Maxwell's Demon: Computation & Physical Entropy

The second law of thermodynamics states that we cannot decrease the entropy of a closed system on average. This is important because if it would not hold, one could in principle build a "perpetuum mobile" (a machine that generates energy out of thin air). However, this is not a trivial law as we will see in the following. Physicist have repeatedly tried to construct machines (on paper mostly) that would violate the second law. The result of these thought experiments was that information (in the form of abstract sequences of bits) is intricately tied to the notion of physical entropy.

Let's first define entropy of a physical system. The usual definition counts the number of microstates contained in a macrostate. This means that we first have to define the macrostate and this usually involves coarse-graining of the micro-world.
The easiest example is a sequence of T coin-flips. Every sequence has an equal probably to occur, however if we study the space of the number of times heads comes up we can easily see that the entropy of the state N=T/2 is much larger than that of N=T. This is because there is only one unique sequence with N=T but NchooseT sequences of N heads (and this quantity peaks at N=T/2).

For a more realistic physical system let's imagine that we have contained 2N molecules in some volume, N of them are colored white, the remaining N are colored black. Let's divide the space in two imaginary halves and also imagine that the molecules randomly move around (as is realistic). Macrostates are now defined as counting the number of black balls in the left half and the number of white balls in the right half, which we will denote with B(l) and W(r). Again, B(l)=N, W(r)=N has a much lower entropy than B(l=N/2), W(r)=N/2, simply because there are many more ways to accomplish the second macrostate in terms of microstates.

If we use macrostates as the actual microstates themselves, then the entropy will always stay constant because there is exactly 1 microstate state per macrostate. However, if we use coarse-grained macrostates, then the entropy will typically increase, because you will tend to observe macrostates states that are more likely to occur. Thus in the case of the molecules, you will approximately find an equal number of white and black balls in the left and right halves. The second law says that you will not move back from this state to the ordered state of all black balls to the left and all white balls to the right. (There are fluctuations around the average, but with many molecules they are very small and the odds against fluctuating into the completely ordered state or enormous.)


All good so far. But Maxwell invented a "demon" that would build a separating wall between the two halves and control a gate. If a white ball approaches the gate from the left, the demon quickly opens the gate and lets the white ball through. Similarly for a black ball coming from the right (see Figure). The end result is that all black balls end up to the left and all white balls to the right.

So did we break the second law? To start with, the system of the molecules is no longer closed because it now includes the demon. But how will we treat the demon thermodynamically? Our first guess is that the act of moving the gate up and down and measuring the white and black balls as they approach the gate increase the entropy by a significant amount (more than can the demon can decrease the entropy of the environment). Although this is what will usually happen in practice, it turns out that the demon's trick can actually be achieved by a computer with vanishing increase of entropy (of course the construction is subtle, but you will have to take it on faith).

However, the demon will have to obtain information from the system and record it (say on a tape) in order to be able to operate the gate. As it turns out, one can actually decrease the entropy of the molecules in a vat only at the expense of recording the information about the system. Note that we are talking about an abstract sequence of 0's and 1's here. One can show that the act of erasing this information again will necessarily increase the entropy by an amount more then was extracted from the environment in the first place. Again, this is a very subtle argument, because the information needs to be erased irreversibly and that may not be as easy as it looks. Anyway, we will take it for granted now.

This then restores the second law before and after we recorded and erased the information, but what happened in the middle? Did we beat the second law for an arbitrary long time (the time information was recorded on a tape)? In a way yes, and physicist have been forced to change the definition of the second law to include the abstract (Shannon type) information of the sequence of 0's and 1's living on the tape. The new law says that the total entropy of the physical system plus that of the bit-string will not decrease over time. Hence, we have moved some of the entropy of the physical system onto a computer memory and froze it there. The boarders between the real physical world and the abstract world of computer science have been blurred.

This story re-enforces the opinion that physics can be understood as computation ("it from bit" according to Wheeler). But it also points to a potentially deep connection between intelligence (learning systems) and physics. The demon had to study the system and record the information in his "brain" before he could lower the entropy of his environment. But according to a Bayesian, the amount of information obtained by the demon has only meaning relative to his own subjective prior about the world. Is entropy then in the end a subjective quantity? There are many more indications that at least certain aspects about the world are rather subjectively defined (for instance the collapse of the wave function in quantum mechanics leading to the Schrodinger's cat paradox). Is there even an objective world out there? That will have to wait for another blog (;-).

