Like a Swinging Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum’s mass causes it to oscillate about the equilibrium position, swinging back and forth.

The time for one complete cycle, a left swing and a right swing, is called the period. A pendulum swings with a specific period which depends on its length. The equation of motion is:

In order to solve this equation, some assuptions are needed. If the initial velocity is small, the angular displacement is small as well. Expanding we find:

The general solution for the harmonic oscillator is . Applying the initial conditions and we get:

It is important to note that the period of each oscillation is independent of the mass of the swinging body. This property, called isochronism, is the reason pendulums are so useful for timekeeping.

If the initial velocity is not small, the error in the Taylor expansion of grows larger and larger. Using advanced mathematical methods, or the conservation of mechanical energy, it can be proven that the period is given by:

where is a complete elliptic integral of the first kind, that can be series-expanded as shown above.

Newton’s Laws of Motion

1. First law: There exists at least one reference frame in which the velocity of a point particle remains constant unless the particle is acted upon by an external force;
2. Second law: The acceleration of a body is parallel and directly proportional to the net force and inversely proportional to the mass m, i.e., ;
3. Third law: The mutual forces of action and reaction between two bodies are equal, opposite and collinear.

The three laws share some features. For instance:

• The first law can be seen as a special case of the second if the acceleration is null:
• The third law for point particles is
• Non-instantaneous acceleration is given by a force integrated in time: (impulse)

 

Newton’s laws are valid in inertial frames only, defined by the First Law. The simplest inertial frame is the one given by the fixed stars (which, btw, are not fixed after all!). In non-inertial frames the force is given by a much more complicated expression:

• 

where is the angular velocity, is the position of the body, is its velocity. The “fictitious” force adds extra terms to the “newtonian” force:

• The Coriolis force ;
• The centrifugal force:
• The Euler force:

Vectors and planes

Date of the lesson: 31 Oct 2011

In order to solve exercises involving vectors, trigonometric identities come very handy. For instance:

And don’t forget the prosthaphaeresis formulae!

One important relationship is the Law of Sines:

Summing vector is easy as long as you project each vector along the coordinate axes. Five steps north, two steps east, three steps south…

How to fly a plane from Bologna to Prague:

http://www.algebralab.org/Word/Word.aspx?file=Trigonometry_ResultantsDotProducts.xml

Wanna try to land your cessna airplane? Click this link for a nice flash game. Why don’t you code an android applet for you mobile phone instead?

Exercises

1. Calculate and , where: ,
2. The resultant vector has length . The angle between and is . Knowing that the angle , find the length of and .
3. Calculate the angle between ,
4. For the die-hard: calculate the area of the triangle defined by vertices A(-1,2,3) B(2,-1,1) C(1,3,2)

Integrals

Calculus in the ancient times: Method of exhaustion (see wikipedia)

User-friendly definition of the integral:

It would be better if the integral would not depend on the definition of the bins. In the limit it doesn’t matter if they are evenly spaced or not:

Fundamental theorem of calculus:

If is a continuous real-valued function on a closed interval

then is continuous and differentiable in and:

as a corollary:

Exercises

(tip: by parts)

Try to solve a more general formula:

(get the solution using Wolfram:alpha )

For the die-hard:

( solution )

tip: make a substitution deplyoing this relationship:

Even and Odd Functions

Even functions: e.g.

Odd functions: e.g.

See also wikipedia

This explains why in the previous post:

Find out what this function looks like using Wolfram|Alpha (click this link)!

Derivatives

Introduced in order to fix the idea of velocity:

See: Zeno’s paradoxes

Working example:

Which is exactly .

 

Enjoy your derivatives

and

And for the die-hard:

You can check them using mathematica or sage (sage is free).

Useful Links

Mathematica for students http://www.wolfram.com/mathematica/how-to-buy/education/students.html

Wolfram:Alpha http://www.wolframalpha.com

Khan Academy http://www.khanacademy.org

The Hunt for the Higgs Boson

Forewords

The hunt officially started on Oct 19th, 1964 when Peter Higgs published a paper called “Broken symmetries and the masses of gauge bosons” on Physical Review Letters (Vol 13, No. 16). Nowadays, his brilliant explanation about the origin of the mass of elementary particles is being tested in two laboratories: at Fermilab, 80 miles south of Chicago (USA), and at CERN, in the outskirts of Geneva (Switzerland).
Friday Jul, 22th marks an important date on the long quest for the Higgs boson.  At the EPS conference in Grenoble (France) the four main experiments in particle physics gathered for a showdown. Has the goddamn boson finally cornered?

