the statistical mechanic

efficiency

2012-01-24T04:56:00.001-08:00

Cosma writes about power laws and psycho killers; The usual story - somebody sees a power law when e.g. a log-normal distribution would be a much better fit.
This suggests that three toed sloths and the internet are not fully efficient distributors of information.

Meanwhile, on intrade I saw (Jan 24, 8am ET) some interesting odds for the Dow Jones Industrial index, suggesting that markets are not always fully efficient either:
Dow Jones to close ON or ABOVE 11750 on 31 Jan 2012: probability = 91.5%
Dow Jones to close ON or ABOVE 12000 on 31 Jan 2012: probability = 93.0%
Dow Jones to close ON or ABOVE 12250 on 31 Jan 2012: probability = 97.0%

So I wonder about the odds in a cliché betting market...

the isle of Elba

2011-12-20T02:42:00.000-08:00

"Frequentists have long been in a kind of exile when it comes to statistical philosophy. ... This may now be changing."

Deborah Mayo has a blog.

on probability

2011-11-13T23:50:00.000-08:00

"We take issue with Kent's arguments against many-world interpretations of quantum mechanics. We argue that his reasons for preferring single-world interpretations are logically flawed and that his proposed singleworld alternative to probability theory suffers from conceptual problems. We use a few thought-experiments which show that the problems he raises for probabilities in multiverses also apply in a single universe."
Guildenstern and Rosencrantz in Quantumland

It seems to me that the debate about many worlds has finally reached the point of asking 'what exactly do we mean with probability?' and I doubt this will be settled any time soon.
To be, or not to be, that really seems to be the question...

harmonic universe

2011-10-23T01:16:00.000-07:00

Yet another attempt to reduce the physics of our universe to the harmonic oscillator; An interesting paper and a pleasure to read. But there is a problem with this approach near the end imho. I do not feel that the proposed boundary conditions psi(0) = 0 and psi(infty) = 0 are natural and honestly I have no intuition what would be natural in this context.
I think that this is a general problem whenever one tries to calculate the 'wavefunction of the universe'. While we have a good intuition for boundary conditions in conventional quantum theory, this intuition is lost for quantum cosmology. But of course those boundary conditions determine everything...

we are all crackpots now

2011-09-26T00:22:00.000-07:00

“If it is true, then we truly haven’t understood anything about anything."
Alvaro DeRejula, theorist at CERN

I fully agree - if the OPERA result is correct (and this is still a big IF) then we are all crackpots now and the only real difference is that some speak the jargon and put their stuff on arXiv and the others post to viXra.

Only a tiny minority was seriously looking at previous hints and of course the inconclusive MINOS result was ignored by almost everybody - it seems that the OPERA experiment was mostly an accident happening on the side.
While we discussed how many angels can dance on a superstring, the real physics 'beyond the standard model' was eventually sitting there, waiting to be looked at during several lost years.

Maybe a small tweak of our theories is sufficient to save the appearances once again. But while I was thinking about just such a tweak I was once again confronted with the fact that QED, currently the best theory we have about light, is more like a patchwork of approximations and loose ends rather than a fully consistent, elegant theory.

Over time physics developed a culture replacing rigorous thinking with the ability to get the next paper accepted by the community.
I am actually hoping that this time reality will knock down this culture a bit.

neutrinos as tachyons, the Scharnhorst effect

2011-09-24T01:46:00.000-07:00

I wonder if neutrinos should travel slightly faster than photons according to standard physics, i.e. quantum field theory as we know it (*).
The reason would be the Scharnhorst effect.
If c denotes the 'bare' speed of light then real photons should actually travel a tiny bit slower than c, due to the interaction with virtual particles in a 'real vacuum': c(physical) < c. The bulk of the effect comes from interactions with virtual electrons and positrons (see e.g. this paper for more details). But neutrinos interact only weakly with those and therefore the velocity of neutrinos should be closer to c; In other words neutrinos would be slightly faster than photons.
I am not sure if the OPERA experiment detected anything real, but if the results are indeed confirmed one should take a second look at the Scharnhorst effect imho; This time considering the difference between neutrinos and photons instead of looking at photons between Casimir plates.

with credit to A.M., a friend of mine and yet another quant interested in physics, who reminded me of this effect, which was discussed already in the 1990s. But if this is all b.s. the embarrassment is of course fully mine.

added later: A very simplified calculation shows that the Scharnhorst effect could have the right order of magnitude!
According to the Scharnhorst paper, equ. 10, the increase of the speed of photons between Casimir plates is to first order approximately
1 + 0.01*alpha²/(mL)⁴.
As we move the plates closer and reduce L (in a thought experiment) we eliminate more of the interaction with the virtual particles and the speed of photons gets closer to the 'bare' speed of light c. However, we cannot reduce L below 1/m (in this approximation) and thus the Scharnhorst formula gives a maximum correction
c = c(bare) = c(physical)*( 1 + 0.01*alpha² ).
If we assume that neutrinos travel at a speed close to the 'bare' speed c, then they would travel faster than photons in the 'real vacuum' by a factor of approximately 1 + 10^-6.
This is pretty much what OPERA measured.

(*) added even later: Heather Logan thinks this explanation cannot work and I changed the sentence to reflect that this is not 'standard' opinion; After all she is a physics professor and my last lesson in QED was about 20 years ago. On the other hand I don't see where the argument goes wrong. Obviously I appreciate any input.

added one day later: After some more thinking and reading I can now better formulate this idea as follows:
i) In QED the Ward identities ensure that after quantization of the electromagnetic field the longitudinal polarization of the photon vanishes.
ii) This implies (by comparison with classical fields?) that the photon mass remains zero and the velocity of photons is the light speed c.
iii) On the other hand, the Scharnhorst effect suggests that photons travel at a 'dressed' speed d and one can increase (in principle) d between Casimir plates (see all the above).
iv) The standard interpretation is that d = c to keep QED simple, but one has then to explain away the acausality from photons travelling at increased speed between Casimir plates.
v) In light of the OPERA experiment, the proposal is to assume d < c and (because the Ward identities hold) for all possible experiments the longitudinal polarization of the physical photon still vanishes, so that it would behave like a massless particle, although it travels slightly slower than the 'true' speed of light c. (By the way, I am not the first thinking along such lines.)
vi) The neutrino would travel very close to the 'true' speed of light c and thus slightly faster than photons. The estimate above suggests that the effect could be in the right ballpark,
see the comments for more ~~confusion~~ clarification.

neutrinos as tachyons

2011-09-21T02:38:00.001-07:00

It is not surprising that there is a debate about the detection of faster-than-light neutrinos before that discovery is even announced...

It is also not surprising that there are already several arguments in favor of tachyonic neutrinos, even if there are good reasons to doubt that idea (*).

What I do find surprising is that slashdot and other news outlets are so far silent about this threat to the well being of grandfathers.

added later: They are no longer silent and OPERA's paper is here...
... as bonus material, let me mention an old joke from my home country, where it is known for a long time that bureaucrats are made from tachyons. How did we know that sometimes they travel faster than light? Because official working hours end at 4pm, but they are often at home already at 1pm.

(*) It seems to me that tachyonic neutrinos could easily resolve the information loss problem and would let us "see" the interior of black holes. They would also explain how information can be retrieved from baby universes and thus support a silly idea of mine. Obviously, this should count in favor of the tachyonic neutrinos. Simple question at this point: So what Bayesian prior should one use???

Eppley and Hannah

2011-08-30T04:13:00.000-07:00

Every now and then somebody asks if it is really necessary to find a quantum theory of gravitation. After all, it is most likely not possible to detect single gravitons, following an argument of Freeman Dyson (because one would need a detector of planetary size for it).
Of course there are many good reasons why one would like to find a way to quantize gravity like all other fields [1, 2, 3]. But I was never worried about this sort of debate, because I knew that there was a thought experiment, published in the '70s or '80s, which settled this issue once and for all: Consistency requires that gravitation must be quantized. I remember that I read the argument and that I found it convincing at the time.

Recently, I was asked about this whole issue and I mentioned the thought experiment and that paper. Finally I promised that I would dig out the reference and I actually did.
K. Eppley and E. Hannah, Found. Phys. 7, 51 (1977)

The reason it was relatively easy to find the reference was that almost thirty years later somebody checked the argument and found that it was flawed. The problem is that the thought experiment asks for a detector so large and heavy that it cannot be built, somehow closing the circle back to Dyson's argument.

order!

2011-08-27T05:26:00.000-07:00

Continuing a previous theme, let me ask this: So why don't they put their house in order? Seriously.
via Cosma

many verses

2011-08-11T02:00:00.001-07:00

The main motivation of no-collapse interpretations is to emphasize that the unitary Schroedinger evolution is all there is.

In order to reconcile this with our experience a description considering 'many worlds' or 'many minds' and the 'splitting' of the universe into different 'branches' is often used: The physicist does not kill the cat but really creates two worlds and then repeating this experiment doubles the number again and again.
It seems that the number N of 'branches' or different 'minds' etc. increases like N(t) ~ w^t where w is some unknown constant. Now one can calculate something like an entropy S from it as S = log(N) and therefore S ~ t; time really is 'm.w.i entropy'.

But how does one reconcile all this with the time-reversible Schroedinger evolution?

asymptotic safety

2011-07-11T22:55:00.000-07:00

Georg v. Hippel blogs about the Lattice 2011 conference and he mentioned a talk by Jack Laiho on Asymptotic Safety and Quantum Gravity.
There is already a paper about that on the arXiv. The main idea is that three coupling parameters are needed in the lattice model and they consider dynamical triangulation in the Euclidean sector with an additional measure term, which comes with the third coupling parameter. The hope is to find a tri-critical point (with 2nd order phase transition) as a candidate for the continuum limit in the asymptotic safety scenario.

balls and urn

2011-06-21T06:32:00.000-07:00

Consider an urn holding N balls which are either white or black. Whenever I take out a ball it is replaced randomly with either a white or black ball, with fixed probability p for the replacement ball being white. I am playing this game for quite a while (so that the start configuration no longer matters) and now I am taking out n white balls in a row (n < N).
What is the probability that the next ball is also white?

If p is (near) 1/2 then one could make these two arguments:

1) If we take out a sequence of n white balls it indicates that there are probably more white balls in the urn than black balls (due to a random fluctuation in the replacement process), so the next ball is most likely also white: P(white) > P(black).

2) If we take out n white balls, the ratio of white to black necessarily decreases, so it is more likely that the next is actually black: P(black) > P(white).

What do you think? And does it make a difference if we actually know p and N ?

added later: I have posted the solution now as a comment, but I have to warn you that this is fun only if you really try to find an answer yourself first.

decoherence

2011-06-07T00:48:00.000-07:00

One can still quite often read or hear the argument that decoherence solves the measurement problem and therefore further discussion of interpretations is unnecessary.
Fortunately, I can keep this blog post short with a link to this paper:
Stephen Adler, Why Decoherence has not Solved the Measurement Problem.

-----

If one wants to read about the role of decoherence within different popular interpretations, I recommend this paper:
Maximilian Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics

It notices that "Decoherence adherents have typically been inclined towards relative-state interpretations ... It may also seem natural to identify the decohering components of the wave function with different Everett branches." and it then proceeds to discuss two important open issues of that interpretation: the preferred-basis problem and the problem with probabilities in Everett interpretations.
If one want to go down that route I recommend this paper for further reading.

the Coleman argument

2011-06-03T03:42:00.001-07:00

In 1994 Sidney Coleman gave the lecture 'Quantum Mechanics in Your Face' which was recorded and the video is available here. Near the end he makes an argument, actually attributed to Dave Albert, which is nowadays often used in debates about the meaning of quantum theory (usually people just link to the video without discussing the argument(s) much further).
But, as we shall see, it does not really work the way people think it does.

Consider a quantum system (e.g. electron in a Stern-Gerlach apparatus) which evolves from an initial state |i> into a superposition |a> + |b> (*). An observer (named Sidney) makes a measurement and evolves from an initial state |I> into a superposition |A> + |B>. How can we reconcile this superposition of observer states with our everyday normal conscious experience?
Well, consider all the states |Sj> of 'Sidney with a normal conscious experience', where j supposedly labels all the different conscious experiences Sidney can have. All those states |Sj> are eigenstates of the 'consciousness operator' C so that C|Sj> = |Sj>.
It is clear that |A>, which here means 'Sidney observed |a>', is an eigenstate of C and also |B> is an eigenstate of C,
C|A> = |A> and C|B> = |B>.

But it follows immediately that |A> + |B> is then also an eigenstate of C,
C( |A> + |B> ) = ( |A> + |B> ), from the linearity of the quantum operator C.

This would mean that the superposition of Sidney states does not really cause a problem; The superposition still corresponds to a 'normal conscious experience'.
So it seems that there is no 'measurement problem' as long as we stay fully within quantum theory. Therefore this argument is very popular with people who want to end the interpretation debate, while I think it may be a good starting point to begin a serious discussion.

In order to see that something is wrong with the Coleman argument, consider that there must be states |Uj> of Sidney where he does not have a 'normal conscious experience', eg. because he is asleep, drunk or worse.
Obviously |Uj> cannot be an eigenstate of C, so that C|Uj> = 0.
The problem is that we have to assume that the initial state of Sidney |I> does not just evolve into the superposition |A> + |B> but it will also contain some components |Uj>, because even for a healthy person there is a small but non-zero possibility to fall into an unconscious state. But as one can check easily, the state |A> + B|> + |Uj> is certainly not an eigenstate of C and due to the superposition of states Sidney will in general never be in such an eigenstate of C. This would mean that Sidney never has a 'normal conscious experience'.

Obviously, there must be a problem somewhere with this whole argument and I leave it as an exercise for the reader to find it and post it as a comment 8-)

(*) I leave out the necessary normalization factors of 1/sqrt(2) etc. in this text and the reader has to assume that these factors are absorbed in the state vectors. E.g. in the state |A> + |B> + |U> we assume that |U> comes with a very small weight/probability, but I do not make this small pre-factor explicit.

modified gravity?

2011-05-23T02:45:00.000-07:00

Lubos suggests an explanation (or actually a replacement) for MOND, which sounds like entropic gravity to me (*). Did he not recently explain to us why such explanations have to be wrong?
The purpose of his proposal is to explain observations which suggest that gravity changes at low accelerations a < a₀ = 1.2x 10^-10 m/s² and it is based on the idea that a₀ could be the inverse size of the visible universe (times c).

(*) I should make it clear that Lubos never mentions 'entropic gravity' in his post, but how would a sentence like "The existence of this center on the hologram may be needed for the usual Kepler scaling laws to emerge." make any sense otherwise? See e.g. this paper on how MOND was derived previously from 'entropic gravity' using "a holographic principle".

added later: When I asked explicitly in a comment, he insisted that his proposal has nothing to do with "the crackpottery called entropic gravity". Alright, I will admit then that I do not understand what he is talking about and leave it to others to sort it out. Feel free to leave a comment to enlighten me. I will leave this post up as it is, because the links could be useful to others.
Perhaps I should make it clear that I am (still) not convinced 'entropic gravity and/or MOND make any sense.

added even later: While Lubos focuses directly on the quantity a₀ = c/T, with T being the age of the (visible) universe, I think it would be more natural to consider the energy E = h/T. The associated temperature is E/k, which is on the order of 10^-28 kelvin and comparable to the critical temperature used to derive MOND in the paper linked above; In fact plugging hk^-1/T into equ. (12) gives a₀ = c/T !

added several hours later: Obviously, there is a straightforward way to test this type of proposal. If one looks into the night sky one sees galaxies at different age T of the universe and the deviation from Newtonian dynamics should be stronger the further back in time one looks. Perhaps there is already enough statistics of galaxy rotation curves to check this.

added much much later: As a counter point to all this speculation a paper [pdf] about an experiment which "finds good agreement with Newton’s second law at accelerations as small as 5 x 10^-14 m s^-2.

statistical theology

2011-05-18T01:31:00.000-07:00

This post is part of a series about important theological issues [1, 2, 3, 4] and this time I want to emphasize the importance of Benford's law in statistical theology. In the recent paper "Law of the leading digits and the ideological struggle for numbers", it was used in religious demography research:
"We investigate the country-wise adherent distribution of seven major world religions i.e. Christianity, Islam, Buddhism, Hinduism, Sikhism, Judaism and Bhah'ism to see if the proportion of the leading digits conform to the Benford's law. We found that the adherent data on all the religions, except Christianity, excellently conform to the Benford's law."

