guess who's back to the future

I have not posted here for several years and the reason was that nobody reads my stuff (I am rounding down a handful of comments I got over the years); after a while I gave up writing if it makes no difference to anything.
But this is changing now with the progress of AI; the machines are reading everything they can get to train their models and there is a non-zero chance that they will pick up some of my arguments and ideas. In other words, I may make a difference to the future after all with the stuff I posted here (and also there).
However, there could be a side effect, if this increases the probability that I only experience an AI simulation instead of real life, as proposed by Scott Alexander.

nothing ever happens

I have previously explained my problems with the "many worlds interpretation", e.g. if one considers the superposition of macroscopically different branches, one would need to know how to handle the branching of space-time geometry.
But let us ignore gravitation for now, assume a flat background, and only consider non-relativistic, simple quantum theory.
We begin with an initial state |i> at t0 and it develops according to Schroedinger's equation, at some point containing the superposition |s(t)> of observer(s) and all that; Schroedinger's cat is in a superposition of dead and alive and so is Schroedinger, if we just wait long enough.

But somebody had to set up the experiment and get it started at t0. We cannot assume a classical observer to do that; instead we have to consider this setup of |i> as just another quantum process and the experiment could have started at t0+d or t0-d. In fact, if |s(t)> is a solution of the Schroedinger equation, then so is |s(t+d)> with d being some arbitrary constant and the many worlds wave function is the superposition of all of them.
Notice that this is very different from standard Copenhagen quantum theory. In Copenhagen there is usually an initial condition (i.e. the beginning of the experiment) that selects one solution; but such an initial condition requires a classical observer who sets up the experiment.
Obviously, if the state of our system is such that |s(t)> = |s(t+d)>, for arbitrarily small d, it follows that d|s>/dt = 0, i.e. nothing ever happens in the "many worlds interpretation"; unless we consider a Hamiltonian that explicitly depends on the time parameter.
So it seems that one needs to introduce classical time and classical clocks somehow externally, otherwise one deals with a system that forever remains in its ground state, H|s> = 0.

escape velocity

This is the copy of a blog post from my other blog, which got some comments there.

CIP asked a question about the entropy during star formation and I think we got the answer, at least qualitatively; but I would like to understand this better.
So let us begin with this calculation of John Baez, which gets the entropy wrong - it would decrease during star formation, i.e. the gravitational collapse of the matter which makes up the star. What the formula leaves out is the entropy of the outgoing radiation, but I would like to stay in a simple Newtonian model with classical point particles only.
In this case the "missing entropy" must come from the particles with velocities above the escape velocity of the star, which leave the collapsing cluster of particles. (The positions and velocities of the particles are actually not bounded, violating an assumption of this calculation, as he noted at the end of his page.) In other words, the formula John uses can only be an approximation, there is actually no decreasing volume V which encloses all particles and if one defines V considering a sphere which encloses all particles which cannot escape, the number N he uses would not be constant. So how does one really calculate the entropy?
A simpler question would be: If the initial number of particles was N, contained in a volume V, what fraction will escape within a small time interval dt? The Maxwell distribution would tell us the number of particles with velocities above the escape velocity and approximately 1/2 of them would escape, if they are within a distance dt*v from the surface ...
But all this seems a bit unsatisfactory; does anybody have the reference to a full calculation of this problem or do I have to run a computer simulation?

added later: A simple simulation of N=1000 particles, initially contained within a sphere of radius 1 and with zero initial velocity, suggests that after long enough time almost all particles escape to a location outside the initial sphere, due to the simulated gravitational interaction. Of course, my program (quickly cobbled together) could be wrong or inaccurate. The chart below shows the fraction of escaped particles on the y axis after so many time steps on the x axis (I have no explanation for the kink after 500 time steps).




The distribution of particles (projected onto a 2d plane) after hundred time steps ...



... one can see a "halo" of escaping particles surrounding the majority of particles in the collapsing star.

virtual information loss

Sabine Hossenfelder wrote a blog post about the information loss paradox, pretty much repeating standard arguments made in this debate. Among them the following:
"Physicists are using quantum field theory here on planet Earth to describe, for example, what happens in LHC collisions. [..] in principle black holes can be created and subsequently annihilated in any particle collision as virtual particles. This would mean then [..] we’d have no reason to even expect a unitary evolution."