The Battlefield

The most successful theoretical explanation of known phenomena in particle physics is called the Standard Model(SM).

Particles of the Standard Model. Matter is made of fermions, grouped in three "generations". Forces are carried by bosons. Mass is given by the Higgs boson

It is a Model, because it tries to make testable predictions but keeps away from making firm statements about the reality itself. And it is Standard because it is widely accepted from the scientific community. In fact, in the last decades, its predictions have always been verified within experimental uncertainties. Anyway, physicists find this model unsatisfactory. Too many free parameters, too many ad hoc assumptions (“it just works fine“) and, last but not least, it makes the worst prediction ever in physics! (see: cosmological constant and vacuum catastrophe)

At the core of the Standard Model lies the concept of field. A field is a an “entity” which is defined everywhere in the spacetime. For instance, weather forecasts show temperature and wind fields across the Country. In our case, a field is associated to every kind of elementary particle: electrons, photons, quarks and the like. Fields are classified in two groups according to an intrinsic property called spin: when measured in units of “reduced” Planck constant h, integer-spin particles are called bosons, and half-integer spin particles are called fermions. It turns out that matter is made of fermions (electrons, quarks) while forces (electromagnetic, strong, weak) are mediated by the exchange of bosons among fermions. Not all the particles are elementary: for instance, the proton is made of three quarks soaked in a sea of gluons and other, fainter quarks. At the LHC, these constituents of the proton interact among each other. We aim to understand the underlying physics looking at what comes out of these high-energy accidents.

Since a long time, experiments show that there are some quantities that are conserved through interactions, such as the sum of all electric charges of a system. To account for this fact, the mathematical description of the fields is scrutinized in the search for invariant (=preserved)  properties. This idea is encoded in a theorem due to Amalie Noether (she was among the first women who taught in a German University in 1915!). According to this theorem, to every symmetry there is an associated conserved current. In turn, thanks to Gauss’ theorem, for every conserved current there is a conserved quantity that we call charge.

In the simplest case, a symmetry called U(1) explains the origin of the electric charge, a property owned by several fields (e.g. electron, quark) but not all. Add Einstein’s Special Relativity and voilá you get a description of the electromagnetic force. A more complicated symmetry called SU(2) gives rise to the so called weak isospin, and its related charge T3 explain why in nuclear beta decays only certain types of particle can be created, and not others at will. Those decays are due to the weak force. Finally, the SU(3) symmetry introduces the color interaction, whose by-product is the strong nuclear force that binds protons and neutrons together inside atomic nuclei. It is so strong (hence its name) that it even overcomes electrostatic repulsion. As the photon mediates the electromagnetic force, so the gluon mediates the strong force. The main difference between the two is that the photon carries no electric charge, and so there is no photon-photon interaction. On the other hand, gluons alwasy carry a color charge: gluon-gluon interactions can happen. This fact brings funny things inside the proton, and it is the main ingredient for the physics at the LHC!

Finally, some theoreticians believe that some other symmetry could explain the gravitational force. Its mediator would be a spin-2 boson called graviton. So far, nor this symmetry nor any graviton have ever been seen (or is it?).

Breaking Rules

Why so many interactions? In 1865 Maxwell published “A Dynamical Theory of the Electromagnetic Field” in which he first proposed an explanation of light as the result of the unification of electricity and magnetism. You can still buy yourself a copy of this book on Amazon! Try to imagine what a breakthrough it was: for centuries, electricity, magnets and light was seen as three completely different phenomena.

Following this school of thought, Salam, Glashow and Weinberg proposed the unification of electromagnetic and weak interactions in what we call now electroweak force. If blending electricity and magnetism explains the origin of light, what could show up merging electromagnetism with the weak force? A long-time unresolved puzzle concerns the typical length of the interactions. While light has an infinte range of actions (photons arrive to Earth from stars millions of light-years away from us!), weak interaction seems to be very short-ranged. A possible explanation would be that the force is mediated by some kind of heavy photon, that fades away in a very short time (a billionth of billionth of a second). As a joke, this is sometimes referred to as “heavy light”!