What does He want to tell us with this exception?

lattice gravity

2011-05-10T07:51:00.001-07:00

You may have noticed the link to quantum gravity on the left hand side of this blog, below the picture of the plastic bag; It leads to several blog posts about "lattice gravity" as well as posts about quantum gravity in general.

If you want to know more what "lattice gravity" is all about, you can browse some of the references provided here; Another good starting point would be Renate Loll's review (in particular sect. 3) and for even more motivation I recommend this paper.

One warning: It is very much possible, and actually quite likely, that these models have nothing to do with real (quantum) gravity. In fact there are several good arguments why a lattice approach can never work. In other words, it is very much possible that this will turn out to be a waste of time.
Just another reason it makes for a good topic on this blog...

is AdS unstable?

2011-04-23T06:24:00.000-07:00

"We study the nonlinear evolution of a weakly perturbed anti-de Sitter (AdS) spacetime by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS spacetime is unstable under arbitrarily small generic perturbations."
Piotr Bizoń, Andrzej Rostworowski

Notice that this study was done in 3+1 dimensional AdS₄, but the authors claim (in the conclusion) that they observed "qualitatively the same behavior" for the 4+1 dimensional AdS₅.
But with all the activity about AdS/CFT, it is it hard to believe that nobody checked the stability of AdS in classical GR before.

the not-so-equivalent principle

2011-04-17T04:12:00.000-07:00

I posted this puzzle a while ago on wbmh, but subsequently it got deleted and I think it is worth reposting it here. Please write a comment if you have an answer (or remember the discussion on wbmh) and win the famous golden llama award.

Perhaps you have seen the video clip of astronaut David Scott dropping a hammer and a feather on the surface of the moon, repeating Galileo's famous (thought)experiment.

But strictly speaking in a high precision experiment the two objects will in general not hit the moon at the exact same time.
Why is that?

---

Let me clarify a few things.
We assume that the shape of the objects makes no difference (we assume the spherical cow approximation is valid) and the surface of the moon is perfectly smooth. Further we assume that there is no trace of an atmosphere on the moon and no electrical charges (attached to the objects). We assume that the presence of the astronaut (and his gravitational field) can be neglected and we assume the sun, earth and the other planets are sufficiently far away. We ignore quantum theory and assume that the many-worlds interpretation and any other conspiracy theories can be neglected...

added later: Furthermore we assume that all objects and observers move slowly compared to the speed of light, so that we can use the standard St. Augustine definition of simultaneity.

update: Akshay Bhat is the proud winner of the famous golden llama award ...

... he will enjoy a free subscription to this blog for a whole year. Congratulations!

If you want to see my own solution to this puzzle click here.

hammer and feather

2011-04-16T00:43:00.000-07:00

In this puzzle we consider a high precision re-enactment of the famous Galileo (thought)experiment and the claim is that hammer and feather dropped simultaneously from a height h will not hit the ground at exactly the same time.

In order to see this, we increase the height h and consider the full 3-body problem with feather (F), hammer (H) and moon (M) approximated as spheres (the famous spherical cow approximation).

Next we increase the distance between F and H and we increase the mass of the hammer H significantly. Therefore the moon will move towards H by a certain displacement a and thus the hammer has to travel the distance h - a until it collides with the moon M, while the feather F has to travel the increased distance sqrt( h^2 + a^2 ), which suggests that F will indeed collide with M slightly later than H.

But we have no full proof yet (notice that the feather is attracted by moon+hammer while the hammer is slightly less attracted by moon+feather and this could compensate for the different distances).
So in order to obtain full proof of our claim (without using too much math) we move the hammer H even further away from the feather F and we increase its mass until it exceeds the mass of the moon M significantly (perhaps it is easier to decrease the mass of the moon until it is more like a hammer).

But now we have transformed this thought experiment into a configuration where F and M are dropped on H, but from very different heights h and 2h. In other words, comparing with the original configuration, the assumption that the three bodies will collide at the same time is disproved by reductio ad absurdum.

I would like to make three more remarks:
1) The equivalence principle is an idealization (notice that in the general theory of relativity we consider test bodies to have infinitesimal mass and distances are small compared to the radius of curvature).
2) If we make the spheres small enough they will in general not collide at all (I leave it as an exercise for the reader to run a 3-body simulation and check this claim), except for symmetric initial configurations like the one in the last picture.
3) The contemporary opponents of Galileo could have made this reductio ad absurdum to counter his argument and discredit his physics. It is interesting to contemplate how science would have progressed in this case ...

the greatest intellectual failure

2011-04-12T05:02:00.000-07:00

We do know how to do the calculations, we can determine the probabilities for various experiments, real or imagined. In fact, an amazing machinery of methods and tools has been developed over decades for this purpose; A cornerstone of modern physics and science in general.
And yet, important foundational questions remain unanswered.

I am talking, of course, about statistics and our theories of probability.

Recently I found this:
"This book is about one of the greatest intellectual failures of the twentieth century - several unsuccessful attempts to construct a scientific theory of probability. Probability and statistics are based on very well developed mathematical theories. Amazingly, these solid mathematical foundations are not linked to applications via a scientific theory but via two mutually contradictory and radical philosophies. One of these philosophical theories (frequency) is an awkward attempt to provide scientific foundations for probability. The other theory (subjective) is one of the most confused theories in all of science and philosophy. A little scrutiny shows that in practice, the two ideologies are almost entirely ignored, even by their own supporters."
Krzysztof Burdzy

But I guess before I buy the book I will browse the blog a bit more...

added later: ... and read some reviews: negative and positive. (I thank Jonathan for the links.)

in search of absolutely no information

2011-03-27T04:20:00.000-07:00

It seems that most readers swallowed my arguments in the previous post, so I think it is time to turn the screw a few more times so to speak.

Again, we are confronted with a deck of 2 cards (we know that there are only black and red cards, no jokers), we take the top card and it is black. What is the probability that the remaining card is also black?

In the previous post we only considered two hypotheses (*):
H1 ... black cards only
H2 ... mixed deck
and we assumed p(b|H2) = 1/2.
This assumption bothers me now.

We don't know how the 2-card deck was put together and it could be that somebody made sure we would always find a black card on top, even for the mixed deck. So I think we need to split H2 into two different hypotheses.
H2a ... manipulated mixed deck: p(b|H2a) = 1
H2b ... random mixed deck: p(b|H2b) = 1/2

Again, relying on the principle of indifference we use an uninformed prior p(H1) = p(H2a) = p(H2b) = 1/3 and find p(H1|b) = 2/5 (I recommend that you actually plug in the numbers and do the calculation).
In other words, the probability that the remaining card is also black (2/5) is now significantly less than the probability that it is red (3/5).

But it is clear that we have not yet achieved the goal of "absolutely no information" in selecting our prior; The problem is H2b.
We have to split it further to take into consideration cases in between outright manipulation and random selection. So we have to consider additional N hypotheses
H2k ... k=1 ... N with p(b|H2k) = k/N
and let N go to infinity (o).
As you can easily check, the large number of hypotheses for mixed decks results in p(H1|b) -> 0 and therefore we have to conclude that using an absolutely uninformed prior the probability that the remaining card is black is (close to) zero.

This remarkable result is due to the fact that there is only one way how the deck can consist of only black cards, but there are many ways how a mixed deck could have been put together. Absolutely no information means certainty about the 2nd card in this case, thanks to the power of the Bayesian method (x).

(*) Nobody complained, but it was a bit sloppy to exclude H0 = 'red only' from the prior (which should be chosen before the black card was seen). But p(b|H0) = 0 and, as you can check, including H0 would have made no difference.

(o) The limit N = infinite would require the use of an improper prior and I think it is sufficient for our purpose to consider the case of large but finite N.

(x) I think this toy model will be very helpful e.g. in cosmology and multiverse research.

in search of no information

2011-03-24T05:24:00.000-07:00

The main purpose of this blog post is to illustrate my ignorance of Bayesian statistics(*), discussing a very simple game with a 2-card deck. By the way, we only consider black and red cards, no jokers etc.

So we begin with the 2-card deck and draw one card - a black card. The question is 'what is the probability that the other card is also black?'.
Fortunately we only need to consider two different hypotheses:
H1 ... both cards are black.
H2 ... a mixed deck (one black, one red).

We update our probabilities using the famous formula:
p(Hi|b) = p(b|Hi) * p(Hi) / p(b)
where b indicates the 'black card event' and p(b) is short hand for the sum p(b|H1)*p(H1) + p(b|H2)*p(H2).
Since we have no further information we use an uninformed prior which does not prefer one hypothesis over the other, in other words: p(H1) = p(H2)
and using p(b|H1) = 1 and p(b|H2) = 1/2 we get
p(H1|b) = 2/3 and p(H2|b) = 1/3. (I recommend that you actually plug in the numbers and do the calculation.)

I admit that there is something weird about this result: We have two cards, we pick one and after updating our probabilities have to conclude that the other is more likely being black than red, actually twice as likely.

The issue seems to be our prior, i.e. the choice of p(Hi).
Indeed, if we draw the 2-card deck randomly then the probabilities that it contains bb, br, rb and rr should be the same. We can throw out rr, which leaves us with a 2:1 majority of mixed decks and should therefore use p(H1) = 1/3 and p(H2) = 2/3. As you can check this resolves the weirdness, we get p(H1|b) = p(H2|b) and thus the probability that the 2nd card is also black would be the same as the probability for red.
But can a Bayesian accept such card counting?

And it gets worse: If the 2-card deck was drawn from an initial deck of 2N cards then the probabilities of bb and br are not exactly the same. The probability of the first b is indeed 1/2 but the probability of a 2nd b is lower, (N-1)/(2N-1), and therefore a mixed deck seems actually preferred, depending on the number N, which we don't know. Should we really conclude that red is slightly more likely after seeing black? Do we have to sum over all possible N with equal weight to get a truly uninformed prior?

But I am sure a true Bayesian would reject all those card counting arguments which smell quite a bit of frequentist reasoning. A truly uninformed prior means that we have no information to prefer one hypothesis over the other. There is a difference between not knowing anything about the 2-card deck and knowing that it was randomly selected. Therefore the symmetric choice p(H1) = p(H2) is the true uninformed prior which properly reflects our indifference and we have to live with the asymmetry of probabilities for the 2nd card.

(*) A true Bayesian would calculate the probability that this post contains sarcasm taking into account the existence of this footnote.

how little do we know

2011-02-20T07:23:00.000-08:00

Assume that you have two finite samples X1, X2, ..., Xn and Y1, Y2, ...Ym independently drawn from two distributions. Don't worry we know that both distributions are normal, no fat tails and no other complications. All we want to know is the probability p, given the Xs and Ys, that the two distributions have the same mean. We don't know and we don't care about the variances.
You would think that statisticians have a test ready for this, an algorithm which takes the Xs and Ys and spits out p, and you would think they have figured out the best possible algorithm for this simple question.
You would be wrong. Behrens-Fisher is one of the open problems in statistics.

micro and macro

2011-01-28T03:43:00.000-08:00

Cosma wrote about the microfoundations of macroeconomics and received several interesting comments [1, 2].
In my judgment the idea that economics is the statistical mechanics of interacting agents has not worked out very well so far. But of course it has some value, if only some entertainment value in the worst case.

I am for various reasons increasingly convinced that the usual tools of statistics are of limited use when dealing with financial systems or the whole economy and recently I came across a book which expresses a similar sentiment.
The Blank Swan: The End of Probability begins with a discussion of 'regime switching', but with the added complication that the number and properties of those regimes change over time. It then turns quickly into a general discussion of probability.
I am not sure yet what to make of it and I don't know if I can recommend the book. I guess most readers would find it rather weird; But perhaps it is the first step in a new direction ...

why this blog is unnecessary

2010-12-04T07:42:00.000-08:00

I assume you noticed already that this blog revolves around a few issues only: the arrow of time [1, 2, 3], the interpretation of quantum theory [1, 2, 3], the meaning of probability [1, 2, 3] and the role of consciousness [1, 2, 3].

But many years ago, H. D. Zeh published a remarkable paper, which already condensed all those issues into one master piece and some thought provoking conclusions, even "assuming that every spacetime point carries consciousness". I really recommend it.

So it seems that all that might be left for me to write about are Ikea chairs and theology.

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added later: Yes I am aware that locating consciousness in single spacetime points does not help us much with quantum gravity and the 'branching' of wavefunctions might be problematic if there is no classical time. But fortunately there is already a realist interpretation of quantum mechanics "suggesting that solving the problem of time in quantum gravity leads to a solution of the measurement problem in quantum mechanics".
And since 'quantum gravity' is more or less a solved problem, this blog really is unnecessary.

the future has ended ...

2010-11-28T07:43:00.000-08:00

... and the past begins when you read this paper about the universal arrow of time. It claims that "if two subsystems have opposite arrow-directions initially, the interaction between them makes the configuration statistically unstable and causes a decay towards a system with a universal direction of the arrow of time."

I think this is just another example of 'initial condition chauvinism' and propose the following counter-example, considering a Newtonian toy model which contains two different types of particles. Initially we assume that there is no interaction between particles of the two different types and we specify initial conditions for the particles of type 1 (red) at t=tI with an associated low entropy. As the configuration evolves for t > tI the entropy increases. Now we specify final conidtions for the particles of type 2 (blue) at tF > tI and evolve the configuration of type 2 particles to decreasing t < tF. Obviously, the associated entropy is low at tF for particles of type 2 and increases as t decreases.

In a second step we turn on a (weak) interaction between the two particle types and can (e.g. iteratively) determine the resulting particle configurations. (We use the configuration we obtained initially without interaction and correct in a first step the particle trajectories due to the weak interaction with the other type. We repeat these corrections as many times as desired.) I claim that this will not reverse the increasing/decreasing entropy of either particle type, even if we finally make the interaction stronger and stronger. Due to the symmetry of the toy model if one could make an argument that e.g the entropy has to change direction for particles of type 2, then one could make the same argument for type 1 in the other direction.

I think it might be interesting to see e.g. a computer simulation of such a toy model.

added later: There are two different ways to think about and simulate such a toy model:
In the first, one specifies 'normal' initial conditions for the red particles at tI and one specifies 'special' or 'correlated' initial conditions for the blue particles also at tI and then evolves the system forward to t>tI. The blue particles would be distributed over a wide region at tI but momentum would be carefully chosen so that they converge towards a narrow region at tF.
In this picture it is natural to assume that interaction between red and blue particles should force a common arrow of time, at least if there are more red particles than blue and if we wait long enough.
I called this assumption 'initial condition chauvinism'.

In the second, one specifies 'normal' initial conditions for the red particles at tI and final conditions for the blue particles at tF, just as I explained above. In this picture it is clear that whatever happens between the red and blue particles cannot change the fact that the blue particles converge into a narrow region at tF, even if there are more red particles and even if the interaction is strong. The only way entropy can reverse for the blue particles is if the red particles would somehow force them into an even more narrow region at tI and I just cannot see how this could happen.

The most interesting aspect of this is, of course, that the two different pictures should be equivalent!