As I said, this is a standard argument, but I have a problem with it:
In any experiment we can do, e.g. at the LHC, the energy of such a virtual black hole would be well below the Planck mass, i.e. far from the quasi-classical limit where the information loss problem is discussed.
In which sense can any particle fall into a microscopic b.h. with a radius much smaller than the Planck length, if its wavelength is much larger? So in which sense would a virtual b.h. pose an information loss problem?
We have to assume that the mass of an off-shell virtual b.h. could be arbitrarily large, but its contribution to any S-matrix element would be strongly suppressed (at least exponentially) for energies much larger than the collision energy, which is well below the Planck mass. Therefore its contribution would for all practical purposes be unmeasurable.


added later: Without a full theory of quantum gravity (and even string theory does not know how to handle black holes yet, see e.g. fuzzballs and firewalls vs. ER=EPR) we can only make some basic estimates.
There are estimates of proton decay due to virtual black holes and the expected lifetime is about 10⁴⁵ years - a factor 10¹¹ higher than what we could currently detect.
But I think even those estimates are too low if information loss requires a black hole of mass > m_Planck (if the surface area is indeed quantized black holes with a small mass m < m_Planck may not even exist). Wick rotation suggests that the contribution of a massive black hole m > m_Planck to any Feynman diagram would be suppressed by a factor exp(-k²) or exp(-(m/E)²) if E is the energy of the collision.
Btw the same exponential factor shows up in a different estimate, suppressing the production of black holes even for large collision energies E.

the many worlds interpretation does not work (yet)

I posted a comment on the Shtetl blog, rejecting (once again) the many worlds interpretation (mwi); it is supposed to solve the "measurement problem" of quantum theory, so let us first consider a simple experiment with 2 possible outcomes.
The main mwi assumption is that after the measurement both outcomes are realized and subsequently two macroscopically different configurations M1 and M2 exist in some (decohered) superposition.

However, we can make the differences between M1 and M2 arbitrarily large and therefore gravitation cannot be ignored. M1 and M2 will in general be associated with two different space-time geometries and so far we do not have a consistent framework to deal with such a superposition (*); should we e.g. use 2 different time parameters t₁, t₂ - one for each observer in each space-time?
In a few cases it has been tried to describe such an evolution but the conclusions are not in favor of mwi.
And how would the branching of space-time(s) work if the measurement is spread out over spacelike events, e.g. in an EPR-type experiment?

This gets worse if one considers a realistic experiment with a continuum of possible outcomes, e.g. the radioactive decay of a Pu atom, which can happen at any point of the continuous time parameter t. Assuming that this decay gets amplified with a Geiger counter to different macroscopic configurations, how would one describe the superposition of the associated continuum of space-time geometries?

The Copenhagen interpretation does not have this problem, because it only deals with one outcome and in general one can "reduce" the wave function before a superposition of spacetime geometries needs to be considered.

A mwi proponent may argue that this issue can be postponed until we have a consistent theory of quantum gravity and simply assume a Newtonian fixed background (or a flat Minkowski background). But if one (implicitly) allows the existence of Newtonian clocks, then why not the classical observer of Copenhagen?

In addition one has to face the well known problem of the Born probabilities (x), the preferred basis problem, the question of what it takes to be a world, the puzzling fact that nothing ever happens and other problems, discussed previously on this blog.

In other words, the mwi so far creates more problems than it solves.


(*) In technical terms: The semi-classical approximation of quantum field theories plus gravitation is ultimately inconsistent and we do not yet have a fully consistent quantum theory of gravitation to describe such a measurement situation.

(x) See also this opinion, which is a variant of the argument I made previously here and here.

entanglement

About five years ago I sent this email to Erik Verlinde:

-----------------------------
1/24/10

Dear Professor Verlinde,

I read your preprint about gravity as 'entropic force' and have the following question/suggestion:
It seems that there are large classes of systems following an area law for entanglement entropy, see e.g. arxiv.org/abs/0808.3773
Should they all show 'entropic gravity' somehow?
If yes, the obvious next step would be to pick a simple model, e.g. arxiv.org/abs/quant-ph/0605112 and check if/how entropic gravity manifests itself.