Leaf Blowers (Suck)

Saturday morning 8:00am. The leaf blowers come. Now, we sleep with our windows wide open. So we wake up. Because leaf blowers make a lot of noise. We call that "noise pollution". Especially Saturday morning at 8:00am.

Now let's review the leaf blower. It moves leafs from A to B. In fact it moves leafs away from places where they belong, namely on the ground between your plants. Once, blown on a pile, they can be shipped off. Now, leafs make humus (degraded organic material in soil). Yes, they rot, but the rotting doesn't smell bad. It's natural (really). And as an additional bonus: it makes your soil fertile.



Leaf blowing removes leafs. So now we have to ship-in degraded, other organic material (that actually does smell strongly and is probably made of your leafs in the first place). We call that mulch. In summary: we use a very noisy machine (at 8am) that uses gas to then remove organic material which is then turned into mulch that needs to be shipped back in from far away, once again, using gas. How stupid is that, given that nature has figured out ways to do that all by itself.

However, what I experienced recently went even further. The leaf blowing humanoid was directing his all-powerful blowing machine into a tree (he was a tree blower). The reason was presumably that he wanted the leafs that had turned a little brown to fall off the tree early. As it turns out, trees have evolved to figure this out all by themselves; they do not need help with that (certainly not at 8am).

One more example of how we have alienated ourselves from nature. Half a year ago or so it actually rained in Southern California. I picked up the following phrase when entering a restaurant: "I am going to move back to the desert". Rain is good for plants. In fact, rain is good for people too. So don't complain if it rains; you only have to endure it about twice per year. Try living in Sudan (no rain) or Ireland (always rain).

I worry: when my oldest child got (consciously) confronted with rain for the first time, and I explained that rain made plants grow she said in surprise: "no daddy, sprinklers make plants grow". Yes, that was actually correct. 60% of our water usage is on watering our lawns. That means that we have to drain the Colorado river to the last drop before it hits Mexico, just to water our lawns (and wash our SUVs). Actually, we also drain the west side of the Sierra's Nevada's resulting in "salt storms" in those regions. So let's get rid of those lawns now and replace them with native plants (almost don't need any watering). Additional advantage: lot's of hummingbirds in your garden!

What Language Does Our Universe "Speak"?

Many profound physicist have come to think of physics at the very tiniest length scale (the Planck length) as a computer or information processor. To name a few: John A. Wheeler, Richard Feynman, Roger Penrose, Gerard 't Hooft. Wheeler expressed this view as "it from bit".

One of the main reasons for this view is the realization that physics at that scale will have to be discrete. If not, it becomes very hard to reconcile relativity and quantum mechanics into one theory. In the continuous domain calculation simply blow up: they cannot be re-normalized. In addition to that, the uncertainty principle of quantum mechanics demands that we can not even pinpoint things down to such precision without creating a black hole which would immediately render any measurement at a scale smaller than its horizon impossible....So these physicist think that physics at that scale is some sort of cellular automaton.

Around the end of every century some people seem the need to make the rather absurd claim that science is coming to an end (I believe we have barely started, but anyway). This century this view is expressed in the book: "The End Of Science: Facing The Limits Of Knowledge In The Twilight Of The Scientific Age" by John Horgan. He argues that there are four recent theories that have shown the fundamental limitations of science:
1. Relativity: anything inside the horizon of a black hole will never get out. So we cannot study the inside of a black hole.
2. Quantum Mechanics: the world is irreducibly random.
3. Chaos: The dynamics of many real physical phenomenon displays extreme sensitivity to initial conditions.
4. Complexity Theory: Godel's theorem of incompleteness of formal systems.