PS: I am aware of the fact that in physics we usually specify initial conditions and not final conditions, but this is exactly the puzzle of the 'arrow of time' and cannot be used to derive it imho.

added even later: It is of course true that "for most mixed initial-final conditions, an appropriate solution (of the Hamiltonian equations of motion) does not exist." However, I am pretty sure that the separation of initial and final conditions for red and blue particles, as described above, ensures that a solution to the equations of motion does indeed exist for this toy world. I would be very interested to see a convincing argument why the iterative procedure (for weak coupling) as described in the text does not converge.

three (actually five) links

2010-10-04T00:09:00.000-07:00

There are better places than this blog to read about probabilities and all that; Here are three examples ...

Terence Tao wrote a non-technical article about universality.
xkcd is not only your favorite web comic, but also posted an interesting probability puzzle a while ago [*].
And speaking of puzzling problems, if you have a question about statistics then maybe you should try StackExchange [x].

[*] If you have problems finding the answer in the many comments, it is here.

[x] added later: There is now also a StackExchange for physics.

a new proof for the truth of string theory

2010-09-27T06:13:00.000-07:00

The proof presented in my previous blog post has meanwhile be examined by many commentators (two) and I have now enough confidence to use its structure in a slightly different context.

1) String theory is the possible 'theory of everything', underlying the physical reality of our world Wo.

2) We know that s.t. leads us to the concept of a multiverse M which contains our world, but many other possible worlds too: M = {Wo, W1, W2, W3 ...}.

3) It is possible that one of those worlds contains evidence for the truth of s.t. (e.g. the energy scales are such that it is easy for physicists to probe the Planck scale).
3b) Therefore M contains at least one world Wst where s.t. is evidently true.

4) But if s.t. is evidently true for one world Wst, then it must be true for all worlds in M.

5) Therefore s.t. is evidently true for our own world Wo.

6) You will notice that the above conclusions are independent of detailed assumptions about the (composition of) multiverse M.

I am aware that this is a physicist's proof and look forward to mathematicians formalizing it in the decades to come.

Also, I am sure that some string theorists already use this argument implicitly, but I still think there is some value in making it explicit.

a new proof for the existence of God ...

2010-09-25T01:45:00.000-07:00

... from assumptions about many worlds.

I believe the following proof is a variation of Plantinga's ontological argument and continues my theological studies. But I think this new argument is sufficiently different from those previous attempts and therefore should be interesting to the reader.

1) We assume the existence of a multiverse M, which contains all possible worlds. [*]

2) M contains our world Wo.

3) It is possible that there is a world Wg created by the omnipotent, omniscient, omnipresent God.
3b) Therefore M contains the world Wg which was evidently created by God.

4) If M = { Wo, W1, W2, ... Wg ... } contains one world created by God then we must assume God created all worlds in M (otherwise God would not be omnipresent).

5) Therefore, if we assume the existence of many worlds as above, it follows that our world Wo was created by God.

6) While the above conclusion is already sufficient, we can go one step further to clarify the meaning of 5):
The existence and creation of our world is independent of other worlds (see first footnote), therefore God created our world regardless of assumption 1). [**]

[*] M is not necessarily the multiverse of string cosmology or related to the many worlds of quantum theory; We only assume that M contains all possible worlds independently and independent of specific theories.
As you will notice, the infinite character of M does not really play a role in the proof, so one need not worry about antinomies related to the set of all possible sets etc.

[**] Consider the sentence s = "If it rains in Australia, then my dog barks here in Vienna." We know that the fact of a dog barking in Europe is independent of rain falling in Australia. Therefore, if we know that s is indeed true, then we know that the dog barks (regardless of what happens in Australia).

added later: It seems that some have a problem with 3) which implies the *possibility* that God exists. I would recommend to re-read this previous blog post, in particular paragraph 2 and 3 and footnote [2], for an explanation why such atheistic doubt is not rational.

shape up

2010-09-12T09:57:00.000-07:00

"I figured you might be able to give me some pointers. I need to shape up."
Lester Burnham

Well, there is this paper which explains 'why decoherence has not solved the measurement problem'. (It is pretty much the argument I used here to state the 'interpretation problem'.)

Then there is this talk about the divergence of perturbation series in QFT.

And finally there is this paper about the strong coupling limit of the Wheeler-deWitt equation.

please can you help me?

2010-09-01T23:54:00.000-07:00

Recently, this problem came up in one of my pet projects:
Does anybody know the current state-of-the-art if one needs to distinguish a weak 1st order phase transition from a 2nd order transition with lattice simulations?
If you have an opinion please please let me know and leave a comment.

added later: It might help if I explain a little bit better what I am talking about.

In my pet project I am doing Metropolis simulations on a 4d lattice and the size is limited so that 32^4 is already 'very large'.
Of course, finite size scaling is an important tool, but I would like to know e.g. if it is still state-of-the-art to use the Binder cumulant, or if there are better ways to do this.

Also, one can try to directly identify meta-stable states, but what is the best technique to do so? I know that e.g. simple histograms were sub-standard already ten years ago.
I am also curious if people use partition function zeros in real problems and if something like this has become a standard tool in recent years.

I would appreciate any input e.g. pointers to articles or books that may be relevant. Please do not hesitate to post a comment (which you can do as anonymous).

Skolem's paradox

2010-08-28T03:05:00.000-07:00

So far I never mentioned how I understand the famous Löwenheim-Skolem result (*):
"no matter how fancy your axiomatic system, which seems to talk about real numbers, complex numbers, geometries, fields etc.,
in the end, all it really does is talk about the countable natural numbers, nothing more and nothing less."

In my opinion it is the most shocking result of the Grundlagenstreit.
But does this tell us something about the true nature of physical reality?

(*) and let me be very clear that I am a layman who read exactly one book about number theory (and I understood perhaps half of it).

down to earth

2010-08-07T03:35:00.000-07:00

I assume you will be relieved that this blog post for once is not some crazy speculation about our universe and it does not contain empty pseudo-philosophical thoughts. It does not even count espressos.
Instead, it is about the down-to-earth topic of quantum gravity. Actually it is just a collection of links to pre-prints; In other words I am cleaning out my to-do list.

B.F.L. Ward: ".. by using recently developed exact resummation techniques ... we get quantum field theoretic descriptions for the UV fixed-point behaviors of the dimensionless gravitational and cosmological constants postulated by Weinberg. Connecting our work to the attendant phenomenological asymptotic safety analysis of Planck scale cosmology by Bonanno and Reuter, we predict the value of the cosmological constant ..."

U. Gursoy: "We propose a general correspondence between gravity and spin models, inspired by the well-known IR equivalence between lattice gauge theories and the spin models. This suggests a connection between continuous type Hawking-phase transitions in gravity and the continuous order-disorder transitions in ferromagnets. ..."

N. J. Poplawski: "The Einstein-Cartan-Kibble-Sciama theory of gravity provides a simple scenario in early cosmology which is alternative to standard cosmic inflation and does not require scalar fields. The torsion of spacetime prevents the appearance of the cosmological singularity in the early Universe filled with Dirac particles averaged as a spin fluid. Instead, its expansion starts from a state at which the Universe has a minimum but finite radius. ..."

A. Strominger et al.: "The problem of gravitational fluctuations confined inside a finite cutoff at radius r=r_c outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff r_c the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. ..."

If this would be a better blog, each one would have its own blog post with interesting explanations etc. - some value added.
But by now you should know that with this blog you will have to make up your own mind about all this ...

empty set

2010-07-30T02:01:00.000-07:00

The empty set {} contains no element, nothing whatsoever.
Next we consider the set {{}}, which contains the empty set as its only element.
Then the set { {}, {{}} } which contains two elements and so on and so forth.
We assign the symbols 0, 1, 2, ... to these sets for convenience.

This is of course the standard definition of the natural numbers N as given by von Neumann; Once we have N then Z, Q, R, C etc. follow from N more or less in the usual manner.

I only mention it because some people believe that all physics is really just math.
But if "external physical reality is assumed to be purely mathematical" then all reality is based on the empty set.

7 x 6 = 41 you little sh**

2010-07-28T02:01:00.000-07:00

Nowadays, Scott A. rarely writes blog posts, so I really enjoyed his latest entry and the discussion which followed. (By the way, Moshe refers to his own blog post which is here.)
It seems that Scott is worried about hyper-computation: "Yes, doing an infinite amount of computation in a finite time using exponentially-faster steps certainly does seem like a cheat to me!"
But I think he should be less worried about the discreteness of space-time and more about the question if one can create artificial black holes and baby universes. Of course, what appears as baby universe from one side, looks like a normal universe from the inside and we don't know if one could create a whole universe just to solve a math problem. (As long as we are not sure about quantum gravity, we are not sure about anything.)
Obviously, the Bekenstein bound would not limit the complexity of problems one could solve using baby universes (the volume of a universe is not bounded) and the only question is if/how one could get the answer out of the universe (perhaps using time travel?).

There are of course several indications that our own universe was indeed created as such a computing device:
1) Our universe seems to be fine tuned for the existence of math teachers.
2) We have reached some sophistication in our studies of math.
3) Life in this world seems to lack a deeper meaning and has a certain tendency towards the boring, uninteresting and annoying.
3b) The creator of this universe seems indifferent to the pain and suffering of its inhabitants.
4) One could resolve the Fermi paradox by assuming that the universe is fine tuned for its inhabitants to hang out at MathOverflow but prohibiting unnecessary inter-galactic travelling.

The only remaining question is this. If our universe was really created as some sort of computation device, is it at least part of a grand scientific project, some kind of ultimate mathematical inquiry?
Or did some ET kindergartener 'borrow' the baby-universe-computation-device of his older sister to solve the home work problem 7x6=?

Ceci n'est pas un blog

2010-06-27T04:23:00.000-07:00

Clarity about these matters ...

2010-06-22T10:08:00.000-07:00

Philosophy, Bayesian inference, statistics and all that.

"Philosophy matters to practitioners because they use philosophy to guide their practice; even those who believe themselves quite exempt from any philosophical influences are usually the slaves of some defunct methodologist."

"We fear that a philosophy of Bayesian statistics as subjective, inductive inference can encourage a complacency about picking or averaging over existing models rather than trying to falsify and go further."

In other words, do not look for a Bayesian super-intelligence, but (try to) learn from your mistakes and fail in interesting ways.

update: The authors blog about it here and there.

arXiv or viXra ?

2010-06-13T08:10:00.000-07:00

Recently, there was some fun with arXiv vs. snarXiv, where arXiv is of course the place where serious physicists store their papers and snarXiv is a random paper generator.
But there is also viXra.org, a self proclaimed 'alternative archive of 1072 e-prints in Science and Mathematics'. As an example, take a look at the paper Logic: a Misleading Concept, which kind of sets the tone for the rest.

In the following I present to you four abstracts and the challenge, should you accept it, to distinguish arXiv from viXra. The first who can assign them correctly (in the comments) wins a golden llama award with a free subscription to this blog for a whole year. Here we go ...

1) Many people believe that mysterious phenomenon of consciousness may be connected with quantum features of our world. The present author proposed so-called Extended Everett's Concept (EEC) that allowed to explain consciousness and super-consciousness (intuitive knowledge). Brain, according to EEC, is an interface between consciousness and super-consciousness on the one part and body on the other part. Relations between all these components of the human cognitive system are analyzed in the framework of EEC. It is concluded that technical devices improving usage of super-consciousness (intuition) may exist.

2) Vafa's (11+1) F theory is extended by means of Bars' 2T holographic theory to yield a 14d Multiverse theory that permeates the brane of a 12d Universe in which both the Universe and the Multiverse have (3+1) spacetimes. Given the 2d toroidal compactification of F theory, we conjecture that the Multiverse has a 4d Cartesian compactification that is filled with 3D+T spacetime via the standard 6d elliptic Calabi-Yau compactification, as in both M and F theory. The result is exemplified using supermassive black hole cosmology.

3) The study of the so called brane world models has introduced completely new ways of looking up on standard problems in many areas of theoretical physics. Inspired in the recent developments of string theory, the Brane World picture involves the introduction of new extra dimensions beyond the four we see, which could either be compact or even open (infinite). The sole existence of those new dimensions may have non trivial observable effects in short distance gravity experiments, as well as in our understanding of the cosmology of the early Universe, among many other issues.

4) We propose a derivation of the empirical Weinberg relation for the mass of an elementary particle in an inflationary type of universe. Our derivation produces the standard well known Weinberg relation for the mass of an elementary particle, along with an extra term which depends on the inflationary potential, as well as Hubble's constant. The derivation is based on Zeldovich's result for the cosmological constant Λ, in the context of quantum field theory. The extra term can be understood as a small correction to the mass of the elementary particle due to inflation.

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The proud winner is cherez who can now read this fine blog for another year for free while looking at the golden llama, knowing that he can tell arXiv from viXra. Congratulations!

Links to the four papers are in the comments.

one pipe, many worlds

2010-06-12T03:22:00.000-07:00

Some people claim that they have seen the image of a pipe on this blog. Obviously, there are many different ways how such an image could have been observed in the many worlds we live in, but in this world there is only one way how it could not have appeared.
So what does this tell us about probabilities?

PS: If you want to think even more about probabilities and many worlds, I recommend this paper and in particular the sections about some toy many-worlds models and about 'the problem of inappropriate self-importance'.

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added later: In my previous example a quantum experiment has two possible macroscopic outcomes, e (espresso) and n (no espresso), and the probability for each is 50%, although e can be realized in N different (macroscopic) ways, while n can be realized in only one way.
Now, one could argue that in both cases the observer does not immediately experience a conflict with the 50% Born probability (the observer in one world e or n does not experience all the other worlds). Therefore we need to consider iterating this experiment.
If the experiment is repeated R times, then observers who see the outcome eeeeeeeee...eeeeee will indeed conclude that something is very wrong. Unfortunately, they are the overwhelming majority with N^R worlds and I would really like to understand how this can be reconciled with e.g. the Deutsch-Wallace interpretation of probability.

one espresso, many worlds

2010-05-30T08:39:00.000-07:00

I walk up to a fully automated coffee dispenser and push the button to get an espresso. What I do not know is that the button is connected to a device which consists of a radioactive source and a detector. The button activates the detector for 5 seconds and if it registers a particle from the radioactive material it will make an espresso and otherwise not (*). The probability to get my espresso, according to text book quantum theory, is exactly 50%.

But this raises the following question. There are many ways I can get the espresso, but there is only one way that it can fail. See, I push the button at 1pm, which here means I push it exactly at 1:00:00. Now the detector could register a particle at 1:00:01 or at 1:00:02 or ... at any time between 1:00:00 and 1:00:05. There is an infinite number of possibilities when and how the detector could register a particle.
But there is only one way it can not register a particle, with the detector shutting down at 1:00:05.
Since we all believe in the many worlds interpretation, this implies that there are many worlds where an espresso has been prepared and only one world where I fail to get it. So how can the probability be 50%?
Unless we assume that somehow the many different worlds are not equally real (x).

So perhaps the many worlds are not really real after all ?

(*) Actually, many automated dispensers in the real world follow a similar design.

(x) added later: There are proposals on how to save the appearances. Let me know if you find them convincing.

weird comments

2010-05-16T05:40:00.000-07:00

Something is weird with the comments to this blog. Now I receive spam (e.g. to the previous entry) which I cannot delete (*). And the comment counters are sometimes messed up too.
So I decided to disable comments (again) and recommend that you do not click on links in comments which consist mostly of strange characters.

(*) Why send spam to a blog which is mostly inactive ?!

update: Meanwhile, it seems that the delete finally worked and I am turning comments back on.

triangles

2010-04-15T23:35:00.000-07:00

In a recent preprint Renate Loll et al. present numerical evidence that causal dynamical triangulation is eventually a discretization of Horava-Lifshitz gravity.
But is this really good news?
In an earlier paper by Christos Charmousis et al. an argument was given that "the original Horava model, and its 'phenomenologically viable' extensions do not have a perturbative General Relativity limit at any scale". Lubos wrote more about that at the time and there is also this paper with a more general argument.
I am neither an expert on CDT nor Horava-Lifshitz gravity and would welcome comments about this. (Of course, I always welcome comments!)

friends

2010-03-24T01:05:00.000-07:00

One reason I do not have a facebook account is that I would probably not have enough friends; Most likely my friends would on average have more friends than I do.
This is just another result of this 'mutant form of math' and is explained e.g. here and here (pdf).