Thank you,
Wolfgang
------------------------------

As it turns out this idea may not have been so stupid after all.

Of course, if entanglement stitches space-time together then the question remains: entanglement of what?
I still think the entangled qubits of a quantum computer are the most natural candidates; if we live inside one it would also solve the measurement problem.
But what would it be computing? We can only speculate.

the sleeping Brad DeLong problem

Brad DeLong was sound asleep for several months, but after he woke up he found an old blog post and quickly calculated the probability that Lubos is an April fool; he wrote a blog post about it, because he had no memory of all the earlier debates about this.
Of course, Lubos disagreed with this calculation, the probabilities are very different according to his argument. So who is right?

I wrote about the Sleeping B. problem almost ten years ago, when it was mostly discussed in academic papers. Meanwhile it has become an internet standard and turned into a political debate: Lubos and the reactionaries vs. Sean and the liberals. Unfortunately, all the nuances of the problem have been lost, as it happens in tribal conflicts, such as the existence of solutions which are neither 1/2 nor 1/3.

In one of the many comments somebody wrote something like "if I would be really interested in this problem, I would code a simulation and see what it does". Unfortunately, if she would actually sit down to implement a simulation, she would quickly find that it does not really answer anything; this is one of the problems which cannot be settled with an experiment or simulation, because it is about the question what we actually mean with "probability". Both physicists and economists are not well trained at thinking through what it is exactly they are talking about, therefore I predict that this debate will not go away anytime soon.

So what is really the issue with this? Well, if the "a priori probability" is 1/2 then S.B. has to bet "as if" it was 1/3 due to the setup of the problem. So if you think probabilities are defined as betting odds (as many Bayesians do) then you will prefer 1/3. If you think probabilities are objective properties of physical systems then you will probably (!) prefer 1/2. (Btw, notice that S.B. has to (implicitly) use the 1/2 "a priori" to calculate the 1/3.)

I actually prefer the 3/8 solution, because it is not so clear how one should understand it. But I realized that I am a tiny minority of one a long time ago ...


-------

Perhaps it is useful to repeat the 3/8 solution here:
If the outcome was Head, she will only wake up on Tuesday; but if it was Tail, she will either wake up Tuesday or Wednesday with "a priori probability" 1/2 for each, since she cannot distinguish the two days.
Therefore, the probability for the just awoken S.B. that today is Tuesday is
p(Tue) = (1/2) + (1/2)(1/2) = 3/4
where the 1st term corresponds to Head and the 2nd to Tail.

Now we can calculate the probability for Head as p(Head) = (1/2)*p(Tue) + 0*p(Wed) = (1/2)*(3/4) = 3/8,
knowing that the "a priori probability" for Head on Tuesday is 1/2.

the chaos computer club

A recent paper suggests a fundamental limit on the chaos in physical systems.
Lubos wrote an easy to read introduction to the main idea.

I think this might be interesting for Scott and everybody else interested in (quantum)computing. If one considers a Turing machine sensitive to initial conditions (i.e. the input string) or if one considers a (quantum)computer simulating chaotic systems, the conjecture seems to imply a limit on computability.

Or think about a device which measures the position x of the butterfly wings, sends the result to a computer, which calculates a function f(x) to determine its output. The conjecture seems to suggest a limit on the functions the computer can calculate in a finite amount of time.

Is it correct to read the result as "the number N of different internal states any computer can reach after a time T is bounded by e^wT where w is a fundamental constant"?

life expectancy

Recently I stumbled upon this math problem:
The positive integer N has a finite value but is unknown to us.
We are looking for a function f(X) which minimizes the sum
E = |f(0) - N| + |f(1) - N| + |f(2) - N| + ... + |f(N-1) - N|
for almost all N.

Notice that we do not know the value of N, so f(X) cannot depend on it, which eliminates the trivial solution f(X) = N among others.
In order to illustrate what I am looking for, consider as first example
1] f(X) = 0, which results in E = N².
However, there is a better solution
2] f(X) = X, which results in E = (N+1)*N/2, which is less for almost all N (except N=1).