A good hint came from the theory that explains why we can make permanent magnets with some metals, but not with others. Every atom can be seen as tiny magnet. When all the magnet axes point toward the same direction, the overall effect is enhanced and becomes macroscopic. Otherwise, if chaos rule, no net effect can be seen in the material. Scientists discovered that there is a temperature above which this property can be imposed, using an external magnetic field. An interesting effect is the discovery that Earth’s magnetic field reversed its polarity several times in the past!

Although it seems irrelevant to the problem of the electroweak unification, it turned out that the same mathematical equations can be used to explain a number of phenomena in particle physics. First, remember that in order to rise the temperature of an object, you have to give it energy, as a kettle on a gas stove. In our case, temperature is proportional to the energy of the interacting particles, the constituents of the proton: quarks and gluons. So, at extremely high temperatures, only the electroweak force exists. Mathematically, it can be described by a symmetry called U(1)×SU(2). When the temperature gets lower (e.g. at room temperature), if this symmetry gets “broken” for some unknown reason, we can see two completely different forces: electromagnetism and weak force. So the question is: what breaks the symmetry?

Enter: Peter Higgs

Well, the answer is not so simple but one man suggested a minimal solution.

Mexican hat potential. Rolling downhill triggers a phase transition

Peter Higgs

Some years before, Landau and Ginzburg proposed a model to exaplain superconductivity, the lack of electrical resistance shown by some materials when the temperature drops under a certain threshold. In their model, the energy of the system is described by a quartic function whose unique minimum is trivially zero at room temperature. Wherever the energy reaches a minimum, the system is said to be in equilibrium.

Superconductivity can appear if the temperature gets low enough: in this case other, more stable minima appear (the function, with properly adjusted coefficients, is called mexican hat). It looks like a ball sitting at the top of a hill: even a slight kick (in this case given by quantum fluctuations) makes the system to go downhill. Physically, this triggrers a phase transition. Such a process is sometimes dubbed spontaneous symmetry breaking.

Peter Higgs worked out how to make use of this process in order to explain why the electroweak force is broken at room temperature.

First, let’s suppose that everything in immersed with a scalar field whose energy looks like the mexican hat. Some of the particles are drenched with this field, other don’t. The ones who do, have weight. The ones who don’t, are weightless. When the mediators of the electroweak force “eat” this scalar field they acquire weight and the symmetry is broken. The leaftover crumbles are the Higgs bosons. Two birds, two stones: this explains why the weak interaction is short-ranged, and why we do not see the unified electroweak symmetry in everyday life! Anyway, it is important to stress that this is just one explanation of the phenomenon. Nature could have chosen a more complicated one, involvin for instance more than one scalar field. Only experiments can wipe out wrong theories.

Merging all togheter, theoreticians predicted the existence of 5 particles: the already-known massless photon (which is in fact…light) plus four massive bosons, never seend befor. They were the three mediators of the weak interactions (W+, W- and Z), and one Higgs boson (H).

The Z boson has been discovered at CERN in 1973 by the Gargamelle experiment (now on display in CERN’s garden). In 1983, again at CERN, the UA1 and UA2 experiments finally catched the charged W bosons. Salam, Weinberg and Glashow were awarded with the Noble prize in 1979. Rubbia and Van der Meer shared another Nobel prize in 1984. The Standard Model was firmly established in 1995 when the top quark was eventually discoverd at Fermilab.

So only one tile in the puzzle is missing.

The Quest for Go(l)d Particle

As the water flows downhill from its spring to the sea following different paths, so a particle can decay through different “channels” until stable grounds are reached. To our present knowledge, only photons, protons, electrons and to some extent neutrinos are stable. All the others crumble down unitl stable particles are eventually produced. The decay process is not completely arbitrary: it must fulfill a set of rules described by quantum mechanics (QM). We can use these rules backward and make an educated guess of what is more likely to have produced the particle we observed. The sum of all the probabilities must give 100%: this fact is so important that it even has got a name: unitarity.

Anyway, with QM we can calculate the probability that a path is followed, but which one is actually taken is a matter of chance and is intrinsecally unpredictable. If you toss a quantum coin you have a 50% probability of getting head or tail. Apparently, Nature prevents ourselves from predicting exactly which side of the coin will show up.