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Adrian Kent thinks that many worlds interpretations are scientifically inadequate.
He is against many worlds.

In this world.
I wonder what he thinks in the other worlds...

And I wonder if I have on average more friends in the other worlds than in this.
By the way, can a whole world have 'friends'? (*).

(*) Friends have to have something in common. So we define that two different worlds or branches of the m.w.i. are 'friends' if they both contain at least one observer with (almost) the same conscious experience. At any point in time we are then dealing with a 'social network' of worlds and the above friendship paradox applies: Most likely the other worlds will have more 'friends', be more popular, than our world.
If I believe in the many minds interpretation and assume that conscious experience constitutes reality, does this mean our world is most likely less real than others?

mean mean

2010-03-20T06:08:00.000-07:00

warning: this blog and in particular this blog post is about a mutant form of math.

Recently, I browsed through Cosma's notebook about large deviations reading that
"The limit theorems of probability theory [..] basically say that averages taken over large samples converge on expectation values."
and immediately my inner contrarian tried to come up with a simple stochastic process where the sample mean does not converge.
Of course, this is not very difficult, since there are many examples of processes where the sample mean is unbounded and does not converge to anything.

It would be much more interesting to find a process where the sample mean is bounded, but 'bouncing around' unpredictably and therefore not converging. In other words, a process where it seems that a 'mean of sample means' exists and yet it does not.

My initial idea was to use the sample mean S of a random variable y itself as a variable in the stochastic process and consider the following:
y(t+1) = -sgn[S(t)]/( |S(t)| + e ) + noise
where e (epsilon) is a small number, S is the sample mean S(t) = ( y(1) + y(2) + ... y(t) )/t and the noise term is a (uniformly distributed) random variable between -1 and +1.
The sgn[S] function is defined to be -1 for negative values of S but +1 otherwise, so that sgn(0) = +1.
Notice that we can write S(t) as ((t-1)/t)*S(t-1) + Y(t)/t , formally making this a Markov process for (y,S).

If the current sample mean S(t) is a small negative (positive) number, the process will generate y with large positive (negative) values, but if the current sample mean is large then the process will generate small y distributed around zero, forcing the sample mean to lower values. This should make for a nasty little process with a really mean mean.

Unfortunately, the sample mean of this process still converges (#). And it converges towards zero, as depicted in the following picture, produced by a numerical simulation with e = 10^-6 (and S(0) = 1 instead of 0).

It is not too difficult to understand that the convergence rate is proportional to the value of e and it is not difficult for a statistical mechanic to fix this problem with a little tinkering. Using (e/t) instead of e does the trick.

As the numerical simulation depicted in this picture suggests, the sample mean does not converge, but seems to remain bounded (x), bouncing unpredictably between positive and negative values (*).

The obvious question is what the expectation value E[y] is and in which sense S converges towards it.
Unfortunately, I have to leave it as an exercise for the reader to find an answer and actually proof the (non)convergence of this process.

(#) does it really?

(x) or diverging very slowly?

(*) By the way, the values of y(t) are finite for all finite t, but obviously are now unbounded; But this is also true for Gaussian noise and should not really bother us.

added later: I convinced myself of the following:
(#) Yes, for the 1st process (with fixed epsilon e) the sample mean S does indeed converge on E[y] = 0.
(x) No, for the 2nd process (with decreasing e/t) the sample mean S(t) is still bounded (by the inverse of e). Due to the symmetry of the system [the fact that I assume sgn(0) = +1 and not 0 is irrelevant by the way] this indicates that the mean of the S(t) values, i.e. 'the mean of the means', converges on zero, which would be compatible with E[y] = 0.
However, it is also the case that S(t) *cannot* converge on zero.

you have to click

2010-03-07T23:47:00.000-08:00

If you read this, it is because you clicked. And I know you need some more links, because you have to click. You cannot stop. Here they are, pure, high quality stuff. Take them, click them.

The Aeolist reviews Sean Carroll's book and concludes that it is "too sloppy to be of much help".
John Baez writes about Algorithmic Thermodynamics and describes a "design for a heat engine powered by programs".
Boris Kosyakov examines the classical physics of a particle at the top of a hill and concludes that it is indeterministic, re-discovering what John D. Norton found out about The Dome.
Last, but not least, an easy-to-read explanation (pdf) of Xia's proof of the existence of non-collision singularities in Newtonian gravity.

interpretations, part 4/4

2010-02-25T00:01:00.000-08:00

I recommend that you read the first part of this series first.

"If the facts don't fit the theory, change the facts." Albert Einstein

If you assume that nothing is wrong with the wave function W and you dislike the interpretations of the previous part 3, then you have to conclude that something is wrong with (your perception of) the reality R.
So, one way out of the interpretation problem, as posed in the first part of this series, is to simply assume that actually both detectors clicked; You are just somehow confused about it.

The many-worlds interpretation and its variants (consistent histories, many minds, etc.) are increasingly popular and solve the interpretation problem by pointing out that the wave function W = |1> + |2> continues to evolve into |1>|D1> + |2>|D2> and finally |1>|D1>|Y1> + |2>|D2>|Y2> , where Y1 indicates you, puzzled why click 1 has been observed but not click 2.

But, as I have argued previously, the full meaning of a many-worlds interpretation can only be appreciated if one goes 'backwards in time', trying to find the origin of W (which cannot evolve from a 'collapsed' wave function). In the words of Matthew J. Donald:

"Each time we pass back (through the appearance of a collapse) we get a better approximation to W. Eventually, we arrive back at the big bang. ...

The quantum state of the universe coming out of the big bang looks - at least in its non-gravitational aspects - very like a thermal equilibrium state. In the Hamiltonian time propagation of that state, the stars and planets which we see now do not exist as definite objects, and certainly neither does any particular measuring device now being used by us on one of those planets. W seems to be a complete mess. However, it does have a great deal of hidden structure, and it is the job of a no collapse interpretation to explain how that hidden structure comes to be seen." (*)

You may think that we have only replaced the original interpretation problem with a much more complicated one. But the beauty of it is that this allows us to continue to think about the meaning of life, the universe and everything... (x)

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(*) While I agree with the overall conclusion, I disagree with the details.
i) Nowadays the multiverse landscape seems very popular (especially among string theorists) and it is not clear at all that we arrive back at one big bang.
ii) In general, I doubt that one could really reconstruct W as suggested or otherwise determine the wave function of the universe, unless there is a new general principle which limits the possibilities (notice that we only know about one branch out of an infinity of possible branches.)
iii) If one assumes that W may contain an infinite number of branches and since those branches (assuming decoherence) are only almost-orthogonal to each other this poses a real problem, If one wants to use W to calculate anything.

(x) As far as I know, Frank Tipler was the first who realized that the many-worlds interpretation opens the door to theological interpretations of quantum theory. If one wants to follow this path (and don't we all want to believe?), then I would use a previous result to conclude that the wave function of the universe is not only invisible but necessarily pink.

interpretations, part 3/4

2010-02-23T00:05:00.000-08:00

I recommend that you read the first part of this series first.

The ensemble interpretation goes back to Albert Einstein, who assumed that W does not describe an individual system, but instead an ensemble of equivalent systems or experiments. (Notice that Chad Orzel emphasizes that we need to use many photons to see an interference effect in the Mach-Zehnder interferometer.)
This resolves the mismatch between W and R, without assuming that anything is wrong with either W or R (*).
It seems to me that this minimalist interpretation is actually quite popular with many physicists and especially experimentalists.
Further, I think it is the philosophical basis of the "shut up and calculate" approach and suspect that some physicists use it who are otherwise convinced that Einstein never understood quantum theory.

In stark contrast, the Copenhagen interpretation assumes that W very well describes individual quantum systems, but emphasizes that we necessarily have to use classical concepts to describe the outcome of experiments. Therefore, one needs to change the 'description' during a measurement and the wave function 'collapses' at some point; Nowadays one could refer to decoherence to better determine that point.

In this interview Werner Heisenberg emphasized that W does not describe (fundamental) reality itself...

"That is just the point; I do not know what the words fundamental reality mean. They are taken from our daily life situation where they have a good meaning, but when we use such terms we are usually extrapolating from our daily lives into an area very remote from it, where we cannot expect the words to have a meaning. This is perhaps one of the fundamental difficulties of philosophy: that our thinking hangs in the language."

... instead he assumed that W does describe what he called 'potentiality'.

'What does a wave function actually describe?' In old physics, the mathematical scheme described a system as it was, there in space and time. One could call this an objective description of the system. But in quantum theory the wave function cannot be called a description of an objective system, but rather a description of observational situations.

The explanations of Niels Bohr were usually even more profound, with complementarity playing a prominent role in his philosophy; But it seems that they were indeed so profound that nobody actually read them (x).

Let me finally mention some results which emphasize that the problem to understand the measurement process results from the impossibility of self-measurement. The observer of an experiment can therefore not use W to describe herself and her own experience. In my opinion one could either use these results as further argument in support of the 'collapse' of the Copenhagen interpretation or as the starting point for a new 'relational interpretation'.

-----

(*) Notice that in many cases a Wick rotation translates between quantum and statistical mechanics.

(x) "In a widely used compendium of papers on quantum theory [...], the pages of Bohr's reprinted article are out of order. This paper (Bohr's response to the famous 1935 Einstein-Podolsky-Rosen critique of the standard Copenhagen interpretation) is widely cited in contemporary literature by physicists and philosophers of science. Yet I have never heard anybody complain that something is wrong with Bohr's text in this volume."
Mara Beller about the philosophical pronouncements of Bohr, Born, Heisenberg and Pauli.

continue to part 4.

la statistique bayésienne

2010-02-22T08:42:00.000-08:00

Andrew Gelman and Cosma Shalizi wrote something about the real philosophical foundations of Bayesian data analysis. Very interesting.
Unfortunately, it is all French to me...

interpretations, part 2/4

2010-02-20T23:16:00.000-08:00

I recommend that you read the first part of this series first.

The idea that the wave function W is only an incomplete description of the reality R is as old as quantum theory itself. Already at the Solvay conference of 1927 Louis deBroglie suggested to add 'hidden variables' with W being only a 'pilot wave'.
During the 1930s Albert Einstein published several thought experiments (the most famous being the EPR paper) to demonstrate that W was obviously incomplete.

Consider the following 1-dimensional thought experiment.
A particle enters a detector of (great) length L at time tI and we determine its momentum p = mv with high precision, knowing that this will lead to large uncertainty dx in its position x. At a later time tF we turn the detector on, which will now determine the position x of the particle with high precision and leave dp large. But although we assume that Heisenberg's uncertainty relation holds for each measurement, we can now reconstruct the path of the particle between tI and tF, knowing v(tI) and x(tF) with high precision and assuming conservation of momentum (just as we can reconstruct the path which the photon must have taken in the Mach-Zehnder interferometer, once we know which detector clicked).
But this reconstructed path R(t) for tF > t > tI is not described at all by the wave function W(t); This suggests that W provides for an incomplete description of reality only.

A consistent theory of hidden variables for non-relativistic quantum theory was later formulated by David Bohm and attempts have been made to generalize it to relativistic field theories (1, 2).
By the way, notice that the 'hidden variables' are actually the particle and pointer positions we observe in a measurement (while we never experience the wave function directly).

---

An even more elegant way to introduce hidden variables is to refer to (human) consciousness as selection principle. This works especially well, because on one hand one cannot deny the reality of our conscious experience, on the other hand it never directly shows up in physics.
John v. Neumann was the first to mention it, Wigner and his friend made the idea popular and Henry Stapp worked out some concrete proposals. Actual experiments, using EEGs to test the influence of consciousness on quantum experiments, have been proposed, as described in this paper.

Of course, in a section about quantum theory and consciousness I have to mention Roger Penrose. He does not only think that W is incomplete, but assumes that current quantum theory is actually wrong, because it ignores the effects of (quantum) gravitation. He also proposed an actual experiment to test his idea.

There are many other proposals to modify quantum theory and I only mention the Ghirardi-Rimini-Weber theory, which assumes real sporadic collapse of the wave function.

continue to part 3.

interpretations, part 1/4

2010-02-13T06:21:00.000-08:00

It is time to write about something truly exciting, in other words it is time to write about the interpretation problem of quantum theory. This is the 1st of 4 posts about it, the introduction if you will.
Some time ago, Chad Orzel wrote a blog post explaining decoherence and it shall be the starting point for us (he later wrote a more detailed explanation as response to a comment).
Consider a single photon in a Mach-Zehnder interferometer, which will end up at one of the two detecors D1 and D2.

If the interferometer is properly set up, the wave function W of the photon will be something like |1> + |2> (normalization coefficients are absorbed in the state vectors). Notice that W is symmetric in the two possible outcomes 1 and 2.
However, we know that when we do the experiment, only one of the detectors will click, either D1 or D2.
And so we have already all the ingredients together for the interpretation problem.

W: The wave function |1> + |2> is symmetric and prefers neither 1 or 2.

R: In reality, only one detector will click, either D1 or D2, and obviously one is preferred over the other.

As Chad explains, decoherence will eliminate to some extent the quantum interference between |1> and |2> and in this sense 'classical behavior' emerges; But this does not really solve our problem.
Even if the product between |1> and |2> or |D1> and |D2> is nearly zero, this does not elimiate the fact that W is symmetric and R is not. (Notice also, that decoherence will in general bring the product between |D1> and |D2> close to zero but not exactly zero and there is no sharp cut-off between quantum interference and classical behavior. But this is really not that important to our problem.)

By the way, please notice that the interpretation problem is a real problem (W does not match R); We are not talking about an 'interpretation problem' in the sense an art critic or a philosopher might use that phrase.

It is evident that every attempt to solve the interpretation problem must fall into one of three categories.

i) The problem is with W. The wave function is not a complete description of R. We will encounter this approach in part 2 of this series as hidden variables, etc.

ii) While W is complete, the problem is about what we mean with 'W describes R'. The Copenhagen interpretation, the ensemble interpretation and others belong here. I will discuss them in part 3.

iii) The most radical proposal is to assume that W is fine and our problem is with R; Reality is just not what we think it is. The many-worlds interpretation is the most important example and I will discuss it in the final part 4 of this series.

One more remark about the upcoming parts; While similar reviews often point to a favored interpretation and describe all others in a negative light, I will try something new and describe all interpretations as convincing and favorable as I can. I hope that this will increase the entertainment value.

continue to part 2.

entropic gravity

2010-01-23T07:31:00.000-08:00

Recently, Erik Verlinde proposed that gravity can be described as entropic force. I am not sure yet what to think about this, but Lubos explains why he is certain this can never work.
Meanwhile, Lee Smolin published a preprint using Verlinde's idea to derive Newton's law from loop quantum gravity. I am not convinced by his argument.
Verlinde considers the change in entropy dS for displacements dx assuming a holographic principle and in his calculation he implicitly assumes the geometry of a smooth and indeed flat geometry.
There is of course nothing wrong about that, but if Lee Smolin wants to use this argument, then he has to first show that there is a reasonable limit of loop quantum gravity which reproduces this smooth and (almost) flat spacetime and I don't see that.

added later:

More discussion Lubos vs. Erik (scroll down through the comments).

Lee Smolin responds to my comment here.