Unfortunately, I do not know the best solution f(X) and this is where you are invited to leave a comment to help me out.
But I do have strong evidence (i.e. a numerical test up to large values of N) that
3] f(X) = X + sqrt(X) is an even better solution.

So what is the motivation for this problem and why the title for this blog post?
Consider a process or phenomenon which exists already for X years and we try to estimate the total lifetime N without any further information. So our estimate can only depend on X and we try to find a function which minimizes the total estimation error E as described above; every year we make an estimate f(X) which is wrong by the amount |f(X) - N| and we try to minimize the sum of those errors. In some sense this is a variation of the infamous 'doomsday argument'.

It is now obvious why 1] is a bad solution and 2] is much better. Btw the function f(X) = 2*X would give the same total error E minus a small constant, so whether we assume that the process ends immediately or estimate that it will last twice as long (as it already did) makes no significant difference.

Btw the solution 3] creates a paradox: The best estimate for the life expectancy seems to depend on what units one uses: We get a different result if we calculate with days rather than years.


added later: If I assume f(X) = c*X, perhaps motivated by that paradox, then I can show that c=sqrt(2) minimizes E for large enough N. However, the function
4] f(X) = sqrt(2)*X + sqrt(X) seems to be an even better candidate, but I do not know how to determine a,b to minimize E for f(X) = a*X + b*sqrt(X) or determine the general solution for f(X).


added later: There are better ways to illustrate the problem; e.g. a box contains an unknown number N of candies. You take out one after another and at each step X you have to guess N. At the end there is a penalty proportional to the sum of your wrong guesses.
Perhaps a "deeper" example considers a Turing machine, which performed already X steps and we try to guess after how many steps N it halts.
The reason these examples are "better" is that there is no change of physical units (e.g. from years to days) that would affect N.

no black holes?

Laura Mersini-Houghton and Harald Pfeiffer published a paper with numerical results suggesting that black holes may not really exist (see also this earlier result). As one would expect, several pop. sci. webpages have already picked this story up.

The paper is of course not a general proof, but describes a particular model using certain assumptions; it considers the spherically symmetric collapse of pressure-less dust and it makes simplifying assumptions about the Hawking radiation: The energy tensor for the Hawking radiation is taken from earlier calculations for (static) black holes, proportional to 1/R^2, and I don't think this is justified if one wants to prove that black holes do not exist. Further it is assumed that most of the radiation is generated by the collapsing body itself (*) and finally assumptions are made about the heat transfer function C which I cannot follow (yet).

The resulting differential equations are numerically integrated until a shell-crossing singularity appears, in other words a naked singularity (presumably an artifact of the model assumptions, i.e. perfect spherical symmetry, so it is only slightly embarrassing in a paper which wants to remove black hole singularities).
The behavior of the dust suggests a rebound near the horizon, but it is too bad the full evolution is unknown, because it raises interesting questions.
What happens to the pressure-less dust in the long run? Will it collapse again after the rebound, perhaps infinitely often?
What does the final state (including Hawking radiation and the "influx" of negative energy) actually look like?

I am sure this paper will generate several responses and eventually more realistic calculations will follow.
Until then I remain skeptical that this result will actually hold in general.


(*) I admit that I do not understand this passage in the earlier paper: "Hawking radiation is produced by the changing gravitational field of the collapsing star, i.e. prior to the black hole formation [..]. Otherwise the surface gravity of the black hole κ, and the temperature of Hawking radiation would increase with time..."
I thought the standard picture is that the "influx" is at the event horizon (not the collapsing body) and the temperature does indeed increase with time...


added later: Supposedly William Unruh was more direct and he thinks that the paper is nonsense.

any good answers to this one?

At the Strings 2014 conference, Piotr Bizon talked about the gravitational turbulent instability of AdS₅.
I became aware of this issue more than three years go and I have to admit that I still do not really understand what it means. As I see it, turbulence is one of the big unsolved problems in physics mostly due to the fact that it prevents us from neatly separating energy scales; the opposite of the clean separation which enables renormalization a la Wilson.
So what does it mean that this turbulence instability shows up on one side of the famous AdS/CFT correspondence?

my derivation of the Born rule

I just read (parts of) Sean Carroll's derivation of the Born rule, but I do not find it very convincing, because there is a much simpler, straightforward derivation available to resolve this problem of "self-locating uncertainty".