Let's see how these theories would fare in the face of a fundamental theory of "Physics as Computation" (PAC). I think the black hole issue is already close to being resolved. A quantum mechanical treatment of BHs will involve BH-radiation (or Hawking radiation). As such, in-falling matter will cause disturbances on the surface of the BH-horizon that encodes the information of the in-falling matter and which will eventually be radiated out again. No information is lost in the process. (Every BH will eventually die in an explosion that is more violent than the most energetic supernova, but it takes a while..) For the observer that stays outside the BH, the BH horizon is the edge of the universe in a very real sense. It will see his colleague that falls into the BH freeze onto the horizon, get disintegrated and eventually be radiated out again in bits and pieces. For the in-falling observer the edge of the universe is not the BH horizon, but a singularity at the center of the BH. In this case we have to deal with a singularity but it seems evident to me that the final PAC theory will describe that singularity not as an infinitely dense point but rather a sensible finite object.

How the irreducibility of quantum mechanics may be resolved in terms of a cellular automaton was described in my previous blog on "Quantum Mechanics is not the Final Theory".

The phenomenon of chaos in nonlinear dynamical systems makes claims on unpredictability of a more every day nature: for instance the weather patterns are unpredictable because a small error in the initial conditions may result in large differences a few days later (except in California where we don't need weather forecasting). The canonical example is this: x[t+1]=2*x[t] mod 1. This means that at every iteration we move all digits one decimal place to the left and set the number to the left of the dot to 0: 0.12345... <- 0.23456... What happens is that something of the order of 1e-10 will be of order 1 after only 10 iterations! Since we simply cannot specify initial conditions to that level of precision, we loose predictive power exponentially quickly. Now, what if the world is a cellular automaton? The issue only seems to be related to the fact that irrational numbers need an infinite string to be specified. In a discrete world no such things exist. Surely, we can have cellular automata with very different behavior. Some of them produce boring sequences while others produce random looking sequences. We can fruitfully think about the complexity of the sequences but predictability is never lost in a digital world. For example, Conway's game of life seems to have the right level of complexity between boring and random in order to support universal computation.

Finally Godel's theorem. It says that within any sufficiently complex formal system there will be true theorems that cannot be proved. I am still thinking about these issues, but I seem to have an issue with the notion of "a true theorem". True can only acquire meaning as an interpretation of the formal system (say mapping sequences to mathematical or physical "truths"). But mathematics is itself a formal system. Truth does not exist outside any axiomatic system and the interpretation that Godel's theorem shows that truth is bigger than formal reasoning just doesn't sit well with me. Anyway, some future blogs will unquestionably be devoted to these deep issues.

It will be very interesting to be able to answer the question: "what is the complexity class of the sequences generated by the cellular automaton that governs our universe". Or phrased more informally: "What language does our universe speak". Here is my prediction: Dutch ;-) (or maybe a language of the same complexity). It seems that Dutch is more complex than context-free languages due to cross-referencing but still decidable in polynomial time. It represents a possible level of complexity where things are not too regular but also not too unwieldy. Anyway, my prediction here should be taken with a huge grain of salt of course.


Soooo, the universe is a huge computer that is computing "something". It is our task as scientists to figure what and how it is computing. Actually, we already know the answer: 42 ;-). But what was the original question? Let's leave that to religion.

Complexity

In the previous blog I argued that a truly random sequence can (should) be defined as an incompressible one. A sequence from which nothing can be learned. Moreover, it can be shown (mathematically) that certain deterministic chaotic dynamical systems can indeed generate these random sequences.

If randomness is a slippery subject, complexity is worse. Complexity has turned into an entire research field of its own but is being criticized for not even having a good definition for the term itself. In fact, there seem to be 31 definitions floating around. Complexity is one of those things for which we seem to have a clear intuition but is hard to capture in math.

Let's try to build some intuition first. Consider the tree-image above and compare that to say a completely white image and an image of white noise (your television screen if it receives no signal). We feel that the white image is boring because it is utterly predictable but the noise image is also boring because there is nothing to predict about it. The tree-image on the contrary seems to interesting. There is lots of structure. We want our complexity measure to peak at the tree-image but vanish at boring images and random images. Therefore, complexity should not be equated to randomness.