Robert Helling comments on entropic everything.

previously, somewhere else

2010-01-17T04:46:00.000-08:00

Just some links to that other blog, which you may or (more likely) may not find interesting..
.. about many worlds (again),
.. lattice gravity models,
.. the impossibility of self-measurements,
.. and a lottery ticket of zero value but with probability to win greater than zero.

here and now

2009-11-24T04:21:00.000-08:00

"How is it that there are the two times, past and future, when even the past is now no longer and the future is now not yet?"
Augustinus, Confessions, Book 11, chap. 14

Meanwhile, we know that in addition to events which happened in our past and events which will happen in our future, there are also events which are space-like (neither in the past nor in the future); And we know that the present is really a single point in space-time.
Of course, there is nothing mysterious about relativity, the math is easy enough that nowadays they teach it in high school, Minkowski diagrams and the Lorentz group as homework exercise.

But still - I wonder about the present and how the activities of my brain fit into a single space-time point when I experience the "here and now" ...

Perhaps I should think more about world lines.

about entropy

2009-11-20T04:13:00.000-08:00

"I thought of calling it 'information', but the word was overly used, so I decided to call it 'uncertainty'. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, 'You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.'"
Claude Shannon

In order to contemplate the definition of entropy, and perhaps gain an advantage in debates about it, I recommend this article about classical thermodynamics. In section 3 'a one-dimensional classical analogue of Clausius' thermodynamic entropy' is constructed, which dates back to Helmholtz. (The properties of one-dimensional classical gas were previously discussed on this blog here.)

And then there is a pdf file available to this text book about 'Entropy, Order Parameters and Complexity'.

I found both the article and the book on Cosma's list of interesting links.

world lines

2009-11-01T02:26:00.000-08:00

I write this blog post about something I have not really thought and don't know much about.
The Aeolist discusses the Dowe/Salmon causal processes account of causation and raises important objections in my opinion. (see also here)
But I think there is a more fundamental objection: In order to define world-lines one needs a concept of space and time, but how could one determine the properties of space and time and measure distances between world-lines?
One would have to introduce a metric, in other words a field, and quickly end up with (something like) general relativity and its complications. As the Aeolist already noticed, the introduction of fields "brings along a whole other suite of problems like how to individuate objects and processes on that conception".

Does it really make sense that philosophers retrace the history of physics (and the inconsistencies of its various concepts) from Newtonian particles to quantum field theory just to define terms like 'process' and 'causation'?

deep or trivial

2009-10-13T01:10:00.000-07:00

I would summarize the previous blog post as follows:
In general, a closed system, which contains e.g. a physicist and/or computer, cannot predict its own future, even if we assume deterministic laws of physics (*).
This is quite easy to understand once you think about it; But is this some deep insight or just trivial stuff?

An equivalent statement would be that, in general, a closed system, which contains e.g. a physicist and/or computer, cannot determine or know its own microstate.
Although equally trivial, it has a bit more 'statistical mechanics flavor' to it and might be interesting if one considers foundational questions about entropy or even more profound quantities.
It also has a certain Zen-like quality...

(*) See also this and that.

Arthur and free will

2009-10-12T03:37:00.000-07:00

In recent days and weeks I read an interesting but rather heavy book about Arthur Schopenhauer, his philosophy and his times. And I think this was the main reason I did not find the will in me to write something on this blog. I still don't have too much to offer, except the following silly story...

-----

This story is about a very smart physicist and her simpleton friend. Among other things the smart physicist entertained herself by predicting his behavior. This was possible because they lived in a Dennett-Newtonian universe, where all thoughts and all behavior was a function of the configuration of molecules constituting a brain and the movement of these molecules was deterministic, following Newtonian laws. The bouncing of molecules was not so difficult to predict with all forces well known.

The physicist had several machines in her laboratory to determine the configuration of huge numbers of molecules to arbitrary precision and a supercomputer (also made of Newtonian particles of course) to calculate the future configuration of molecules ahead of time. Since her friend was made of a large but finite number of molecules, all she had to do was e.g. to use her machines to measure the configuration of molecules at 9am (he did not even notice it), feed the result into her supercomputer and read out the calculation which predicted his behavior at 10am. And when she determined that he would say "I am bored, let's go for a walk" this was exactly what happened, like clockwork. Easy as pie and quite funny.

Unfortunately, she made a mistake. She wanted to show him how smart she was and wrote down her prediction so he could see it and for some reason all of a sudden it failed to work.
Of course, on one hand it was immediately clear what happened. As soon as he read that at 10am he would "go to the window and open it" he decided to open it earlier and at 10am already closed it. There was nothing mysterious about it, actually it was a completely deterministic process, with Newtonian photons carrying the prediction to his Dennett-Newtonian brain, which was not very complex, but smart enough to do simply the opposite of what she wrote on the paper. He did it just to prove a point. And of course it was quite irritating.

On the other hand, she did not understand this at all. Her machines could measure the configuration of all molecules in the room (including herself) and the supercomputer calculated this forward to arbitrary precision. So the calculation 'knew' that a prediction was written on a piece of paper and the Newtonian photons carrying the message and his simpleton brain receiving it and doing the opposite of what was written etc.

So how could this prediction go wrong? Everything was deterministic! And still, no matter how many times she tried, her simpleton friend with his simpleton stubbornness did the opposite of what she wrote down. Every time. Was Newton wrong after all? Or Dennett?

She found a solution, of course, it was easy enough. Get another friend. But still...

-----

If you have a good explanation of what caused this 'failure' of determinism then please post a comment. The first to solve this silly puzzle will win a Golden Llama award for major contributions to the blogosphere of physics, which includes a free subscription to this blog.
Of course, if you just want to debate the whole thing feel free to post a comment too, or if you want to let me know just how silly this silly story really is.

added: And the winner of the Golden Llama Award is Chris, who pointed out that the problem is with the supercomputer (trying to) predict its prediction. (See the comments for more details).
Congratulations!

heretic stuff

2009-09-20T06:22:00.000-07:00

Some evildoer at ScienceBlogs asks "Is String Theory an Unphysical Pile of Garbage?" and references this paper (in particular p.54 to 57) and other heretic stuff.
"... in string theory, we don't know both the variables and the equations. In fact, unless another theory (...) comes along that encompasses and expands upon string theory, string theory isn't a fundamental theory at all, due to instabilities."

I think this is yet another job for SuperLumo, defender of the one true string theory!

While we wait for SuperLumo, let me add a few remarks about this (but keep in mind that I am not a string theorist).
The blog post and the paper it references basically complain about the lack of a fundamental, non-perturbative formulation of string theory. But it seems to me that the Maldacena conjecture provides for such a non-perturbative description, currently at least for AdS.
A while ago string field theory has been proposed as a more fundamental theory and the referenced paper suggests that it suffers from instabilities. I cannot judge the argument in detail, but it seems to me that string field theory is indeed no longer such a hot topic. Most of the effort seems today focused on generalizing and understanding the AdS/CFT correspondence.
However, I also think it is a mistake to underestimate 'string theory as perturbation theory'. After all it provides for the only known way to deal with quantum gravity but at the same time keep local Lorentz invariance (i.e. a smooth spacetime) and quantum theory as we know it. And it gets surprisingly far, including consistent calculations of black hole entropy, making use of the amazing string dualities.
Many questions remain unanswered and perhaps early hopes that string theory will answer all questions of particle physics turned into consternation about the multiverse, but I think the answer to the "incendiary title" of the blog post obviously has to be "No!".

no quantum resolution ...

2009-09-14T05:03:00.000-07:00

This paper is a really interesting comment to the one proposing a "Quantum resolution to the arrow of time dilemma", discussed here previously.
"In this note we show that the argument is incomplete and furthermore, by providing a counter-example, argue that it is incorrect. Instead of quantum mechanics providing a resolution in the manner suggested, it allows enhanced classical memory records of entropy-decreasing events."

Bayes vs Kelly

2009-09-08T05:09:00.000-07:00

A Bayesian statistician understands probabilities as numerical description of 'subjective uncertainty'. In order to make this a little bit more concrete, Bruno de Finetti considered betting odds and order books (see e.g. this paper [pdf!] for a more detailed discussion).
Meanwhile this approach is usually introduced and shortened as follows (*):
"we say one has assigned a probability p(A) to an event A if, before knowing the value of A, one is willing to either buy or sell a lottery ticket of the form 'Worth $1 if A' for an amount $p(A).
The personalist Bayesian position adds only that this is the full meaning of probability; it is nothing more and nothing less than this definition."

But there is a problem with that.

As every professional gambler knows, if one places a series of bets one needs to follow the Kelly criterion to avoid certain ruin in the long run. But the Kelly fraction and thus the amount one should be willing to bet is strictly zero for the case of a fair bet (when one would be willing to take both the buy and sell side).
In other words, it is wrong to pay the price suggested in the above definition, if one wants to survive in the long run.

Now, it would seem that this is one of those 'technicalities' that one can easily sweep under a rug or into a footnote. But I think it would be difficult to somehow incorporate the Kelly criterion in the above definition of probability, because in order to derive the Kelly fraction one needs to know already quite a bit about probabilities in the first place.
It may make more sense to emphasize that, of course, an order book is really a process and Bayesian probabilities are found in the limit when such order books approach the fair price and bet sizes tend towards zero. But unfortunately, in general we don't know much about the convergence of this process and real world examples of order books do not exhibit such a limit [x].

There is one additional problem, because the Kelly criterion is actually the limiting case of a more general rule. Indeed, if the estimated probabilities contain just a small amount of noise, bankruptcy still looms even if the Kelly criterion is used. Therefore professional gamblers know that one must bet 'less than Kelly'; In general a rational agent will bet 'less than Kelly' due to risk aversion and this means that the direct relationship between bet sizes and probability, as proposed in the above definition, is lost completely [x].

Therefore, I suggest that Bayesians simply refrain from using order books etc. (free markets are becoming quite unpopular anyways) to define probability and simply state that 'subjective uncertainty' is a self evident concept. After all, we know that self evidence is a very reliable kind of rug.

(*) I do not want to pick on this particular paper, since similar 'definitions' are nowadays used in many cases. But since the paper is about the foundations of quantum theory I would like to point out that I have it on good authority that God does not play dice.

[x] If risk aversion and probability estimates of the participating agents are correlated, this would be already one reason why the order book would not converge towards the 'fair price'.

managed world views

2009-09-06T04:01:00.000-07:00

We already have managed accounts and managed healthcare, now we also have managed world views, thanks to Scott. A very interesting application and I recommend that you try it and check your views on quantum mechanics, strong AI and other topics.
Of course, the devil is in the details and several commenters have already detected ambiguities in the statements proposed by the world view manager and the detected tensions are often already accompanied by comments disputing their validity.
But where Wittgenstein ran into a brick wall, Scott may yet succeed. His manager could indeed be the first application of Web 3.0 a.k.a. 'the semantic interwebs'.
In the future, bloggers and other opinionators will check the consistency of their views and rationally resolve tensions and perhaps we will have, in addition to virus and spam filters, consistency scans of web pages and blog posts.

e.g. "wvm just detected an unresolved tension between this and that blog post and recommends to shut down this blog until the tension is resolved."

I am sure that Web 3.0 will be a better place, with about 95% of the current blogs removed from the interwebs due to inconsistencies.

... and peace reigns once more

2009-09-02T08:26:00.000-07:00

"Nothing could be simpler. It is merely a small wooden casket, the size and shape of a cigar box, with a single switch on one face. When you throw the switch, there is an angry, purposeful buzzing. The lid slowly rises, and from beneath it emerges a hand. The hand reaches down, turns the switch off and retreats into the box. With the finality of a closing coffin, the lid snaps shut, the buzzing ceases and peace reigns once more."
The 'ultimate machine' of Claude Shannon.

memories

2009-08-24T00:31:00.000-07:00

Cosma links to this article about a 'quantum solution to the arrow-of-time dilemma' and suggests to read it carefully. I think I read it already a year ago as this preprint and it has several interesting ideas; In particular I liked the passages about Borel's argument.

But I was also thinking about the hidden assumption of this paper.
As far as I understand it, the author (implicitly) assumes that memories are about the past and in a paper about the 'arrow-of-time dilemma' this should be explained not assumed.
In other words, the author suggests that we could live in a time-symmetric world and we just do not have memories about "phenomena where the entropy decreases"; This is all fine, but in my opinion one needs to explain then why we only have memories about the past but not the future, without "an ad hoc assumption about low entropy initial states".
E.g. in the conclusions we read that "we could define the past as that of which we have memories of, and the future as that of which we do not have any memories" (*), but the author does not ask or answer the question how there can even be such a thing as the future, of which we have no memories, in a time-symmetric world.

update: Sean wrote about the paper also and in the comments the author responds.
I am glad that at least one comment agrees with my own reading; And it seems that Nick Huggett has some interesting papers I should read.

update2: The author responds to Huw Price (who raised the same objection I did).

update3: Last, but not least, Dieter Zeh comments on the paper as well. I wrote about his book about 'the direction of time' previously.

PS: The question "if the world would run 'backwards', would we even notice it?" was raised and discussed previously on this blog.

(*) As it stands this statement is of course contrary to the usual convention(s). From what we know about physics, most of those events of which we have no memory are space-like to us and not in our future. And of course there are even events in our past of which we have no memories. E.g. we know for sure that Philip Augustus of France spent a night with Ingeborg of Denmark, but we have no documents or memories about the mysterious events of that night.

Fermi event

2009-08-14T10:51:00.000-07:00

The Fermi experiment detected a 31 GeV photon emitted by a short gamma-ray burst and "this photon sets limits on a possible linear energy dependence of the propagation speed of photons (Lorentz-invariance violation) requiring for the first time a quantum-gravity mass scale significantly above the Planck mass".
In other words, the observed event suggests that Lorentz invariance holds up to (and in fact above) the Planck scale and thus provides an empirical argument to rule out several proposals for quantum gravity. As far as I know, this is the first time direct empirical evidence about the Planck scale was obtained!
Via Lubos.

So what does this tell us about string theory?

quantum Hamlet

2009-08-12T01:36:00.000-07:00

The arXiv blog reports about a new proposed effect called Quantum Hamlet Effect. According to the author "It represents a complete destruction of the quantum predictions on the decay probability of an unstable quantum system by frequent measurement".
But I think there is a problem with it. The Hamlet state is prepared by a series of subsequent measurements, happening at decreasing time intervals tau/sqrt(n), with n going to infinity. This leads to a divergent sum in the probability, which then leads to the "complete destruction of predictability".
But, of course, in reality the limit n to infinity cannot be taken (e.g. as the author notices himself he neglects time - energy uncertainty!), so we have to assume the procedure stops at n = N, with some finite but perhaps large N. Unfortunately, the divergent Hamlet term is the harmonic series, which increases only with log(N), i.e. it increases much slower than sqrt(N).
Therefore I doubt that the Hamlet effect will "turn out to be more useful and famous than [the Zeno effect]" as suggested on the arXiv blog.

truth

2009-08-10T09:27:00.000-07:00

The statement "p is an unknown truth" cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of all truths must not include any of the form "p is an unknown truth"; Thus there must be no unknown truths, and thus all truths must be known.
At least according to Frederic Fitch.

Later, after a heated debate about the question if all true logical statements are indeed tautological, one of the philosophers finally screamed "enough is enough" and stormed out of the room.

summer and hot and beach

2009-07-15T02:51:00.000-07:00

It is summer, it is hot and you should be at the beach.
All I have are some links to stuff you have probably seen already:
A paper about improved mean field approximations.
It turns out that 1-d classical gas is known as Jepsen gas in the literature [1, 2].
A link between birds on a wire and random matrix theory.
A paper about non-renormalizability of quantum gravity (via Lubos).
A talk by Steve Carlip about quantum gravity at short distances [pdf].
Last but not least, an important contribution to an old problem of philosophy.

Really, you should be at the beach...

1-dimensional gas

2009-07-04T05:22:00.000-07:00

In this episode we study the amazing properties of 1-dimensional classical gas.

