient for "solving the measurement problem", etc. Here, though, I'm exclusively interested in the narrower issue of whether or not such a decoherence source exists.

Gambini et al.'s basic idea is easy to explain: quantum-gravity considerations (e.g., the Bekenstein bound) very plausibly put fundamental limits on the accuracy of clocks. So when performing a quantum interference experiment, we can't know exactly when to make the measurement---and of course, the energy eigenstates are constantly rotating around! So, for that reason alone (if no other!), we can think about any pure state we measure as "smeared out a little bit" into a mixed state, the off-diagonal entries in the density matrix a little bit less than maximal.

More precisely, Gambini et al. claim the following rough upper bound for the magnitudes of the off-diagonal elements. Here, T is the elapsed time between the beginning of the experiment and the measurement, Tplanck is the Planck time, and EA-EB is the difference in energy between the two things being kept in superposition (so that $\frac{E_A - E_B}{\hbar}$ is the Bohr frequency).

(1) $\left| \rho_{offdiagonal} \right| \lt \exp \left( -\frac{2}{3} T_{planck}^{4/3}T^{2/3} \left( \frac{E_A - E_B}{\hbar} \right)^{2} \right).$

If EA-EB were equal to (say) the mass-energy of a few million protons, then (1) could certainly lead to observable effects over reasonable timescales (like a second).

Now, it might be that there's an error in Gambini et al.'s analysis or in my understanding of it, or that the analysis relies on such speculative assumptions that one can't really say one way or the other. If so, please let me know!

If none of the above holds, though, then my question is the following:

Can the bound (1) really do anything like the work that Gambini et al. claim for it---that is, of preventing "macroscopic interference" from ever being observed? More concretely, is it really true that anything we'd intuitively regard as a "macroscopic superposition" must have a large EA-EB value, and therefore a relative phase between the two components that rotates at unbelievable speed? In principle, why couldn't we prepare (say) a Schrödinger cat in an energy eigenstate, with the alive and dead components having the same energy (so that EA-EB=0)? Would such a state not constitute a counterexample to what Gambini et al. are trying to do?

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## 4 Answers

Jorge Pullin sent me an email responding to my question and to Ron Maimon's criticisms. In case people are interested, I'm posting it here with Jorge's kind permission. I particularly appreciated his clarification that the Montevideo interpretation is not really an "interpretation" at all, but new physics that makes new and different testable predictions. In particular, in Jorge and Rodolfo's view, the argument they give involving clocks is "motivational" in character; they're not claiming it as a derivation from accepted principles of quantum gravity. --Scott

Let us start with your own question. We believe it is answered in this paper of Paz and Zurek.

Quantum limit of decoherence: Environment induced superselection of energy eigenstates. Juan Pablo Paz (Buenos Aires U.), Wojciech Hubert Zurek (Los Alamos). Nov 1998. 4 pp. Published in Phys.Rev.Lett. 82 (1999) 5181-5185 DOI: 10.1103/PhysRevLett.82.5181 e-Print: quant-ph/9811026 [quant-ph] PDF

In fact our approach may be considered as a completion of the environmental decoherence approach. Paz and Zurek have shown that the pointer basis is defined by the eigenvectors of the interaction Hamiltonian and the self-Hamiltonian. It is not clear if pointer states can be defined among the eigenstates belonging to the same eigenlevel of the dominant Hamiltonian. If that were the case our axioms would not lead to the production of events for a Schroedinger cat that would involve states of that type.