So it is for the Higgs boson. The first thing we want to do sounds quite simple: just count how many of them we see! The quest is twofold: theory predicts a number of different way to produce it, and several decay paths it can follow. What is actually predicted is a quantity called cross-section. It can be loosely interpreted as the probability to generate a desired particle given certain conditions. This probability depends strongly on the type of particles you collide, on the energy of the collision and on the mass of the particle you aim to produce. While you can control the first two using different kinds of accelerators (e.g. Tevatron or the LHC), the third is set by Nature. In our case, we don’t know which is the mass of the Higgs boson: at the end of the game, we will discover its value by inference.

Once you managed to produce some Higgs bosons (congratulations!), you still have to calculate the chance to see them decaying into each possible channel. Most importantly, it turns out that the probability of each channel depends on the mass of the Higgs boson. But this is not the end of the story: the Higgs is likely to produce intermediate particles, which in turn decay until stable ones are generated. A definite set of stable particles describing the end of a decay chain is called final state. Some final states are considered extremely appealing from an experimental point of view: they are easy to study and so they are dubbed golden channels.

A quite strong assumption is that the production and decay probabilities are independent of each other. This means that daughter particles have only a loose memory of their parents. If you think about it, a lot of animals behave this way! So far, it always proved to be true for all know particles: there is no reason to believe that the Higgs might have a different opinion. It cannot say “I accept to decay to a couple of Z bosons, but only if they promise me they will decay into four charged leptons!”. But you never know…

Can you see the peak due to the decay of the Ypsilon meson?

Unfortunately, the Higgs boson shares its possible final states with other less interesting processes. In jargon it is said that what you’re looking for is the signal, while all the other stuff is the background. Most of the intersting signals are very tiny, and most of the events are rubbish. The proportion is usually less than 1 in one billion.

Being an experimentalists consist mostly in trying to extract signals out of the background. There are many techniques to do it. The simplest one is to apply “cuts“: final states are sifted and only the very interesting ones end up in your histograms. If your analysis is set up correctly, some peak should rise above the background. For the die-hard nerds, more advanced techniques involve neural networks and other pattern recognition techniques.

Pushing to the Limits

Now that you have all the tools of the trade, you can join the Search for Gold. Pick your preferred final state, calculate its probability as a function of Higgs boson’s mass and start looking for it digging into data. Some suggestions are:

• A very light Higgs is more likely to produce two photons, or two b-quarks, or two tau leptons;
• A moderately heavy Higgs could produce a couple of W bosons. In turn, the can decay into two pairs of charged leptons and neutrinos;
• A heavy Higgs could decay into two Z bosons. The final state with four charged leptons is the golden channel;
• An extremely heavy Higgs breaks unitarity so it’s not believed to exist!

Do not jump into conclusions too fast! The golden channel has a very clean signature (few processes end up with four charged leptons!) but it is very very rare and it works fine for a heavy Higgs. Other channels have a higher probability in some mass ranges, but they suffer from a larger background – so it’s harder for you to teach your computers how to recognize this particle!

If the Higgs boson really exists, with an infinite amount of data we would be able to see it in every channel. However, with a limited number of events we must play smart. The key to success in the search for the Higgs boson is in the combination of all channels. This is done using statistical procedures that take into account what is shared among the channels, and what is specific of each one. At the end of the day, one hope to see a combined signal rising above the background. But what if we see no signal at all? Again, do not jump into conclusions too early. The calculation of the cross-section, along with machine parameters such as luminosity, center-of-mass energy and detector efficiency tells you how many events you must collect before you can say Eureka.  If you have acquired enough data and you still have seen nothing, perhaps the particle you are looking for does not exist at all! Othere possibilities are that its production rate is lower than you thought, or it’s just that you have been unlucky: a downward statistical fluctuation produced less Higgs bosons than expected and you just happened to see none. In these unfortunate cases, what’s done is to set limits, which is to say: “I collected a certain amount of data, but the signal did not show up. I conclude that its strength s* must be f times smaller than that as predicted by the theory s_th“. In order to take into account statistical fluctuations, f  is presented along with an uncertainty. This range of possible values gives you a confidence level or credible interval. Their meaining is roughly this: “The strength of the signal has an unknown but fixed value s*. If I could run an infinite number of experiments, 68% of these would say that s* lies within the quoted uncertainty“. Most common intervals are 68% (stringent), 90% and 95% (loose) confidence levels. If you test your hypothesis about f as a function of the mass of the Higgs boson, you obtain a graph that looks like this:

Combined ATLAS limits on Higgs mass (EPS conference)

The observed data are tested against two hypotheses for each mass point: background-only and signal+background. Before taking data, one runs the simulation for the background-only scenario, obtaining the dashed curve, surrounded by it uncertainty (the green-yellow “Brazil band”).When the dashed line drops under the horizontal level (fraction = 1 that means s* = s_SM ) theory predicts that you are supposed either to find or not the signal. Once these data are taken, limits on the signal strength are calculated (the solid black line). If there is no signal at all, the black line is supposed to be very close to the dashed line. If this is not the case, something is going on! The more the two lines are separated, the higher the probability that what you are seeing is not due to a statistical fluke!

So, what do you see in this graph? The black line drops under f=1 between 155 and 190 GeV. We saw no Higgs there. However, there is a broad excess between 120 and 150 GeV (the black line is still over f=1, but it is clearly outside the yellow band). How to interpret this observation? At the moment, the only thing that we can say is that experiments seem to contradict the background-only hypothesis. Whether it is consistent or not with the presence of the Higgs boson is a matter of debate. The statistical significance is low (it’s still too close to the yellow band to tell), but at the end of the 2011 there should be 5 times more data to be analyzed. The answer is near!

Conclusions

After almost 40 years we will see if Peter Higgs’ explanation of the electroweak symmetry breaking is right or wrong. Either way, this answer will be a turning point in the history of particle physics. For some theoretical reasons and astrophysical observation, the Standard Model is already in dire straits. Understanding what lies beyond is LHC’s business.

The Computational Universe

The Laws of Nature

When you first encounter physics at school, you are initiated to the common idea that at least in principle every phenomenon could be explained if we knew which are exactly the “Laws of Nature”. This idea is pretty simple, but not free from conceptual problems.

First of all, you are assuming that there is a finite set of mathematical relationships that for some reason has been selected among all the possible ones. For instance, Albert Einstein believed that in the end one and only one simple and beautiful Master Equation is meaningful. There is no other way: with this equation you can describe every possible process in the only possible Universe. However, string theory suggests that there could be a “landscape” of possible set of Laws, each set resulting in a particular Universe. Is there any selection principle out there that excludes all but one? Or are we in just one of the possible Universes? This question suggest that there could be other Universes out there (perhaps stemming one from another), each one being the implementation of a set of Laws. All together they form the so-called Multiverse.

Seth Lloyd pointed out that these Laws, at least how know them at present, are based on calculus. Calculus is applied on real numbers, and real numbers have infinite precision. There are good reasons to doubt that an infinite precision could be obtained in our Universe even in principle, and the key point is: Information.

You can associate information to every particle. It is estimated that there are some 10^90 particles in the Universe, thus the maximum infomation contained in the know reality is about 10^120 or 2^400 bit. Indeed a very large number, but not infinite. Even Richard Feynman was puzzled about this: how comes that Nature calculates Feynman graphs with infinite precision if there is an infinite number of graphs for every interaction that occours?

Digital Reality

One possibility is that there are no Laws of Nature at all: they dynamically arised when the Universe came into being. You can do something similar with your computer (or even a sheet of paper!) using cellular automata.

Cellular Automaton following Rule 110 in the Wolfram classification

First, you need to define an N-dimensional grid (one or two dimensions represent a good choice to start with). Every cell can have one out of two possible states: on or off, 1 or 0 (this represents 1 bit of information!). Then, you decide a set of rules. At each iteration the status of all the cells is updated according to these rules, and the status is a function of the status of the surrounding cells at the previous iteration. The rules are intended to be a function of the status of the surrounding cells at the previous iteration. In 1D, each cell has 2 neighbors, in 2D it has 8 neighbors, in 3D geometric it has 26 neighbors (in general, it holds the relationship N(D) =3*N(D-1) +2 ).

Stephen Wolfram

The simplest grid is made of just 1 spatial dimension plus the time dimension. Even in this case you can reach an incredible level of complexity, such as in the case of “Rule 110” in the Wolfram classification. Since there are 2x2x2=8 possible binary states for the three cells neighboring a given cell, there are a total of 2^8=256 elementary cellular automata, each of which can be indexed with an 8-bit binary number (in our case 110_10 is written in binary 1101110_2). Rule 110 is of particular interest since it has been shown to be computationally universal, i.e. it can simulate any Turing machine in polynomial time. Moreover, Rule 110 has “class-4” behaviour, producing non-repetitive nor completely random patterns.