There are N 'molecules', each with the same mass m, in a narrow cylinder of length L, moving back and forth, colliding with each other and we assume that the collisions are perfectly elastic. The cylinder is closed on both ends, but at the right end the rightmost molecule can escape if its energy exceeds the threshold Eo, providing a 'window' for a physicist to observe what is happening inside.
How many 'molecules' do we expect to leak out from the 'window' and what can we learn from observing them (like peeking into a 1-dimensional oven)?

It turns out that the statistical mechanics of this system is remarkably simple, once we consider what happens when two molecules collide.

Assume that 'molecule' A moves with velocity v1 and collides with B moving with velocity v2. Using momentum and energy conservation (and the fact that both 'molecules' have exactly the same mass) we find immediately that after the collision A moves with velocity v2 and B moves with velocity v1.

But this is equivalent to assuming that we are actually dealing with two (quasi)particles 1 and 2, which move with constant velocities v1 and v2 and do not interact with each other.

I guess a string theorist might call this a duality; We have two equivalent pictures of the same situation. In one picture we are dealing with interacting molecules, bouncing off each other, and in the other picture we are dealing with non-interacting molecules moving with constant velocities. As long as we cannot distinguish the molecules (they have the same mass) both pictures describe the same situation.

Of course, statistical mechanics is quite simple in the non-interaction picture and we can immediately answer the question from above: If initially there are n molecules with kinetic energy E > Eo, we will observe n molecules leaking out and the time this takes will be less than 2*L/v, with v = sqrt(2*Eo/m). We only need to consider a freely moving (quasi)particle with E = Eo (+ epsilon), traveling from the right end to the left, getting reflected and moving back to the right end where it leaves the cylinder; All other (quasi)particles with E > Eo will leave even earlier.

And what do we learn about the gas (remaining) in the cylinder from observing the escaping molecules? The answer is exactly nothing, because of the independence of the (quasi)particles. We won't even know the temperature of the remaining gas, but we would know for sure that the kinetic energy of each remaining molecule is less than Eo.
Notice that the molecules of the 1-dimensional gas will in general not follow a Boltzmann distribution, instead the probability distribution for energy and momentum remains whatever it was initially - the 1-d system does not 'thermalize' as one would expect from experience with real 3-d gas.
It is left as an exercise for the interested reader to determine if (or in which sense) the 0th and 2nd law still hold.

I assume somebody must have studied the properties of 1-dimensional classical gas already, but I am just not aware of it. Please let me know if you have a reference.

decreasing entropy

2009-06-28T09:50:00.000-07:00

I was thinking about the director of this previous post and how she could use a mechanical device to generate the fire alarm.

A long and narrow cylinder has a slowly moving particle A on one side and a particle B at rest on the other (*). She sets everything up on Sunday and simply waits until she hears the 'click' of the two particles colliding, which then triggers the fire alarm.
Here is my problem: It would seem that the entropy of this closed system decreases until we hear the 'click'; The entropy due to the unknown location of particle B is proportional to lnV, but the volume V decreases with time as A moves from left to right.
Notice that she can make the cylinder (and thus the initial V) as large as she wants and she could use more than one particle at B so that the initial entropy kNlnV can be arbitrarily large(**); And if the velocity of particle A is very small this decrease can take a long time ...

added 1 day later: I think I (finally!) figured out where the problem is with this puzzle. See the comments if you want to know my solution (instead of finding out for yourself.)

(*) While the initial position and momentum of A are well known (the particles are heavy enough that we don't have to worry about quantum effects), the position of particle B is unknown (but we do know that it is at rest).

(**) Of course the effort to set up the device will increase entropy by an even larger quantity, but all this occurs already on Sunday.

added later: I am no longer sure about that. She might have simply picked a cylinder of unknown length, but shorter than 5m. The (right) end of that cylinder and the particle B would be identical. Now she sets up particle A on the left side to move with a (constant) speed of 1m/day and when A hits the other end (=particle B) it triggers the alarm (at which point she then knows the length of the cylinder).
I don't see how the act of picking up a cylinder of unknown length increased the entropy on Sunday.

reading

2009-06-24T13:21:00.001-07:00

I just came across the book Information, Physics and Computation, by Marc Mezard and Andrea Montanari, which was published just recently. The draft is still available as pdf files here. Now you know what I am currently reading.
And there is also this paper about MAP estimation of Hidden Markov processes; I mention it as a follow-up to earlier posts.
"We reduce the MAP estimation to the energy minimization of an appropriately defined Ising spin model..." Sounds interesting.

time and uncertainty

2009-06-14T00:06:00.000-07:00

I am sure you know this one already, but ...

The director announces that next week will be a fire drill. In order to make it more realistic the day of the drill will be a surprise.
Here is the problem: The drill cannot be on Friday (the last day of the work week), because everybody would know on Friday morning that if the drill did not happen yet it will have to be this day, so it would not be a surprise.
But for the same reason it cannot be on Thursday, because everybody knows it cannot be on Friday and on Thursday morning, knowing that it did not happen yet one would have to conclude it will be this day, so it would not be a surprise. etc. Therefore the fire drill cannot be on any day.

But on Tuesday the alarm bell rings and of course nobody knew it would be that day...

C.F. v. Weizsäcker discussed the puzzle in his book 'Aufbau der Physik', assuming that it tells us something about the nature of time.
According to Wikipedia no consensus on its correct resolution has yet been established despite significant academic interest (*).

Maybe we should try to assign Bayesian probabilities. Obviously, we have p(Fri) = 0 but then it follows that ...

(*) Notice the citation of the famous remark made by Defense Secretary Donald Rumsfeld!

tbfkatbfka...tbfkaTSM

2009-06-04T02:32:00.000-07:00

I assume that you pay attention and noticed that several weeks ago this blog changed its name to 'the blog formerly known as The Statistical Mechanic', which one may abbreviate as tbfkaTSM.
Yesterday it occurred to me that it is time to change the name once more, this time to 'the blog formerly known as the blog formerly known as The Statistical Mechanic' or short tbfkatbfkaTSM. Of course, thinking ahead, it was clear that a more future proof name would be tbfkatbfka...tbfkaTSM.
But then it dawned on me that tbfkatbfka...tbfkaTSM is actually equivalent to tbfkaTSM in a strange way. And so I had an opportunity to appreciate the axioms of logic, which allow one to compress unnecessarily long statements.
Like the axiom S5 of modal logic. I only mention it because Alvin Plantinga used it in his ontological proof, which is a variant of Anselm's proof.

brains

2009-06-02T01:08:00.000-07:00

In yet another post Lubos Motl writes about Boltzmann brains and makes the following argument.

"The Boltzmann Brain hypotheses should already be expo-exponentially suppressed relatively to sane hypotheses. Since the people began to think about the world, they have made so many observations of the ordered real world that had to look like miracles from the Boltzmann Brain viewpoint that whatever the Boltzmann Brain prior were, they were already suppressed essentially to zero."

You have 10 sec to figure out what is wrong with this argument.

tossing biased cyber coins

2009-05-30T07:32:00.000-07:00

I decided to test this comment to the previous blog post with a numerical simulation and the result was quite surprising (at least to me).
I wrote a simple C program, which generates n-state HMMs randomly (*) and then runs them N times (generating a sequence HTHH...), followed by another N cyber coin tosses.
The question was if a 'simple strategy' can predict from the first sequence the bias of the second sequence. In the following, I performed the experiment for n = 10, 20 ... with N = 100.
The 'simple strategy' predicts a bias for Head if the number of Heads in the first sequence exceeds 60 = 0.6*N and I registered a success of the prediction if the number of Heads was indeed larger than 50 = 0.5*N in the following sequence. The experiment was repeated 10000 times for each n and the graph below shows the success rate as a function of n.

Notice that the success rate is well above 50% for n < N, but even for n > N it seems that the first sequence can predict the bias of the second to some extent. This is quite amazing, considering that for n > N the first sequence has not even encountered all the possible states of the HMM.
Obviously, as n increases (and N stays fixed) the success rate approaches 50% and if one does not know n this leads us back to the questions raised in the previous post. But the success rate for n < N and even n > N is much higher than I would have expected.
The next task for me is to double check the result (e.g. against the literature) and to do some more experiments.

------

(*) An HMM is really described by two tables. The one stores the probabilities for H vs. T in each state s = 1, ..., n. The program initializes these probabilities with uniform random numbers between 0 and 1.
The other table stores the n x n transition probabilities and the way my program assigns these probabilities is to first assign uniformly distributed random numbers and then normalizing probabilities by multiplying the probilities p(i,j) to get from state i to j so that sum_j ( p(i,j) ) = 1. There is some ambiguity in this procedure and I guess one could choose a different measure.

homework

2009-05-17T05:09:00.000-07:00

In order to illustrate some comments made to my previous post, I suggest the following homework problem:

We are dealing with a Hidden Markov model with n internal states, which produces as output a sequence of Hs and Ts. We know that it is ergodic (*), we are told that it is biased such that either p(H) > 2p(T) or p(T) > 2p(H) (if it just runs long enough) and we observe a sequence HTHHT... of N ouptputs.

How big does N have to be as a function of n to determine with some confidence if we are dealing with one or the other case?
If we do not know n (**), how large would N have to be?
And how does an uninformed prior look like in this case? Will it have to be an improper prior as long as we do not know n or should we somehow weight with the (inverse of) the complexity of the HMMs?

(*) Every internal state can eventually be reached and every internal state will eventually be left, but the transition probabilities might be small.

(**) This makes it somewhat similar to the case of the previous blog post, but of course, in the previous post we do not know anything about the underlying model, other than that there is a strong bias.

biased

2009-05-12T07:26:00.000-07:00

A ~~simple~~ coin toss and all we know is that it is strongly biased. What is the probability to get Head at the first throw?
Since we have no further information to favor Head or Tail, we have to assume p(Head) = p(Tail) and since p(Head) + p(Tail) = 1 we conclude that p(Head) = 1/2. Easy as pie.
But kind of wrong, because the only information we actually do have about the coin is that p(Head) is certainly not 1/2, because the coin is biased.

esse est percipi, part 4

2009-05-07T01:04:00.000-07:00

As I tried to show in the previous blog posts, the interpretation of quantum physics is to a large extent a debate on how to understand "psychophysical parallelism" and how to assign (our) conscious experience to a wave function and/or its components. This is one reason why most 'real physicists' usually stay away from this topic.
But if you want to read even more about it, I recommend the following as starting points:

H. D. Zeh: "Epistemological consequences of quantum nonlocality (entanglement) are discussed under the assumption of a universally valid Schroedinger equation in the absence of hidden variables. This leads inevitably to a many-minds interpretation."

plato about many minds: "... one might well conclude that a single-mind theory, where each observer has one mind that evolves randomly given the evolution of the standard quantum mechanical state, would be preferable."

Dowker & Kent (also here): "We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communication, prediction and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as yet unknown theory of experience."

Bernard d'Espagnat: "The central claim, in this paper, is that the Schroedinger cat – or Wigner’s friend – paradox cannot be really solved without going deeply into a most basic question, namely: are we able to describe things as they really are or should we rest content with describing our experience?"

Last but not least, the webpage of Peter Hankins is not a bad reference for the traditional discussion of conscious entities and the mind-body problem.

esse est percipi, part 3

2009-05-05T02:58:00.000-07:00

The previous blog post (I recommend that you read it first) ended with Sidney Coleman's argument in favor of the 'many worlds interpretation', which really is an argument that a 'collapse' of the wave function is not necessary to explain our usual conscious experience. As we shall see there is a problem with that argument.

Max Tegmark provides for a good (and easy to read) explanation of the 'many worlds interpretation' in this paper. In the last section he discusses the issue of 'quantum immortality', considering a quantum suicide experiment and in my opinion this raises an important question.
In general we have to assume that the wave function |Y> of You the Observer always contains components associated with an alive human being, even thousands of years in the future and even if classical physics would describe you as long dead; The wave function never 'collapses' and preserves all components, even those which describe absurd freak events. It is an important question what conscious experience is associated with such states.

But in order to discuss this further I prefer to modify Tegmark's thought experiment so that the experiment does not use bullets which may kill you, but pills (the "red pill or the blue pill") which may or may not contain drugs to knock you unconscious for a while. We may want to call the thought experiment Schroedinger's Junkies and instead of his cat we place You the observer in an experiment where you have to swallow a pill which either contains harmless water or a strong drug (e.g. LSD), depending on a measurement of the quantum state |s>.
If |s> = a|u> + b|d> we have to assume that after the experiment You are best described by the wave function |Y> = a|U> + b|D> , where the component |U> means that you are unharmed, while |D> means that you are heavily drugged.

Again we consider Coleman's operator C, but this time we have to assume that C|D> = 0 (heavily drugged you will not have a normal conscious experience) while C|U> = |U>. The problem is that now C|Y> = aC|U> + bC|D> = a|U> and the state |Y> is no longer an eigenstate of C. In other words, Coleman's consciousness operator indicates that after the experiment You are not in a normal conscious state [*]; This contradicts the fact that for a = b in approximately half of the cases Schroedinger's Junkies will always experience a normal conscious state (and for a >> b almost all of them !).

Does this counter example to Coleman's argument indicate that something like a 'collapse' (e.g. decoherence) from the superposition |Y> to either |U> or |D> is necessary after all?
I have to admit that trying to understand quantum physics feels like trying to find the solution to x² + 1 = 0 in the real numbers!

[*] One could argue that there is no problem if the total state is not an eigenstate of C, since the "psychophysical parallelism" of m.w.i. assigns consciousness to the components of the wave function only. However, we can split any component into subcomponents and even if C|U> = |U> we can split |U> e.g. into two subcomponents |U> = ( |U> - |D> ) + ( |D> ) so that none of the subcomponents (..) is an eigenstate of C.
Coleman's argument seemed to provide for consistency across different ways to split the wave function into components, but indeed it fails in general; In order to rescue "psychophysical parallelism" for m.w.i. one would have to find a preferred basis and it has been argued that decoherence might just do that. However, I have explained earlier why I am not convinced.

esse est percipi, part 2

2009-05-04T01:05:00.000-07:00

The previous post referenced a rather crude attempt to use our conscious experience as the foundation of (quantum) physics. Usually, consciousness does not even make an appearance in physics and some sort of "psychophysical parallelism" (different states of a [human] brain correlate with different conscious experience) is the only (hidden) assumption.

An interesting example is the notorious measurement problem in quantum physics. (A slightly related classical example was provided earlier.)
Just to quickly recapitulate the main issue: Assume that a quantum system |s> can be in two states |u> and/or |d>. An Observer, initially in the state |I>, subsequently interacts with |s> in such a way that |u>|I> evolves into |u>|U>, while |d>|I> evolves into |d>|D>. With |U> we denote an observer who is sure to have observed the system as |u>, while |D> is the observer in a state with conscious experience of |d>.
The measurement problem arises if we consider the interaction of this observer with a state |s> in a superposition a|u> + b|d>, which then leads to an observer being in the superposition a|U> + b|D>; Schroedinger's Cat, Wigner's friend and all that.
The argument can be made much more precise, see here and here, and one does not have to assume that |U> or |D> are necessarily pure states (and the observer state will in general incorporate entanglement with the environment etc.).

We find this superposition of observer states absurd because we assign a specific conscious experience to a specific observer state following the assumption of "psychophysical parallelism"; But while we know what sort of conscious experience one would assign to |I> , |D> and |U> , we do not know what conscious state should be assigned to a state a|U> + b|D>. We would assume it has to be some experience of confusion, a superposition of consciousness, which we normally do not experience.
At this point physicists introduced various assumptions about a 'collapse' of the wave function to eliminate such superpositions of observers, they threatened to pull a gun whenever Schroedinger's Cat was mentioned and even worse began a long philosophical debate about various interpretations of quantum physics.