As for the points raised by Maimon, there are several, some explicit, some implicit, so we will try to treat them separately:

a) His first point is that there is no relative phase in two interacting systems because underlying the theory is ordinary quantum mechanics. We think this is a misunderstanding of what we do. Or perhaps we did not explain it clearly. Although one talks of an "interpretation of quantum mechanics" the Montevideo interpretation has new physics, described by the modified Schroedinger equation we have. This is clear, for instance, when we formulated it axiomatically. Incidentally, a convenient place to find our papers is at http://www.montevideointerpretation.com

As a result systems lose relative coherence. One can motivate the new physics in the impossibility to observe precisely ordinary quantum mechanics, and we did so in some of the papers, but at some point one has to admit it is new physics. The new physics emerges naturally from the relational descriptions that are suitable for generally covariant theories like general relativity. We showed in the paper by Torterolo et al. that in such a context the only thing one can compute are conditional probabilities between observables that evolve in the unobservable time. One could extend that calculation including a second clock. But even if one prepared both clocks in the same initial state, after some time the clocks will disagree on the times assigned to each measurement, contrary to what Maimon claims.

b) As for his point of the photons, if one could put the screen very very far away, we contend that indeed interference would disappear. But obviously in any feasible experiment the effect is too small to observe. We wish it would be easier, then it would make the whole paradigm experimentally testable. Unfortunately the only way to see the effect is to have Schroedinger cat type states with huge energy differences compared to what is available at the atomic level. Those are not easy to find in the lab.

c) As for the kool aid part, we agree with you as well. I doubt very many people, apart from some string zealots, will claim that string theory solves the problem of time in generally covariant systems. Let alone the fact that the real universe is not AdS!

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I did not read these papers as "quantum-gravity considerations" but about limits imposed by (quasi)classical gravitation on quantum experiments. They discuss mainly clocks (*) but I think for this discussion it is easier to consider distance measurements:

In an interferometer the distances between mirrors etc. have to be controlled with great precision to see an interference pattern. In order to control the distances between mirrors one needs to make them heavier and the limitation from (quasi)classical gravitation is that the mirrors cannot form a black hole.

So I think it misses the point to state that "you get a pattern that doesn't give a hoot about what time it is".

If one would perform an interference experiment with macroscopic bodies the effective wavelengths would be at or below the Planck length and one could not place the mirrors with sufficient precision to see anything. I think this is the kind of issue they are talking about and not a new quantum gravity theory. The fact that the S-matrix can be calculated in string theory is great, but does not really have anything to do with this imho.

(*) I guess the reason they consider clocks is because they want to get universal quantitative bounds on decoherence times and not just qualitative statements about a particular interference experiment.

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The issue I have with this interpretation (which is close to what they suggest, replacing time with space) is that it assumes that there is a decoherence caused by the uncertainty in the gravitational metric background. If this background is coherent vacuum, there should be no uncertainty at all. I understand the experimental setup makes it hard to measure interference for macroscopic objects, but if you have a zero-temperature mesoscopic grain, you should be able to observe interference when it is at rest, and then boost the whole system to a velocity where the wavelength is trans-Planckian – Ron Maimon Aug 29 '12 at 13:10
If they suggested the cosmological horizon has thermal gravitons that you can't shield, I would be on board--- I do think that the deSitter horizon makes it difficult to define quantum coherence for a deSitter universe, and I believe the proper description of dS space is by a thermal statistical ensemble, and no pure state. But the bounds from gravity alone in flat space are not consistent with an interpretation that gravity leads to fundamental decoherence. – Ron Maimon Aug 29 '12 at 13:12
In my example the issue is the uncertainty in the position of the mirror(s) (which has to withstand the recoil of the photon or Schroedinger's cat) and not uncertainty about the metric. – wolfgang Aug 29 '12 at 13:44
Also, if you move the particle (or Schroedinger's cat) very slowly through the interferometer to increase the wavelength, then the payoff is that the experiment takes much longer and you need to fight decoherence that much longer. If G & P did their calculation correctly (which I did not check) then there is a limit to what you can achieve by slowing down the particle or cat. – wolfgang Aug 29 '12 at 13:47

No it cannot. This is not an interpretation, nor a new theory, it is a misunderstanding. The papers are vacuous, and not in an interesting way.