An attractive definition was given by Murray Gell-Mann (Nobel laureate and inventor of the quark). I will discuss a slightly adapted version of his story. We return to the MDL principle (minimum description length). Given a string of symbols we wish to optimally compress it. That means, we want to build a model with a bunch of parameters, P, and use that model to predict most of the symbols in the sequence correctly. The symbols that were wrongly predicted will have to be stored separately. One can imagine a communication game where I want to send the string to you. Instead of sending the string, I send the model and only those symbols which where predicted wrongly by the model. If you use my model on your end to generate the predictable symbols you have all you need to generate the original string (use the model at the locations where predictions are correct but use the separate list of corrections where the model was wrong). The trick is now to find the model that has the optimal complexity in the sense that the total information necessary to encode the model and the unpredictable symbols is minimized. In other words, I want the procedure that maximally compresses the original string.

There is something fundamentally important about this procedure because it guarantees that the model will be optimally predictive of future data. If your model was too small, there was more structure that you could have predicted in the test string. If on the other hand you use a more complex model, you will have "modeled" the randomness of the input string (called overfitting) and this is obviously not helpful either. Living things survive because they are able to predict the world (and leverage it). It is our business to compress the input stream of our senses!

We are now ready to define complexity: it is the information necessary to encode the model (parameters), excluding the residual random bits of the string that remained unpredictable. A white image has very little information to start with and even though it can all be predicted by our model (no random bits) so the total is still very small. On the other, a random string is completely unpredictable and even though it carries a lot of information (a la Shannon) the model part is zero. Both cases have vanishing complexity as we wanted to, and the tree-image will have lots of complexity.

Complexity is a relative quantity. For instance, it depends on factors such as the length of the string. For small strings it doesn't pay off to use a complicated model. This is not so bad in my opinion, because imagine a primordial world where all that has ever been created is the string "GOD". The string are just three symbols embedded in nothingness. It will only acquire its structure (meaning) after we read the bible.

Another more serious objection is perhaps the fact that the true complexity is in-computable because it can be shown that one can not decide whether the model you have is really the optimal one in general. Imagine the fractal image below.
The image looks wonderfully complex. Who would have thought that there is a really simple algorithm to produce it? So, practically speaking, complexity is defined relative to how good of a modeler you are.

Then there is the issue of scale. If we view the world at a large scale, it may look very predictable. However, the world is composed of quadrizillions of elementary particles that are moving in chaotic motion. At the small scale the world is random, but at a larger scale order emerges through averaging. This may be best illustrated by going back to the coin-toss. If we are asked to predict the next outcome of a coin-flip, we will be at a loss. However, if we are asked to predict the total number of heads over the next 100 coin tosses we can make a good guess (50 for a fair coin). This is because even though every sequence of 0's and 1's is equally likely, there are many more sequences with 50 heads and 50 tails than that there are sequences with only heads for instance.

Finally there is the issue of the time that it needs to compress and decompress the string. We may have found a great model that does a wonderful job at compressing the input string, but that requires a very long computation time to use for encoding and decoding. This seems important at least biologically. If I have a model of lions that takes 5 hours to use (i.e. make predictions with) then it is not very useful in terms of my chances of survival. As another example, consider the Mandelbrot fractal again. There is very small prescription to generate it but one needs considerable time to actually compute that image (in fact, this connects to the previous point because you would have to first specify the scale at which you want to stop otherwise it would take infinitely long).

In machine learning researchers have also started to realize the importance of computation in addition to prediction quality. They acknowledge that modern day problems are so large that a really good model may require too long to train up and use for predictions. So, the computational budget is factored into the total equation, favoring faster algorithms and models over slower ones even though the latter may be more accurate (it is in this sense that machine learning is really different from statistics and in fact the reason why machine learning is in the computer science department and not the mathematics department).

Should we adapt our definition of complexity to accommodate running time? Gell-Mann thinks so and he argues for a bound on the computation time. There are indeed many definitions of complexity, such as "logical depth" that do just that. But if we include computational complexity, shouldn't we also include the amount of memory we need during computation (since they can be traded off)? Uh-oh we are descending in the bottomless pit call complexity research with its 31 definitions. This may be a good point to stop.:-)

Is A Coin Toss Random?