1) We shall use a "hardcore" many worlds interpretation, assuming that the world splits into a quasi-infinite number of branches at any time, which realizes all possible outcomes of quantum theory. We assume that those branches are all equally real and a simple counting argument shows that the Born rule does not hold for almost all of those branches. It follows that we do not live in one of those generic branches, which solves the first part of our self-location problem.

2) It is reasonable to assume that some of those infinitely many branches contain at least one quantum computer capable of simulating human life. Those computers will have to simulate quantum theory, but we can further assume that they will only keep one branch at a time in order to save resources. It is straightforward to assume that they are programmed to use the Born rule to select this branch randomly.

3) We observe the Born rule to great precision and it follows that we are the human beings simulated in one of those quantum computers. This finally resolves the self-location problem.

I would add that (some of) the simulated human beings will use the Copenhagen interpretation to explain what they experience; i.e. an interpretation which emphasizes the importance of the observer and her 'conscious experience'. Obviously, the simulated human beings are unaware that their 'conscious experience' is indeed a side effect of the procedure which selects the simulated branch randomly.

effective altruism

I mentioned Jess Riedel in the previous blog post. Here I want to highlight his list of organizations related to effective altruism.
While we contemplate how many worlds there are, we can try to improve the one we know - beyond posting hashtags on twitter.

preferred basis

I did write some comments on Scott's blog which might be interesting to those bothered by the 'preferred basis problem'. It begins here and references are made to this answer at physics.stackexchange by Jess Riedel and this paper by Dowker and Kent.

While I'm at it, I should also link to this paper about 'entanglement relativity' and a (claimed) inconsistency of the Everett interpretation. I am not sure if the argument is correct (decoherence might appear different for two different decompositions, but does this really prove anything?) and would appreciate any input.

added later: The back and forth in the comment thread ended (for now) with a homework exercise for mwi proponents.

added later: Btw another interesting paper from an Austrian team about decoherence due to classical, weak gravitation (i.e. on Earth).

----

Btw this unrelated comment Scott made about the "arrow of time" was a bit shallow imho. My own view of the problem begins with this thought experiment.

a probability puzzle

No paradox and nothing profound here, just a little puzzle to pass the time until Monday.
I have two reasons for posting it: i) It is similar to some problems I have to deal with at work (*) and ii) it gives me an opportunity to link to the blog where I got it from (after the solution is revealed).

So Alice and Bob like to play a certain (card) game (if they are not busy with encryption problems and black hole entanglement). Everybody knows that Alice is slightly more skilled at this game and wins with probability 55%; However, she really likes to win and so Alice and Bob always play as many games as it takes for Alice to be ahead in winnings (x). So sometimes they play just one game (if Alice wins immediately) and sometimes many, but what is the expected number N of games the two will play (after a fresh start)?

(*) A similar problem I would be dealing with could be e.g. of the form "if I have an order sitting at the bid, how long will it take on average to get filled".

(x) Added later, just to clarify: Alice and Bob play N games until Alice wins one game more than Bob. E.g. Alice wins the 1st game; Or Bob wins the 1st and Alice wins the next 2 games; Or ...


------------------


This puzzle is equivalent to a biased random walk of the difference D in winnings between Bob and Alice. It begins at D=0 and if D>0 it means that Bob is ahead; The random walk ends at D=-1 i.e. when Alice is ahead by one. So what is the expectation value E = E[N] of the length N of this random walk?

There are two ways to solve it. One can (try to) sum up all terms in the series of all possible events as described here. I assume this is how John von Neumann would have solved this puzzle.

Fortunately, there is a much easier solution for the rest of us and you can find it in the comments.
It gives us E = 1/(2p - 1) and with p=0.55 for Alice to win a single game we get E=10.

Notice that E diverges for p=1/2 and I find this somewhat counterintuitive, knowing that an unbiased random walk will visit every point D with probability 1.

the strange result(s) of Frank Tipler

I met Prof. Tipler in 1992 during a seminar in Vienna about relativity and cosmology, he was a visiting professor for a year and I remember very 'normal' discussions e.g. of the Reissner-Kerr solution.
Two years later he wrote about The Physics of Immortality and I thought that his book was quite interesting, although I disagreed with his conclusion(s) and I remember that I felt uneasy about the certainty with which he expressed his unconventional views.
He jumped the shark with his next book about The Physics of Christianity and I am not sure in which Lalaland he found himself after this jump ...