### What time is it?

The basic idea is that we don't know what time it is, really. So we make a probability distribution for what time it is, and this changes the phase of the wavefunction by an amount proportional to however much we don't know the time. Then the authors claim that this uncertainty introduces a decoherence phenomenon into the Schrodinger equation, because the phases of energy eigenstates are shifted by an uncertain amount.

This is just plain wrong. The reason is that although we don't know what time it is, we know from the author's assumption that there is a consistent Schrodinger time (this is one of their axioms) that whatever time it is, it's the same time for any two things in the theory. So while there is an uncertainty in the phase of an isolated system from not knowing what time it is, there is no uncertainty introduced in the relative phase of two interacting systems, and there is no decoherence caused by this uncertainty, except through mistakes in analysis.

### Mistakes

These mistakes are subtly introduced by making a separation between "observer" and "system", and introducing the probability distribution for the clock reading only in those cases where the observer is interacting with the system, and when observers are interacting with each other, not when systems are interacting with systems.

For example, suppose you have a photon split by a beam splitter, with one part going through some glass, then through a double slit to measure interference. This will work, the beams stay coherent, because the photon that gets through doesn't excite any irreversible quantum in the glass. Notice that the glass is large and macroscopic, though.

Anyway, once the photons interfere, you get a pattern that doesn't give a hoot about what time it is, just on the relative phase-difference between the two photons, the path-difference in the optical system. So the photon didn't care that we don't know what time it is for the glass, because it just goes through the glass and not, and whatever time it is, it interferes with the other photon, which doesn't care what time it is either, because it's the same time as the other photon.

Note that the photon interacted with this enormous glass, and all the atomic absorption and emission events had to coherently come together even though for the glass, we don't know what time it is. The relative coherence is maintained throughout.

So when does the problem of time show up in these papers? It shows up when the observer entangles with the system, and at this point, the authors declare that the uncertainty in what time it is shows up as an uncertainty in the phase of the system that the observer measures.

If there is a second observer measuring something else, they introduce an uncertainty in the second observer's time. But then when the observers come to talk, the authors pretend that the two observers phase-shifts are uncorrelated, when in fact the uncertainty in what time it is is exactly the same for the two observers, because it is an uncertainty in the same global t variable that they both don't know.

So whatever the t-uncertainty for each observer, the coherence effects between the two observers are not washed out, unless you make the assumption that the actual global t is different for the two observers, an assumption that is at odds with the postulates of the theory, that there is a global time ticking down there underneath it all.

### There is no problem of time in S-matrix theory

Constrary to the authors' claims, string theory solves the problem of time definitively and for good, that's the whole point. The solution was the motivation for Heisenberg to introduce S-matrix theory in the first place, it allows you to make a theory in cases where space and time are unreliable.

An S-matrix theory doesn't give a detailed history of the events in the interior of spacetime, it only relates things on the boundary to other things on the boundary. It doesn't have a real local non-asymptotic time variable at all, so it can't describe time-dependent phenomenon, like the formation and evaporation of a quantum black hole in detail. This is why we are in the embarassing position of having essentially exact quantum description of forming and evaporating black holes while at the same time not being able to answer some of the simplest questions about this process.

So if you make a string scattering calculation, or an AdS/CFT calculation using boundary states, you don't have a problem of time on the interior, you can't, because time on the interior just doesn't appear in the description. It is at best reconstructed approximately from the quantum state on the boundary.

You might say "but then what about the problem of time on the boundary!", but the boundary theory is non-gravitational, and it doesn't have a time problem either. This is the miracle of string theory, and this is what makes it the only plausible candidate for quantum gravity--- the philosophical problems completely evaporate in S-matrix, it is as if they never existed.

You might object that there is a t-variable on the perturbative string world-sheet, but this is an artifact of the perturbative theory, of describing the string scattering process in detail using intermediate states, which you then interpret as localized in time. This interpretation is not completely good, you can't associate local operators to the string. If you do string field theory, you need to do it in light cone, and then the string story becomes more or less local along the light-front, but the light-front time variable is going diagonally in space time, and the string field is only telling a local story in the transverse coordinates to the light-cone pair. It stays nonlocal in the light-cone pair (time and one other coordinate).