Randomness is a slippery concept. If you teach it in an AI class you are quick to point out that what we call random is really a matter of ignorance: because we don't know and/or model all the details of the world we lump together everything that is outside the scope of our models and call it a random effect. Under this reasoning the toss of a coin is *not* really random in a philosophical sense.

Unless of course, quantum mechanics has anything to do with it. Imagine we toss a coin through a quantum measurement. Now we are confident to claim true randomness. The quantum world is with a nice world "irreducibly random", meaning that there is no deeper theory from which quantum mechanics would emerge and where the coin toss is once again the result of our ignorance. However, as I reported in an earlier blog, there may even be ways out of the seemingly "irreducibly random" nature of quantum mechanics. In this view we are back to square one: what then is randomness?

There is a beautiful definition of randomness in a mathematical discipline called algebraic complexity. To understand this definition let's consider a very long sequence of coin flips. Now let's try to compress this sequence of "HTTTHHHTTHTTTTHHHTHTHTHHH" into a smaller sequence by building a model. If the sequence where just "HTHTHTHTH..." things would be easy: "I will simply state that head and tails alternate and the sequence starts with H (I will have to count the symbols in this sentence as the cost of encoding it). The sequence is not random because I could compress it. The definition of randomness is thus: a sequence is random if it cannot be described (encoded) using a smaller sequence. In terms of learning theory: one cannot learn anything from a random sequences.

Using this definition, a sequence can be random yet and at the same time be generated by a fully deterministic process. This process will have to be "chaotic" which we will not define here but has nothing to do with quantum mechanics. In fact, it snugly fits the definition of a random coin toss. For instance, if you toss a coin in a deep tank of moving water and record whether it landed head or tails on the bottom, then I am pretty sure that the details of the trajectory of the coin are chaotic. Hence, there is no model that can predict anything about the final outcome. This is not because we don't know how to model moving water in a tank, but because trajectories of coins in these tanks have something inherently unpredictable about them.

I therefore conclude that the flip of a coin toss can be truly random and deterministic at the same time. We don't need quantum mechanics for true randomness, we have something far better: chaotic dynamics. In fact, I predict that the so called "irreducible randomness" of quantum mechanics will in the end be reduced to randomness derived from chaotic dynamics.

On Intelligent Design

Consider the following reasoning I witnessed a while back at a parent meeting. Some trees are so tall that there is no scientific explanation for the fact that these trees can pump water from its roots all the way up to its top (I actually seriously doubts that there is no explanation but let's ignore that for now.) Therefore there exists a upward force that pushes things in the opposite direction as gravity...

What is the problem here. I claim there are two fallacies in the logic. Firstly, the fact that we currently don't have an explanation for some phenomenon in terms of ordinary physics, doesn't mean there is none. A scientist should readily admit that many measurable phenomenon remain unexplained today. It doesn't mean that we have to introduce a new mysterious force. Secondly, the "upward force" feels like a valid explanation, however we have just given the unexplained phenomenon a new name. Instead of saying "I don't know the answer to this question" we say "I know the answer, it is X". It turns out this is just another way of saying the same thing. Giving names to things generates the illusion that we understand it. And people really don't like to not understand things. Our survival depends on it, so it to give us peace of mind to simply delude ourselves.

Isn't this is even true in real science? Apparently, when Feynman asked his father about the reason why objects tend to persist in their straight motion, his father said something of the sort. That's because momentum is conserved. But he warned the young Feynman that giving something a distinguished name isn't the same as explaining it.

When does something become an explanation then? Isn't all we are doing just giving names to things? Well no, the distinguishing factor is prediction. When you can genuinely predict a new phenomenon, then you have found useful structure in the world. Whatever your naming conventions, that predictive power is useful and real.

Now to intelligent design (ID). In ID, the most scientific concept seems to be "irreducible complexity (IC)". This idea should be given a chance. If someone can compute the IC of some complex system as the gap in complexity between the system under study the next simplest version of the system that still performs some useful function, than that seems like a genuine advance. Note though that this is extremely difficult because one would have to search over all possible complex systems that can lead to the system under study and one would have to have a notion of what it means to be useful. I believe these problems basically make the concept of irreducible complexity a non-starter, but we want to be extremely open minded here.