But following an argument made by Sidney Coleman (in this video clip [at about 40min.]), this is really not necessary. Consider again the wave function |Y> of You the observer and assume that there is an operator C which tells us if You the observer has a normal classical conscious experience, so that C|Y> = |Y> if you are in a normal conscious state and C|Y> = 0 if not [*].
Returning to our measurment problem, we have to assume that C|D> = |D> and also C|U> = |U>, but then it follows from the linearity of quantum operators that even if |Y> = a|U> + b|D> we have C|Y> = aC|U> + bC|D> = |Y>. In other words we have to conclude that You the observer has a normal classical conscious experience even in a superposition state after the quantum experiment.

This argument is at the basis of the 'many worlds interpretation' [x], which assigns a normal classical conscious experience to the components of the wave function |Y> and then shows that this does not lead to contradictions with our everyday experience for superpositions of such components. A subtle shift in our assumptions of "psychophysical parallelism" with drastic consequences.
A 'collapse' of wave functions seems no longer necessary (and the act of pulling a gun would only create yet another superposition of states 8-).

[*] Obviously we do not know what such an operator would look like, but if we believe in "psychophysical parallelism" we have to assume it can be constructed. Notice that if we would not believe in "psychophysical parallelism" then there would not be a 'measurement problem' either.

[x] It is obvious that 'many worlds interpretation' is a really bad name and should be replaced e.g. with 'many experiences'.

esse est percipi

2009-05-02T01:34:00.000-07:00

I would not have thought that Bishop Berkeley was perhaps one of the founding fathers of modern physics.
But we have to consider this: "In the present work, quantum theory is founded on the framework of consciousness, in contrast to earlier suggestions that consciousness might be understood starting from quantum theory. [..] Beginning from our postulated ontology that consciousness is primary and from the most elementary conscious contents, such as perception of periodic change and motion, quantum theory follows naturally as the description of the conscious experience."
Could the phenomenalism of e.g. Ernst Mach make a bit of a comeback after all?

I really like the phrase "in contrast to earlier suggestions", which sums up about 250 years of physics as "earlier suggestions". 8-)

In the appendix (section 10) a mathematical model of consciousness is presented, as a process which tries to find the solution to x² + 1 = 0 in the real numbers, which reminds me of The Confusions of Young Törless.

living with ghosts

2009-04-25T01:38:00.000-07:00

" We conclude that quantum gravity with fourth order corrections can make sense, despite apparently having negative energy solutions and ghosts. In doing this, we seem to go against the convictions of the last 25 years ..."
Hawking and Hertog, 2001

It is well known that the perturbation theory of quantum gravity is not renormalizable, but one can 'fix' this problem by introducing higher order terms ( R² ) in the action. Unfortunately, it is also well known that higher order derivative terms (appear to) come with dangerous ghosts, threatening the S-matrix with states of negative probabilities. However, in their (very clear and easy to read) paper Hawking and Hertog provide for a convincing argument that one should not be afraid of such ghosts.

In a related paper Bender and Mannheim, 2007 showed that "contrary to common belief .. theories whose field equations are higher than second order in derivatives need not be stricken with ghosts. In particular, the prototypical fourth-order derivative Pais-Uhlenbeck oscillator model is shown to be free of states of negative energy or negative norm."

Last but not least, Benedetti, Machado and Saueressig, 2009 "study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques" and argue that "asymptotic safety also resolves the unitarity problem typically haunting higher-derivative gravity theories."

In other words, if (for whatever reason) you don't like string theory, you could try to get used to living with ghosts...

rational or real

2009-04-19T01:14:00.000-07:00

Physics as we know it is based on real (or complex) numbers, but it is interesting to ask what (if anything) would change if we would consequently replace real valued variables with rational numbers. After all one could make the case that any measurement can only result in rational numbers and probabilities derived from counting outcomes are rational numbers as well.
Obviously, it would not be an easy change, e.g. one would have to replace differential equations with difference equations (with arbitrary base). One would have to consider if and how it affects e.g. the Lorentz group and one would have to deal with the fact that the rational numbers do not constitute a Hilbert space. In other words, it is not immediately clear that one gains anything using Q instead of its natural completion R.
However, if one thinks that it is obvious that using Q instead of R would only complicate things but not make any real difference, I suggest to read this paper: "We explicitly evaluate the free energy of the random cluster model at its critical point for 0 < q < 4 using an exact result due to Baxter, Temperley and Ashley." The authors find that the free energy of the system depends on whether a certain function of the continuous parameter q is "a rational number, and if it is a rational number whether the denominator is an odd integer".
This is one of the weirdest things I have ever seen in statistical mechanics.

update: RZ (see comments) points out that it is not immediately clear from the paper that the numerical value of the free energy is indeed different for rational numbers, just because the form of the free energy function is not the same. However the sentence "This implies that the free energy of the random cluster model, if solved, would also share this property" on p.3 would then be highly misleading.
In any case the quantum kicked rotator, also mentioned by RZ, might be a better example of what I had in mind.

theology and probability, part2

2009-04-05T00:30:00.000-07:00

In the following we shall finally consider one of the more serious issues. Similar questions have bothered the serious thinkers for several centuries and perhaps I can finally make an important contribution. What is the probability for the existence of the invisible pink unicorn?

It may be necessary to clarify a few terms first. With "invisible" we mean here that one cannot test or detect the supreme being with currently available scientific methods. With "pink unicorn" we mean that a true believer may (or may not) experience the supreme being through direct revelation as pink and a unicorn. It is obvious that this believe system is consistent [see footnote 1] and furthermore it seems to be reasonable as we shall see in the following.

It is also immediately obvious that an atheistic position (which assigns a probability p(i.p.u.) = 0 to the existence of the i.p.u.) is problematic and indeed inconsistent with the usual rules of reasoning.
In general one should not assign a zero prior to a consistent and eventually reasonable model, but furthermore if one is certain that the i.p.u. does not exist, then there can be no facts directly related to the i.p.u. and thus no facts to make a rational decision that p(i.p.u.) = 0. [2]

The agnostic position, which assigns a probability p(i.p.u.) = A with 0 < A < 1 seems reasonable at first (the A stand for either agnostic or arbitrary), however, it really is problematic as well.
The obvious question is what value A to use, which does not have a good answer. But it gets much worse once we consider the fact that an agnostic (and only an agnostic!) must assign the same probability to the "invisible yellow dragon", the "invisible green parrot", the "invisible red herring" etc. In other words, the agnostic has to deal with a discrete but obviously infinite set of possibilities, which are a priori equally likely, and assigning any probability A to one of them would mean that the sum
p(i.p.u.) + p(i.y.d.) + p(i.g.p.) + p(i.r.h.) + ... = A + A + A + ...
necessarily diverges instead of adding up to one [4].

This seems to leave us with the true believer, who assigns p(i.p.u.) = 1, as the only one with a consistent and reasonable position [3]. It is also the only one with a chance to gather real evidence through direct revelation.

-----------

[1] In order to better understand that believing in the "invisible pink unicorn" is perfectly consistent, we consider a model which assumes that the world around us with all its possible experiences is just the result of an elaborate computer simulation. The supreme being would then e.g. be the administrator of this simulation and she would be "invisible" to all the beings in the simulated world. However, she may (or may not) choose to reveal her existence to true believers every now and then by programming the direct experience of a "pink unicorn". Of course, to have true faith in the "invisible pink unicorn" does not include believe in this particular model and indeed a true believer would most likely understand it as heresy by limiting the potential mode of existence of the supreme being.

[2] If one makes the scientific statement that "pink unicorns do not exist", then it really means e.g. that a careful and exhaustive scientific search has not detected "pink unicorns" or that a particular well established theory excludes "pink unicorns". In other words "pink unicorns do not exist" is really a statement e.g. about the scientific search or the well established theory.
However, in the case of the "invisible pink unicorn" such a scientific search is meaningless and no relevant theory can exist.
Furthermore, the fact that one did not experience the supreme being as "pink unicorn" through direct revelation is of course irrelevant, because such revelation requires true believe.

[3] True faith in the "invisible pink unicorn" sets the probability for all other possible supreme beings to zero, thus avoiding the problem of the agnostic.
Different to an atheist, the true believer in the "invisible pink unicorn" can indeed set the probability for the existence of the "invisible green parrot" etc. to zero, p[i.g.p.] = p[i.y.d.] = ... = 0, because she can use the fact of her own strong faith as justification; The atheist has no such fact available.

[4] added later 4/6/09
Yes, I am aware that a true Bayesian may use an improper prior in this case. (I also admit that I learned about it only two weeks ago and it was actually one reason to write this post.)
But how would she update this prior to get to real probabilities? Notice that reports of revelation will only become available if true believers are around, but this means Bayesian updating would depend on the existence of people not using the Bayesian method. How can one rely on the testimony of such irrational people?
One could also (try to) argue for a non-uniform prior. E.g. all invisible supreme beings are described as "the invisible X1 X2 X3 ... Xn" using n words X1, X2... , Xn. It could make sense to weight the probability with the inverse of the complexity of the supreme being, approximated by n, e.g. such that p("the invisible X1 X2 ...Xn") = exp( -f(n) ) and f(n) is chosen so that the sum of all probabilities converges. But there are a few problems with this.
It is obviously a very crude method to approximate the complexity of a supreme being in such a way, e.g. some of the descriptions might be of the form "the invisible being which is ... but is very simple indeed", etc.
Also, notice that the "invisible blue dolphin which likes yellow fish and ..." is known as "iok Aum" in the Zaliwali language, so it would have a much higher probability than for an English speaking Bayesian. But of course, probability is about subjective uncertainty 8-)
I guess there are many clever ways a Bayesian could "fix" this problem, but I am afraid from my point of view it would only get us deeper and deeper into nonsense land.

a quantum of solace?

2009-04-04T00:37:00.000-07:00

I cannot deny that the quality of this blog is declining rapidly (actually it is more accurate to say it did not improve as quickly as I hoped for). Thus I changed the title to reflect this sad fact.
In this spirit I shall write a few lines now about quantum gravity and the links I collected recently.

In her last post Sabine asks if quantum gravity has been observed by GLAST/Fermi in the gamma ray burst GRB 080916C. Interestingly, Lubos wrote about the same result(s) but with exactly the opposite conclusion. (Did I mention that the two don't like each other?)
Previously, Peter wrote once again about string theory being useless for ever predicting anything. At about the same time Lubos mentioned a recent paper about 'The footprint of F-theory at the LHC' and concluded that the validity of string theory is a settled fact and indeed string theory is highly predictive. (Did I mention that the two don't like each other?)
Lubos also discussed Boltzmann eggs, mostly to attack a straw man used to stand in for Sean Carroll and his upcoming book. (Did I mention that the two don't like each other?)
CIP picked up on it and in comments to his post Lubos made some confusing or confused statements, which you may or may not find interesting.

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

And even more links.
John Ellis et al. also discussed the GLAST/Fermi and related results.
Craig Hogan proposes that excess noise observed at the GEO600 interferometric gravitational-wave detector could be direct evidence of holographic quantum gravity.
But Igor Smolyaninov thinks that this is unlikely.
Renata Kallosh found an argument why d=4 N=8 supergravity is finite for all loops.
Simon Catterall et al. put N=4 SYM on a 4d lattice (see also here).
Last, but not least, Aaron Bergman asks an important question about fretless guitars.

theology and probability

2009-03-27T07:58:00.000-07:00

If you are a member of The Church of Bayes (*) I recommend you read this.
If you are a physicist considering to become a member I recommend you read that.

(*) Wikipedia: Thomas Bayes (1702-1761), British mathematician, statistician and religious leader

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

I wanted to write about a nice illustrative example of Cosma's 1st exercise as the 2nd part to this post. But it has already been done before I could even begin with the typing (I was travelling) and I guess it is much better than what I would have achieved (but I would have left out the 'waterboarding', which is not that funny).
I recommend the comments to Brad DeLong's example if one is interested to see some members of the church argue (with each other). But then, perhaps you have something better to do...

The main lesson from all this is very simple. If the set of considered models does not contain the true model then Bayesian updating can go very wrong. But how does a Bayesian know that her process includes the true model without leaving the reference frame of her church?

Of course, we should not expect a true Bayesian to agree that there is a problem (e.g. in the comments to DeLong's post) - after all we know (now) that their procedure does not always converge on the truth...

Ikea chairs

2009-03-04T04:43:00.001-08:00

In a review article about agent-based models Dietrich Stauffer once wrote
"Physicists not only know everything, they also know everything better."
In the spirit of "this indisputable dogma" Lee Smolin recently published his thoughts about "time and symmetry in models of economic markets", advocating to formulate "economics in the language of a gauge theory".
If I would be in a generous mood, I would perhaps only point out that worse stuff has been published on the arxiv and in particular its econophysics section. But I am not and thus I will actually quote some of the deep insights of the author:
"..publishers have a simple motivation to cut costs by only printing the books that will sell, but it seems very difficult to predict accurately which books will sell and which won’t. One rule of thumb-to which there are exceptions-is that books that are not in book stores don’t sell."
" Combining space, time and uncertainty, there is then a vast explosion in the number of goods. Rather than having a particular model of Ikea made chair, we have a vast set of chairs..."
"Consider the adage, well known to sailors, ”The two happiest days in the life of a boat owner are the day they buy their boat and the day they sell it.” Imagine coding this in a utility function." (*)

If one is interested in similar deep insight then this paper offers plenty of it.
If one is actually interested in the application of gauge theory in finance I would recommend the book of Ilinski and this review.
But if one wants to read a well written critique of econophysics, I would recommend this essay instead.

(*) Actually, I can confirm the adage from personal experience as an obvious truth.

anti-block

2009-02-17T07:01:00.000-08:00

Bryan lists the possible philosophical positions on the passage of time.
Currently I think 'anti-block' is the most convincing, but probably I just read too much Ernst Mach recently...

groundhog day

2009-02-06T04:38:00.000-08:00

Warning: This blog post is neither entertaining nor informative. I recommend that you just skip it.

Once again we are floating in a Newtonian universe described by its microstate S(t), but of course we are just ordinary observers who do not necessarily know it. At t0 we perform a little experiment, the equivalent of tossing a coin, with two possible outcomes H(ead) or T(ail); At t0+D we know the result.
The omnipotent demon watches the outcome too and in the case of T reverts the Newtonian universe to its previous state S(t0-D), which explains the title of this blog post; In the case of H she does nothing. However, the omnipotent demon is not infinitely patient and therefore the universe would loop only N times through the same state(s), then it continues.
We know about all this, because the omnipotent demon was so nice to inform us about it in advance.

Now we try to calculate the probability for H and there are two different ways to do it:
i) We do not know the microstate and both H and T are equally likely for what we know about S(t0). The fact that the demon will play some tricks at t0+D does not change anything, thus p(H) = 1/2.
ii) There are the following possible cases for this silly game: This is the 1st time we observe the experiment and the outcome is H, which we denote as 1H. Then there is 1T and also 2T, 3T, ... NT.
Notice that there cannot be a 2H, 3H etc. because the universe is deterministic, if T was seen the first time it must be seen the 2nd time etc.
Since we cannot distinguish between those cases, we have the probability p(H) = 1/(N+1).

Obviously, this has some similarity with the sleeping beauty problem. But I am not asking which of the two probabilities is 'correct' and I am not even interested if this little thought experiment tells us anything about a relationship between time and probability. I am asking a different question.
How do you understand the limit D -> 0 ?

Told you so...

constraints

2009-02-01T00:34:00.000-08:00

The field equations of general relativity can be separated into hyperbolic evolution equations and elliptic constraints [ADM]. The evolution equations propagate an initial field configuration 'forward' in time, similar to other theories of classical fields.
However, due to the constraints one cannot choose the initial configuration freely and this is very different from other classical fields. In some sense the constraints 'connect' spacelike points and thus one could call general relativity 'holistic' if this would not be such an abused word.