If one were given the correct exact string S-matrix in our vacuum, there would be no t-variable in the S-matrix, only the S-matrix in and out state which does not reference any clocks at all. You might object that the S-matrix gives you phase shifts in outgoing waves, and to measure these phase shifts you might think you need a clock, but this is not so, since the relative phase between two states can be determined in principle by perfoming a second much-later scattering which can be approximated as two separate scatterings, and allowing the scattered products in different direction to interfere with each other to make fringes. The phase shifts of the original scattering now show up in the k-directions of the bragg diffraction of the two waves, and you can reconstruct the phase shift information from complicated scattering matrix data in principle without needing a clock on the interior, just by considering a more complicated in state.

This is exactly what you do for a photon--- you scatter it off a double slit to turn the difference in phase shift into a spatial diffraction pattern. This is not obscure at all, although it is hopeless to describe the appropriate s-matrix elements in detail for any realistic experiment.

This means that the problem of time cannot even be stated in S-matrix theory, and time is not treated differently from space, because neither is treated at all. This is the greatest virtues of string theory, and it is the reason that the S-matrix program was able to make such surprising progress in quantum gravity, which was not its original goal.

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Thanks, Ron. Offhand remark: to those who haven't fully drunk the Kool-Aid yet, not being able to state the problem of time, since you don't treat space or time at all, might not sound like such a wonderful "virtue"! I.e., if you want to say various things we care about are just "illusions," then the question shifts immediately to explaining those illusions and figuring out their basic properties, which you admit hasn't been done yet in the case of AdS/CFT. – Scott Aaronson Aug 29 '12 at 11:47
Moving on to your criticism of Gambini et al.: I imagine their response would be, sure, in the case of the photon travelling through glass, you'll see the interference for exactly the reasons you say. But when trying to superpose two macroscopic bodies, the situation is substantially different because of the huge energy differences involved (cf wolfgang's answer). Is there any merit to that claim? (For this question, let's forget about whether everything is "really" unitary in some boundary description to which we don't have access; I'm only interested in limits on actual experiments.) – Scott Aaronson Aug 29 '12 at 12:00
@ScottAaronson: The problem is that their hypothesis of fundamental decoherence is not justified by the mechanism they give--- not knowing what time it is. For observing coherence, you only need to know that there is a global time in which all phenomena are coherent. This is a postulate in their method, and if they omitted this postulate, and said that there was a real fundamental decoherence due to gravity, it wouldn't be necessarily inconsistent, but it would probably conflict with experiment (but it might be possible, it's just not what they do). – Ron Maimon Aug 29 '12 at 12:34

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... regarding string theory kool aid, which I drink with no reservation, reconstructing time and space from boundary data is indeed the main sticking point in accepting S-matrix theory, and it is why it took two decades from 1940-1960 before the theory could really take off. The 1960s literature is entirely devoted to the problem of reconstructing local physics from S-matrix data, and it is possible, although extremely difficult and philosophically involved. Although the "illusion of space" is not completely worked out in AdS/CFT, it is worked out enough to know for sure it works. – Ron Maimon Aug 29 '12 at 12:36
OK, thanks. I understood them to be saying: when you superpose two bodies with a huge difference in energy, the Bohr frequency condition implies that the relative phase between them really is rotating at an enormous speed, which then means that you really do need to know what time it is to great precision if you want to measure anything about the phase. This argument might still be wrong, but I don't see how one can refute it by pointing to their assumption of a global time ticking underneath everything. (And incidentally, I didn't see where they ever discussed two observers.) – Scott Aaronson Aug 29 '12 at 12:44
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Can the Montevideo interpretation of quantum mechanics do what it claims? Yes.

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Please back it up! – Qmechanic♦ Oct 26 '12 at 20:41
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