But he continues to write papers about quantum physics as a proponent of a 'hardcore' many worlds interpretation (m.w.i.) and this post is about one of his conclusions:
His interpretation is actually based on the Bohm interpretation, assuming a deterministic Hamilton-Jacobi evolution of a distribution of hidden variables. While the original Bohm interpretation considers particle positions, in the Tipler interpretation the different possible universes are the hidden variables. He understands the Bohr probabilities as Bayesian probabilities obtained by the many real observers in the multi-verse of all those universes. I think at this point his views are still compatible with m.w.i. a la Everett and he argues that the Heisenberg uncertainty principle follows from his proposal arising "from the interference of the other universes of the multiverse, not from some intrinsic indeterminism in nature". So far so good ...

But then he claims to have a test of his interpretation by measuring pattern convergence rates: Frequencies of events measured in real experiments with sample size N will converge as 1/N to the Born frequencies. And I think this has to be wrong.
He even notices that in classical statistics the frequencies of events following e.g. a Gauss distribution converge slower, i.e. 1/sqrt(N), and I wonder why this does not bother him. After all, it is not difficult to set up a simple quantum physics experiment which reproduces classical convergence. Consider a (weakly) radioactive source which triggers a Geiger counter with 50% probability in a certain time interval. Now we let the Geiger counter tick along and we can be quite sure that the sequence 100101011111000010101000... that we will obtain obeys the well known laws of conventional statistics.
What am I missing?
Can one use Tipler's result as (another) example that m.w.i. does not reproduce the properties of Born probabilities correctly?


Btw if you wonder why I wrote this post now ... I saw Tipler's name on this diagram and remembered that I always wanted to write something about his strange result.

the phase structure of CDT

In my previous post I criticized the description of the CDT phase diagram on a popular physics blog.
In this post I want to actually talk about the numerical CDT results.

The phase diagram depends on two coupling constants K and D (in the text they use kappa and delta). While K corresponds to the gravitational coupling, D measures the ratio of 'timelike' and 'spacelike' edges; I use quotes ' ' because the simulation is actually done in the Euclidean sector, but edges fall in different categories, depending on what kind of distances they would correspond to after Wick rotation. There is a third coupling parameter, which corresponds to a cosmological constant, but it is fixed for technical reasons.

As I already explained, one looks for a critical line in D,K corresponding to a 2nd order phase transition and the reason is that long-range fluctuations are associated with such a transition, so that the details of the discretization do not matter any more.
So this is what I find weird: The parameter D describes a detail of the discrete model and the hope is to fine tune D, as a function of K, in order to find a critical line where the details of the discretization no longer matter...

The authors notice that D has "no immediate interpretation in the Einstein-Hilbert action" and thus the critical value D(K) does not correspond to any feature of the continuum limit - unless the continuum limit is not Einstein-Hilbert but Horava-Lifshitz gravity. This is what the authors propose and discuss in section 4 of their paper: HL gravity breaks diffeomorphism invariance of EH gravity, just like CDT does, and the parameter D would have a 'physical' meaning in this case.

It seems that the authors hope that EH gravity will be restored somewhere along the critical D(K) line, however, it is unlikely imho that there is such a path from HL gravity to real gravitation.

backreaction

It seems that an internet tradition is emerging, whereby a blog remains dormant for a while, until something so outrageously wrong appears on the interwebs that one has no choice but to respond to it.

In my case, Sabine Hossenfelder wrote about the phase diagram of CDT on her popular physics blog and I just have to set a few things straight:

1) We read that "... most recently the simulations found that ... space-time has various different phases, much like water has different phases".
But, of course, the phase structure of various lattice gravity models has been studied (implicitly and explicitly) since the early days of lattice gravity simulations, i.e. the 1980s. If one wants to find a reasonable continuum limit for such a model, then one has to examine the phase structure of the lattice model; In general, if the model has one or more coupling parameters then it will (most likely) exhibit different phases, just like water.