We don't really know what the quantum theory of gravitation is, but one would assume that the classical theory reflects the properties of the underlying quantum theory and indeed the Wheeler-deWitt equation is nothing but the operator version of one of the constraints.
I think one needs to keep this in mind when discussing thermodynamics of general relativity, the information loss problem or the entropy of black holes. E.g. if one specifies the metric near the horizon of a (near spherically symmetric) black hole, the constraints already determine the 3-geometry; Therefore I do not find it surprising that counting microstates provides for a holographic result which differs substantially from the naive expectation.
I would also think that an approach to the information loss problem which emphasizes locality as 'conservative' is misguided.

the physics of immortality and the direction of time

2009-01-16T00:55:00.000-08:00

Recently, Sean wrote a blog post about Frank Tipler, who described in his book, The Physics of Immortality, what he calls Omega point theory; Wikipedia has enough about it that I do not need to elaborate much further. The main idea in one sentence is that the 'big crunch' of a re-collapsing universe which contains intelligent life (necessarily) generates a point of infinite complexity, capable to process an infinite amount of information in a finite amount of time [x]. As I mentioned previously, the book contains a lot of interesting physics, but also large sections comparing the Omega point to the God of various religions and as a whole the book is a bit odd.

In a section near the end of his book, Tipler discusses quantum gravity, the wave function of the universe and in particular the boundary condition(s) for such a wave function. The best known example for such a condition is the no-boundary proposal of Hawking, which corresponds to 'creation from nothing'; A different proposal was examined by Vilenkin and others [e.g. here].
Tipler proposes as a new boundary condition the requirement of an Omega point. In other words, he replaces the usual initial condition with a final condition on the allowed physical states. In his own words:
"In my above description of the Omega Point Theory, I used past-to-future causation language, which is standard in everyday life, and in most physics papers. This may have given the reader the impression that it is life that is creating the Omega Point (God) rather than the reverse. Nothing could be further from the truth. It is more accurate to say that the Omega Point, acting backwards in time, via future-to-past causation, creates life, and His multiverse."

This is of course a main difference (if not the main difference) between science and religion.
Science assumes an initial condition (usually of high symmetry and low entropy), with everything following afterwards according to the laws of physics, with no purpose, intention or meaning.
Religion on the other hand assumes that there is a point to the world and our experience, a desired goal and final explanation, which determines everything.
Once Tipler assumes the final Omega point condition, he leaves science as we know it and opens the door to 'explanations' like this:
"I will say that an event is a "miracle" if it is very improbable according to standard past-to-future causation from the data in our multiverse neighborhood, but is seen to be inevitable from knowledge that the multiverse will evolve into the Omega Point."

While his book 'The Physics of Immortality' is vague enough, suggesting that perhaps one may be able to have it both, science and religion, the subsequent development of Tipler's thoughts makes it immediately clear where his proposal leads:
"I shall now argue that the Incarnation, the Virgin Birth, and the Resurrection were miracles in my sense. The key to understanding why these events HAD to occur is the recently observed acceleration of the universe."

I will only add that in my opinion Sean's word crackpot is misplaced in this case; I think 'tragedy' would fit better.

[x] According to Tipler, the discovery of an accelerating expansion of the universe (dark energy) does not necessarily affect his main assumption, as he explains in this interview.

the measurement problem

2009-01-08T00:25:00.000-08:00

Shortly after Newton proposed his new mechanics, the "shut up and calculate" approach of Newton, Halley and others produced the first astonishing results. However, it did not take long until the foundational debate about the interpretation of the new physics began. In particular, the true meaning of the position coordinates x(t) was heavily discussed. The x(t) were of course projections onto a holonomic basis in the 3-dimensional Euclidean vector space. But how exactly would they be determined in a measurement process?

It came down to measuring distances between point masses (1). But how does one actually measure such a distance? Suppose we use a ruler in the simplest case (2). We have then only replaced one distance measurement with two distance measurements, because instead of measuring the distance between two mass points we need to measure now the distance of each mass point to the markings on the ruler (3).
Now we could use another two rulers to measure those distances etc. - an infinite regress. (Notice the superposition of rulers at 3!)

There were soon two main groups of opinion. The first was known as realists, assuming that the x(t) represented the real position of a mass point, and even if human beings had a problem to comprehend the infinite regress of the measurement process, the omniscient God would necessarily know it.
A small subgroup proposed that the infinite regress is the position, but could not really explain what this means.
The other group insisted that the x(t) were only a subjective description of reality but not part of reality itself. They emphasized the important role of the conscious observer who would terminate the otherwise infinite regress of the measurement process; This introduced the issue of subjective uncertainty into the debate.

Careful analysis showed that x(t) was only known with finite uncertainty dx and in general this uncertainty would increase with time. Astronomers noticed that the dx for some planets was larger than the whole Earth! The realists assumed that there was still one true x(t), even if we do not know it, while Sir Everett 1st proposed the stunning interpretation that *all* positions within dx were equally real, rejecting the idea of random measurement errors. The world was really a multitude of infinitely many worlds and the infinite regress of the measurement problem reflected this multitude!
Subsequently, this type of analysis became known as decoherence program: The position of a mass point can be determined only if the mass point interacts with other mass points. But this means that in order to reduce the uncertainty dx, one necessarily increases the uncertainty of the position of all mass points in the environment.
While it was not clear if decoherence really helped to solve the foundational problems, the complicated calculations were certainly very interesting.

In a devilish thought experiment, a cat was put in a large box and then the lid closed. Obviously the cat would move around inside the box (some would even suggest that the cat moved around randomly, since no law was known that could determine the movement of the cat!), but one could not observe it.
The stunning question was, and still is, if the cat had a position x(t) if one waited long enough.
The realists again insisted that the position of the cat was a real property of the cat, even if it was unknown to everybody. But others insisted that it made no sense to assign a position, since the rays emitted by the eyes of the observer were not able to reach the cat; Furthermore, the animal itself has no conscious soul and thus cannot determine its own position.

While the "shut up and calculate" approach celebrated many more successes, the foundational issues of the new physics were never resolved.

spin echos

2009-01-06T00:37:00.000-08:00

In my previous post about a hypothetical 'reversal of time', I should have mentioned the spin echo effect. In real experiments, first performed by E. L. Hahn, a configuration of spins evolves from an ordered into a disordered state, but subsequently the initial ordered state is recovered by application of magnetic pulses.
The spin echo effect is described in this text (sect. 11) and further discussed here.

Obviously, this effect raises several interesting questions about the foundations of statistical mechanics, e.g. the definition of entropy via coarse-graining; But as noticed by the Aeolist "many of the terms used in the debate, beginning with the all-important definition of entropy, and including terms like ‘preparation’ and ‘reversal’ (and its cognates), are still used in so many different ways that many of the participants are speaking at cross purposes".
By the way, I did not see a discussion of the 'backreaction' of the ensemble of random spins on the magnet (and its entropy) that is used to trigger the reversal; It may or may not be important in this debate.

A somewhat related model is the swarm of n free particles moving with random but fixed velocities on a ring, as discussed by H. D. Zeh in appendix A to his book about the direction of time.

thermodynamics

2009-01-03T02:03:00.000-08:00

It seems that there is some confusion about several issues in thermodynamics, so the following might be helpful.

1) If a system is not in thermodynamic equilibrium, certain macroscopic quantities may not be well defined, e.g. temperature as mean kinetic energy. However, entropy as a measure of our ignorance about the micro state is in general defined even far away from equilibrium. Otherwise we would not have the 2nd law of thermodynamics, because dS/dt ~ 0 if a system is in equilibrium.

2) The heat capacity of a gravitating system (Newtonian gravity) is in general negative. As an example consider a star radiating energy away, which will cause it to heat up due to gravitational contraction. This can be confusing, but there is nothing wrong with thermodynamics if one includes Newtonian gravity.
In general, the 0th law does not always hold and things can get funny, but this does not affect the 1st and 2nd law.

3) If we consider Newtonian mechanics carefully, we find that no classical system is stable and thus no purely classical system can be in thermodynamic equilibrium. This was historically the reason for Bohr to propose the first version of quantum mechanics.

4) In general, we do not know how to calculate the entropy of a particular spacetime. There is the proposal of Penrose to equate it with the Weyl curvature; However, there are problems with this proposal.
Things can get quite funny if one considers a spacetime which contains a naked singularity or closed timelike loops. Unfortunately, current state-of-the-art is still that one has to remove such geometries by hand on the grounds that things get quite funny otherwise.

5) In quantum theory, if a system is in a pure state the corresponding entropy is zero. If one assumes that the 'wave function of the universe' was initially in a pure state, it would remain in a pure state, assuming unitary evolution for quantum gravity (as suggested by the AdS-CFT correspondence). There is thus a problem for (some) many worlds interpretations in my opinion.

backwards or twice as fast

2009-01-02T00:43:00.000-08:00

Recently I came across an argument about 'reversal of time' and our conscious experience (I am sure this type of argument must be at least hundred years old) and I thought I should mix it with an old idea of mine. I am curious what others think about it; So here it goes:

Imagine that we can describe the world as a Newtonian universe of classical particles so that x_i(t) , where x is the position(vector) of the i-th particle and t is the classical time parameter, determines the configuration of our world for each moment of time. I am pretty sure that the following argument can be generalized to a quantum mechanical description, but it is much easier to stick to Newton for now.

We assume that the world evolves according to the laws of Newtonian physics up until the time t0. At this moment an omnipotent demon reverses all velocities: v_i(t0) = x'_i(t0) -> - x'_i(t0), where ' is the time derivative, and the Newtonian evolution continues afterwards.
Obviously, for t > t0 everything moves 'backwards'; If a glass fell on the floor and shattered into many pieces for t < t0, it will now assemble and bounce back up from the floor etc.; If the entropy S(t) increased with t for t < t0, it now decreases for t > t0.
One can also check that x_i(t0+T) = x_i(t0-T) and x'_i(t0+T) = -x'_i(t0-T) for every T (as long as we rule out non-conservative forces).

The interesting question in this thought experiment is "what would an observer experience for t > t0 ?".
If we assume that the conscious experience E(t) of an observer is a function of x_b(t), where b enumerates the particles which constitute her brain, then we would have to conclude that the observer does not recognize anything strange for t > t0, since x_b(t0+T) = x_b(t0-T) and it follows immediately that E(t0+T) = E(t0-T). So if all the experiences E(t0-T) contained only 'normal' impressions then the same is true for E(t0+T). In other words, while the sequence of experiences is 'backwards' no single experience contains the thought "everything is backwards" and nobody feels anything strange.
But this would mean that no observer is able to recognize 'backward evolution' with entropy decreasing and distinguish it from normal evolution!

One way to avoid this strange conclusion is to assume that E(t) is a function of x_b(t) and v_b(t). Of course, we do not have a physical description of conscious experiences and how they follow from the configurations of our brain (yet). It is reasonable that our conscious experience depends not only on the position of all molecules in our brain but also their velocities.
Unfortunately, this leads us into another problem. If we rescale the time parameter t as t* = s*t, this would rescale all velocities so that v(t*) = s*v(t) and thus E(t) = E[x(t),v(t)] -> E(t*) = E[x(t*),s*v(t*)]; But if the function E is sensitive to v_b then it would be sensitive to the scale s too. I find this to be quite absurd, our experiences should not depend on an unphysical parameter.

The summary of my argument is the following:
i) If the world evolves 'twice as fast' we should not notice a difference (the molecules in our brains would move twice as fast as well).
ii) However, if the world suddenly evolves 'backwards' we would like to be able to recognize this (otherwise how would we know if the 2nd law is correct).
iii) But it seems that one cannot have both i) and ii) if one assumes that our conscious experience is a 'natural' function of the material configuration of our brain, e.g. if we follow Daniel Dennett and assume that consciousness simply is the material configuration of our brain: E(t) = [x_b(t)] or E(t) = [x_b(t),v_b(t)] (*).

Perhaps one can solve this puzzle by assuming E depends on higher derivatives x'' and/or perhaps one can find some clever non-linear function. But I think this would introduce other problems (at least for the few I tried ) and I don't find this very convincing [x].
Of course one can challenge other assumptions too. I already mentioned quantum mechanics instead of Newton or perhaps we have to assume that our conscious experience is not a function of the particle positions in our brain. But still, none of these solutions are very convincing in my opinion.
What do you think?

(*) Dennett is never that explicit about his explanation of consciousness.
In general, one could imagine that E is some sort of vector in the 'space of all possible conscious experience' - whatever that means.

[x] e.g. E could depend on v_b/N with N = sqrt(sum_b v²_b) instead of v_b. But where would the non-local N come from and also there would be a singularity at N=0, i.e. when all velocities are zero. One would not expect a singularity of E for a dead brain (with all molecules at rest) but rather zero experience.

Nerds on Christmas Eve

2008-12-25T02:25:00.000-08:00

Only a true nerd would post Nerd Self-Help on Christmas Eve. And look how many comments there were that evening!
One of those comments is such a beauty that I have to re-post its core argument:
It’s not very helpful to assign a predicate E(x) to mean “x exists”, since you are forced to conclude ∀x E(x).
After all ~∀x E(x) is equivalent to ∃x ~E(x), a contradiction.

Hours of philosophical dispute resolved with a two-liner!

Also, Scott mentions Kant’s refutation of Anselm’s ontological proof of the existence of God, which gives me an opportunity to link to Goedel's ontological proof, which fascinated me for quite a while when I first learned about it many years ago. (Nowadays it is on Wikipedia and you can easily read all the pros and cons arguments.)

a first post which is not the first post

2008-12-23T07:21:00.000-08:00

As I understand it, quantum theory consists of two parts.
... the 2nd part is Born's rule and using it one can follow the 'shut up and calculate' approach which is so successful. On the other hand, trying to really understand or even derive it can be quite confusing and I admit that I am confused [*].
But in my case, this is just one of several issues in physics which I do not understand. To be honest, I already have a problem to explain what exactly 'probability' is supposed to mean. And why do we use it most often when considering the future but not the past? I feel like Augustinus and if I wish to explain, I recognize that I do not know.

Whenever I considered myself to be a physicist, I suffered from the suspicion of being a fraud. Like most physicists I used (and abused) various mathematical concepts, but quite often I really had no clue what I was doing. But it seems that I am perhaps not the only one [x] with this problem...

Later, I found a solution to my doubts - I simply do not consider myself to be a real physicist any longer. At least I can feel free now to ask stupid questions and make silly proposals...
This is why this blog exists. Again.
As some readers know there is that chance that I may delete it as soon as the feeling of being a fraud creeps up again. What is the probability of that? A good question, which I may be able to answer as soon as you help me figure out what probability means .-)

If you found this first blog post by chance and want to get a better idea in advance what this blog will be about, I suggest you take a look at this page.
Welcome to the statistical mechanic and let's hope it will be an interesting journey.

PS: Perhaps you noticed that this text was somehow written in reverse order. It is not that I try to be confusing on purpose - but often it just comes out that way...

[*] A good starting point is this paper by Zurek and I also recommend these comments. Notice that the comments appeared earlier than the original paper on the arxiv 8-)
And there is a video of a lecture by Sidney Coleman on quantum theory, which I recommend. (If you want to jump to his treatment of the 'measurement problem' move to min. 38 and probability is discussed at min. 55)

[x] I should clarify that in my opinion the interesting part of the Bogdanov Affair is not played by the two brothers, but the community of professional physicists. E.g. consider this statement of Roman Jackiw: "It showed some originality and some familiarity with the jargon. That's all I ask."

first post

1972-08-01T04:52:00.000-07:00

There are many places to discuss the 'hot topics' of our time.
If Nixon wins re-election, what does it mean for science in the US?
Do you think it is a good idea to end the Apollo program or should we continue to fly to the moon?
Is Fortran 66 really so much better than Fortran IV ?
etc.

But this blog will be a place to discuss general questions about physics, statistics and all that.

PS: Currently I am thinking about appropriate little pictures as icons for my blog.

But why would I use a llama or the image of a plastic bag?