Where things go desperately wrong is when we think we found a system that has a high irreducible complexity and then exclaim that therefore it must be designed by an intelligent designer (aka God). This is like the tree example: we have shown that contemporary science cannot explain the phenomenon and therefore we give the problem another name, namely God. However, stating that God designed certain things in the world does not make us understand and thus predict the phenomenon any better. Our attitude should be: 1) maybe Darwin's version of evolution needs to be improved to properly explain structures with high irreducible complexity, or 2) perhaps our estimate of the irreducible complexity was wrong and we need to look harder for simpler functional structures that could evolve into the "problematic" structure.

Personally, I am very tolerant to religion. It brings some good and some bad things to people, but everyone should make up their own mind on these issues. It would be arrogant to claim there is no God. My hope is that we can teach religion in schools on an objective level. I favor teaching all religions to our children, not just one. However, I do not believe science and religion should be mixed. They live on different planes. Trying to prove the existence of God is a lost battle. Trying to prove that someone performed three miracles and then declaring that person saint is simply silly (at the level of Santa Claus). Declaring the existence of God based on the fact that you (supposedly) cannot explain some natural phenomenon using modern science is equally backward.

Jared Diamond's Psychohistory of the Human Race

In his Foundation Trilogy, Isaac Asimov describes a new kind of science: "psychohistory". The idea of this discipline is to predict the course of human history through mathematical analysis. The fundamental assumption is that the impact of individuals is "washed out" due to the law of large numbers: if you have enough elements it is only their average behavior that counts. The same idea underlies thermodynamics and statistical mechanics: if we have a *very* large number of individual atoms behaving chaotically, we will have no chance in predicting properties of individual atoms. However, new emergent properties such as temperature and pressure are predictable.

Humans are no atoms of course, and although in Asimov's universe there are many more humans to average over than the 6 billion that live today, the appearance of "The Mule" does cause a breakdown of the predictions. Interestingly, one can relate this idea to recent insights in physics and mathematics that societies are in a state of "self criticality". At its core this means that small disturbances can have large consequences that propagate through the entire system. To give an example from human history: the invention of the atomic bomb by a few scientists changed history radically.

Despite these potential objections, Jared Diamond's book "Guns, Germs and Steel" is the best psychohistory of the human race I have read so far. In fact, personally I find this book the best popular scientific read on my list, right next to the "Selfish Gene" by Richard Dawkins.

What are Diamonds claims? He claims that despite the sometimes large impact of individuals there are also very important and predictive regularities of human history that depend on geography. Why did the European colonist basically wipe out the native Indian population in America and not the reverse? The core reason, so he argues, lies in the fact that massive food production was first invented in the Fertile Crescent (the Iraque, Iran region). The climate was ideal for many species of plant and animal to become domesticated. Moreover, the East-West orientation of Eurasia made it easy for inventions to spread to Europe (and as far as China, although China seemed to have invented food production around the same time). Food production made it possible to switch from a hunter-gatherer life style to a farming lifestyle which in turn made it possible for many people to specialize in other things than farming. This way, larger cities and states started to emerge with a specialized fighting caste. These "successful" states then spread either by conquest or by simply producing more offspring.

But surprisingly, even more important than having powerful new weapons was the fact that cattle generates diseases and thus living among cattle causes a population to become resistant to lethal diseases such as small pox, measles and so on. (of course the price paid for this resistance was a high death toll because for a population to become resistant an aweful lot of cruel "selection of the fittest" will have to take place first.) In the America's, food production was much tougher due to climate issues and due to the North-South axis which prevented effective spreading of inventions. Also, very few large animals were available for domestication (perhaps they were all killed when the first people started settling a long time ago). So when the Spanish arrived they did not only have superior weapons and administration skills, they carried many more lethal diseases with them as well.

Diamond goes on to tell the amazing story of many continents: how the Polynesians were former Chinese evicted from the main land, how the aboriginals from Australia were decimated by Europeans, how the Buntu people spread over much of Africa by using superior farming practices. If you need to remember one thing it is this: it is massive food production that has caused all the major migrations and colonizations of populations. And perhaps equally important, the fact that one civilization ended up dominating another has nothing to do with race, it has to do with geography.