If you take a closer look, the patterns look like particles in a bubble chamber, travelling through a foamy medium and occasionally interacting among each other, sometimes producing new patterns that look like new kind of particles.By the way, since the rules are forward-only, no travels back in time are allowed and the grandfather paradox is easily solved.

With two dimensions, even new patterns appear. Probably the most celebrated one is the “Game of Life” by John Conway, in which a large set of “objects” show intrinsecally complex behaviours such as rotations, emssions and pulsations. And the rules are simply:

1. Any live cell with fewer than two live neighbours dies, as if caused by under-population.
2. Any live cell with two or three live neighbours lives on to the next generation.
3. Any live cell with more than three live neighbours dies, as if by overcrowding.
4. Any empty cell with exactly three live neighbours becomes a live cell, as if by reproduction.
   

   John Conway's Game Of Life

Could it be that the Universe is an enourmous cellular automata running on a massive “supercomputer”? It is suggested that each cell would be as large as a Planck lenght squared, which is the smallest meaningful length in our Universe: lp = 1.616252(81) x 10^{−35} m, and each iteration happens  every Planck time tp = 5.39124 x 10^{-44} s (think of it as a quantum of time). The number of cells in the Universe would be atonishingly large, and so would be the computational power of the hypothetical supercomputer that should be able to calculate the status of all these cells in a while. It is worth remembering here that quantum gravity effects are expected to appear for interactions happening at a scale in the order of the Planck length.

One interesting point is that we can interpret every pattern of cells as a kind of particle. In this paradigm, interactions are forced to be local, since they are defined by the nature of the Rules: the status of a cell depends only on the status of the surrounding cells. This has an interesting corollary: there must exist a maximum velocity given by a cell with status 1 that in the next generation makes a step forward (or in diagonal). Perhaps a pattern with this property could represent a physical particle travelling at the maximum speed, i.e. the speed of light. In fact, this velocity does not depend on the velocity or the direction of the pattern, but is an intrinsic property of the Game, as specified by the Rules. It is suggested that the physical space could somehow arise from the computational one. The existence of a maximum velocity of the propagation of information is the second postulate of Special Relativity, the other one being the principle of relativity itself (the laws of physics must be identical for all observers – uh uh, these Laws again).

However, the picture gets quite complicated when we want to take quantum mechanics into account (no surprise).

For Bohm (John) Bell tolls?

The birth of this topic is the publication of the paper called “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?” by Einstein, Podolski and Rosen (1935). It concerns a characteristic quantum phenomenon called entanglement. Two particles are said to be entangled when the properties of one of them are completely correlated with the properties of the other one. For example, two electrons can be described by a wave function f = ( |up>|down> + |down>|up>)/sqrt(2). If we now measure the spin of electron 1 (we have 50% change of getting up, 50% down), the wavefunction collapses to either |up>|down> or |down>|up>. What Einstein pointed out is that by measuring the spin of particle 1 we obtain immediately the information about the spin of particle 2, wherever it is! It has been proved experimentally (Gisin in 1997) that this holds true even in cases when the spin of the two particles are measured when they are space-like separated (i.e. by a distance that can’t be covered by a signal travelling at the speed of light). Einstein dubbed such a signal a spooky action at distance. He provocatively asked: how does particle 2 know the spin of particle 1 even if they are 1,000 light-years away? (my second question would be: is the knowledge inside the particle or inside our mind?)

David Bohm

In the ’50s, David Bohm worked on an deeper explanation of quantum mechanics. In this picture, “hidden variables” guide the two entangled particles so that the wave function does not collapse at all, their properties are present in advance and not given by chance at the time of measurement. Reality would not be probabilistic but deterministic. However, we have no access to this variables, that are thus “hidden” from us. The net result would be that QM is incomplete and probabilistic, but reality has no such weird behaviour. So the question is: can bohmian mechanics reproduce exactly every prediction of quantum mechanics?

John Stuart Bell

In 1964 John Bell proposed a way to test the existence of these hidden variables, encoded in his famous theorem. It is based on an inequality (Bell’s inequality), that states that:

• The number of objects which have parameter A but not parameter B plus the number of objects which have parameter B but not parameter C is greater than or equal to the number of objects which have parameter A but not parameter C.