2) The holy grail to a physically interesting continuum limit is the existence of a non-trivial fixpoint, which appears in the phase diagram as a 2nd order phase transition. IF such a transition exists for CDT, it will be located on (one of) the critical lines and perhaps at the tri-critical point. The continuum limit will not appear in the area C of the diagram; There you certainly cannot "share images of your lunch with people you don’t know on facebook".
As far as I know, the existence of such a 2nd order transition has not been demonstrated yet, although intriguing hints have appeared in other lattice models previously. Of course, even IF such a 2nd order transition could be demonstrated, one would still not know if the continuum limit has anything to do with gravitation as we know it.

3) This 2nd order phase transition is prerequisite to a consistent continuum model and all 4d geometries would be generated with the same critical parameter values. It is therefore misguided to imagine that this phase transition happened at or near the big bang.
Indeed, the coupling parameters depicted in the phase diagram are bare, i.e. un-renormalized, coupling parameters and while the diagram may indicate existence and location of a non-trivial fixed point, almost all of the phase diagram is actually non-physical.
Therefore one cannot expect that this phase transition may be an alternative and/or replacement for inflation (as Sabine discussed in the comments).

Knightian uncertainty

I was thinking about a good example of Knightian uncertainty and this is my proposal: We cannot know what theorems a particular mathematician or a group of mathematicians will be able to prove.
As a concrete example, consider the collaborative effort to lower H from Zhang's 70 million; Currently, the best confirmed value for H is 60,726. But will H drop below 1000 by the end of this year?

Obviously we cannot know for sure (otherwise we would have a proof for H < 1000 already) and I think that any attempt to come up with a 'Bayesian probability' in this case would be inappropriate.
But this means that the state of our world by the end of this year is unknown. (If you wonder how to reconcile such a statement with physics, I recommend these links: 1, 2, 3, 4).
Btw it is possible that some Martian mathematicians, with far advanced math capabilities, would already know the answer to the above question, but in this case I have to assume that they have theorems in their advanced Martian mathematics which they cannot prove yet.

So the strong form of my example is this: In our universe there is at least one mathematician, who is unable to predict if she will be able to prove the theorem she is currently working on - and nobody else is able to predict it either, because she is the smartest/most advanced mathematician.
Therefore the future state of our universe is unknowable.

many Turing machines

We randomly implement a Turing machine Tm with N states, using a Geiger counter plus radioactive material as 'quantum coin'. We proceed as follows: First we use the 'quantum coin' to determine (with probability 1/2) if the Tm has N=2 (head) or N>2 (tail) states. If we got tail, we use the 'quantum coin' again to see if N=3 or N>3 and so on and so forth.
Once we have determined N in that way, we then construct the transition table(s) of the Tm, using the 'quantum coin' again and again, so that all (8N)^N possible transition tables could be realized.
Once this construction is finished, we start the Tm on a tape with a few 1s sprinkled on it and then watch what happens.

This experiment is easy to understand if we follow the Copenhagen interpretation. The Tm we will build is most likely quite simple, because the probability for a complex N-state Tm decreases rapidly as 1/2^N-1. Once the Tm is put together, its evolution is not even a quantum mechanics problem any more. If we want, the transition table(s) of this particular Tm can be inspected before it runs, to determine if it will halt.

But this experiment is much more difficult to understand within a many worlds interpretation: Every use of the 'quantum coin' splits the world and thus we are dealing with a wave function of the universe which contains every possible Tm in one of its branches. The amplitudes assigned to worlds with N-state machines are quite small if N is large, but all the worlds are equally real.
Unfortunately, due to the Halteproblem, the evolution of this wave function is uncomputable. In other words, the wave function of the universe does not have a mathematical description at all.

The best part of this experiment is that I do not even have to do it myself. Somewhere in the universal wavefunction there is already a branch where somebody, somewhere in the universe, has decided to do this experiment (*). This means that the amplitude of my branch is already uncomputable (x).


(*) This also takes care of the counter argument that on Earth resources are finite and thus the experiment has to terminate at a certain (large) N₀. Since we have to consider the wave function of the multiverse (of infinite size), this argument is not convincing, because we cannot know N₀.

(x) Notice that the overlap between different branches of the wave function is very small, due to decoherence, but in general non-zero.