So it seems some form of psychohistory is still possible, even though we are averaging over tiny numbers compared to the numbers Asimov had in mind. I warmly recommend this book to anyone who is even vaguely interested in how the world has become what it is today.

Quantum Mechanics is not the final answer.

The first thing I was told when starting my course on quantum mechanics (QM) was: "if you think you understand QM, you don't understand QM". It turned out to be true. I could master the rules of the game, but things always struck me as fundamentally weird. Information is carried by wave-functions that would collapse into definite states when you decide to look at it?? "Oh well, that is because we were not evolved to think about the very tiny" I kept telling myself.

For decades very few serious physicist would dare to propose something more interpretable and fundamental than QM. The main reason is the Bell inequalities that say that QM can not be explained by a deterministic hidden variable theory, i.e. a more fundamental, deterministic, causal and local theory at a smaller scale that would give rise to QM at a larger scale. It turned out that in order to do that one would need a non-local theory or a non-causal theory.

Now all of that seems to change. It takes a genius with a Nobel prize at the end of his career to stick out his neck. Gerard 't Hooft from the university Utrecht has started a lonely uphill battle to come up with, yes, a deterministic hidden variable theory of QM. So how does he propose to deal with the Bell paradox? The gist of the argument is that QM is an emergent theory, in a similar way as thermodynamics is an emergent theory of many particles that move chaotically. The concept of pressure, temperature, entropy etc. only make sense if we are talking about the average behavior of very many particles that move about in a way that is unpredictable at the level of an individual particle. However, treated as a group new structure emerges. That is what we call thermodynamics.

In a similar sense, QM can emerge from a more fundamental, deterministic and interpretable theory at a smaller scale. A first draft of such a theory is based on the idea of cellular automata. Basically, imagine one can split the world up in small boxes and every box can be in a certain state. Time evolution of a particular box is based on rules that generate a new state (deterministically!) as a function of the states in its direct neighborhood. However, there is a twist. New states are created on two-dimensional surfaces such as the horizon of a black hole. The theory of nonlinear dynamics states that "information" can be created using what is known as "chaos". This basically means that any information about a system, e.g. where the constituent particles are located, is lost very quickly under the evolution of the system. In 't Hooft's universe information is lost in the three dimensional interior. This happens because two different states can evolve into a single state. This idea, that the information of a universe is encoded at two-dimensional surfaces is known as the "holographic principle", another brain-child of 't Hooft.

So how then do we beat the Bell inequalities? Here is the intuition. The large scale theory known as QM defines variables that are functions of the more fundamental states. However, the way that this works out according to 't Hooft is that a QM state at time "t" is an aggregation of all the states that will eventually evolve into a single state. This definition however is non-causal because it involves knowing which states in the future will collapse into a single state. Hence while the fundamental theory is causal and local, the emerged theory at a larger scale can exhibit strange features because the variables defined by us mix up the present and the future.

I may have missed a point or two in trying to translate these ideas, but I certainly think this is a very exciting new development. Einstein may turn out to be right after all in saying that "God does not play dice". These bold ideas definitely give me a sense of relief that it was OK to feel dissatisfaction with the strangeness of QM. Between string theory and deterministic QM, I will bet on the latter. Has 't Hooft ever been wrong?

His Holyness the Pope

I cannot say that I am a very religious man, although under the influence of my wife I do start to appreciate the teachings of Buddha. I am much less inspired by the current pope though. In fact the Vatican's history is one littered with crime and murder: witch burnings and torture by its inquisition. On a lesser but still significant scale: I found it "curious" to see President Mugabe being invited at the funeral of Pope John Paul II and to see the Vatican in a state of denial when it concerns its cowardice actions in the second world war. How does anyone embedded in such a history claim any moral leadership?

Let's have a look at today's pope Benedictus XVI. Recently he declared the use of a birth-control pill immoral apparently because it pollutes the environment. More damaging, he reiterated the churches condemnation of the use of condoms and he flatly denies that it helps in the fight against AIDS. This was contradicted by the Lancet recently. Then there is his unfortunate decision to reinstall bishop Richard Williamsen who thinks the holocaust is being highly exaggerated. This is now reverted after much critique. He also seems to lobby for sainthood of pope Pius XII (who had a very doubtful role in dealing with the nazi's in the second world war). The current pontiff also believes that homosexuals should be "cured" and protected against self-destruction.

What was the church's attitude when cases of child abuse in the US started to surface? A comprehensive study finds that child abuse of some form happens in as much as 4% of priests. I would have expected that the only morally justifiable action is to immediately remove these priests from the church and let the law bring them to justice. No,in many cases the priests were allowed to move to another parish and keep practicing there. In fact, pope Benedictus seemed to be involved in a secret document that instruct how to cover up child abuse, putting the churches' interest always ahead of child safety. I cite: "The document recommended that rather than reporting sexual abuse to the relevant legal authorities, bishops should encourage the victim, witnesses and perpetrator not to talk about it. And, to keep victims quiet, it threatened that if they repeat the allegations they would be excommunicated."
For more reading see this Wikipedia article.

The Vatican has enormous power over people, and its not helping to solve the challenges of this world.

I like to end with a positive note. At least the Roman church inspired (well, perhaps just pay) artists such as Michelangelo to make beautiful art...

The Infinite Universe that Started as a Point

The beginning of our universe is shrouded behind a huge paradox. Recent observations seem to indicate our universe is open. Which basically means that it has infinite volume. Yet, and here comes the paradox, it started in a single point: the Big Bang.

Sure enough there are alternative solutions to the equations. It could have a closed universe. It's much easier to convince oneself that a closed universe can start at a single point. Imagine you have lost 1 dimension, so you live on a sphere. (there is no such thing as space outside of this sphere: you have become 2-D yourself.) At the moment of the big bang the sphere come into existance and expands fast. whereever you are on the sphere, if you look around you you see other galaxies move away from you. But, since the volume of this space is finite, it's not hard to imagine that at some point in the past it was zero, which was the time of the Big Bang.

Alas, observations say we are not in a closed universe, but in an open one. Logic seems to demand that if space is infinite now, it must have been infinite when the universe got created (unless you believe there was a time after the Big Bang that it suddenly became infinite). It was filled with matter everywhere, and moreover, this matter was expanding outward. In fact, it doesn't matter who you would ask, everyone would tell the same story: the universe is exanding outwards and galaxies that are far away are expanding faster than the galaxies that are close by.

I find that paradoxical. Yet there is a way out. It is described very well here. The basic idea is this. You can switch to another frame of reference. In this view the universe has finite volume and you can visualize it as an expanding sphere again. However, galaxies are packed into this sphere in a peculiar way. Due to the so called Lorentz contraction, distances become contracted for galaxies flying away at very speeds. Since the galaxies at the outer edge of the expanding sphere fly at lightspeed, radial distances are infinitely contracted there. And so we can pack infinitely many galaxies in an infinitely thin slice of space.. In fact, because there is infinitely mass at that rim, it forms a sort of wall (a singularity) beyond which there is simply nothing. So it is not useful to think of this sphere as expanding inside something else. Spacetime is not defined outside this wall. Btw note that you can not visit this wall to peek over it, because it is receding at the speed of light. Also, there is noone really at this edge because the picture looks the same for everyone!

Ok, we have now packed infinitely many galaxies inside a finite volume. Why is space then infinite? For that we need to realize that not only does space contract at high speeds, it also slows down time. So watches on those distant galaxies are moving at a slower rate relative to your watch. Again, this situation is perfectly symmetric: people on those distant galaxies don't notice their watches going any slower (that's because their brains go slower too, so we reason) and moreover, they observe our watches going slower too. Now, we introduce a new cosmic time. This is the time where every observer in the galaxy sets their watch at 0 at the Big Bang and takes their watch with them when they fly outwards. If we now measure the volume of space defined when all these watches display some constant time space becomes infinite. This is because time is standing still at the rim (according to us) and so the infinite space contraction and the infinite time dilatation cancel.

It's head-spinning, but the conclusion is that space was created infinitely large in one point.