Bell’s inequality is satisfied by a hidden variables theory. This means that the inequality holds if:

• Logic is valid (at least, classical logic)
• Electrons have a definite value of the spin independently of our knowledge (hidden variables exist, the truth is out there)
• No information travels faster than light (principle of locality)

Does quantum mechanics fulfill this inequality as well? No! So the two theories are intrinsecally different, the Bell’s theorem states that no hidden variables theory can reproduce all the results of quantum mechanics. Now, that’s why we run into troubles: cellular automata looks like a hidden variable theory, so even in principle they could not reproduce quantum mechanics. If hidden variables exists, in order to fulfill Bell’s theorem there must be non-local interactions – but at this point the very core of the Universe-as-a-cellular-automaton  drops (at least, in my opinion). Unfortunately for our beloved cellular automata quantum entanglement does exist, and they can’t reproduce it.

Or do they?

Quantum Cellular Automata: Work in Progress

Gerard 't Hooft

Cellular automata look suspiciously like a hidden-variable theory, but it seems that there is a way to make them fulfill the Bell’s inequality, as shown by the Nobel laureate Gerard ‘t Hooft (they say you could win a lot of money if you manage to pronounce his name correctly!). The keyword here is complexity: cellular automata are interesting mainly because they can generate complex beahviour from very simple rules. Their large-scale behaviour could be so complicated that one needs a stochastic treatment to figure out what’s going on – ‘t Hooft’s idea is to use quantum mechanical techniques for addressing their statistics. In his own words:

Thus, we found sufficient motivation to proceed along the path chosen here: take some classical cellular automaton, use quantum operators to describe its time evolution, write the hamiltonian as an integral of an Hamilton density and treat the resulting theory as a full-fledged quantum field theory. The quantum states that serve as its basis will have to be interpreted entirely in the spirit of the Copenhagen doctrine. As such, there is no reason to expect these states to obey Bell’s inequalities.

So far so good. Unfortunately, we are still far from having a set of rules that give rise to the observed Universe, or even a part of it. I believe that the path pointed by ‘t Hooft is a serious one. Even if low-level rules (the ontological level) do not show the same symmetries as the large-scale Universe (e.g. translational and rotational invariance), the phenomenological level does in virtue of the possibility to create an Hilbert space. Space symmetries, energy and entropy are emergent properties of the cellular-automaton Universe. By the way, entropy has a dark relationship with black holes: is there a deeper explanation of why the entropy of a black hole is proportional to its area? Are cells “eaten” and information lost?

Who breaths fire into equations?

Stephen Hawking and Lt. Commander Data

We are not finished yet. As Stephen Hawing asked:

What is that breathes fire into equations and makes a Universe for them to describe?

There are equations or rules on one side, and there is “reality” on the other.

What’s the link between them? The very idea that the Laws of Nature and Nature itself lie on two separate planes of being could prove to be wrong at some point. Event the Rules could emerge from an even deeper level. We might even ask: who decided the Rules?

A very intriguing solution has been given by John Archibald Wheeler:

A picture by John Wheeler representing the idea of a participatory Universe

It is not unreasonable to imagine that information sits at the core of physics, just as it sits at the core of a computer. It from bit. Otherwise put, every ‘it’—every particle, every field of force, even the space-time continuum itself—derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. ‘It from bit’ symbolizes the idea that every item of the physical world has at bottom—a very deep bottom, in most instances—an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes–no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe.

With his view (which is not free from criticism) we can catch two birds with one stone: there is no Universe if no one is looking at it, and the Rules/Laws look the way we see them because they are the ones that make possible the creation of sentient beings that in turn observe the Universe. Mind blowing, isn’t it?

Conclusions

The road to reality is winding. I really appreciate Stephen Wolfram’s idea of complexity emerging from very simple rules, such as rule 110. Cellular automata can do that for sure even in very simple topologies, but the question is: can they reproduce our reality, our complexity? From an experimental perspective, I would turn the question into a more practical one: will someone be able to make testable predictions out of all this sooner or later?

Sunday Afternoon TED Talks



Stephen Wolfram: Computing a theory of everything

Brian Cox on CERN’s supercollider

Patricia Burchat sheds light on dark matter

Brian Greene on String Theory

Jill Tarter’s call to join the SETI search

The LHC @ home: How to build an electromagnet

At the LHC, the main experiment at CERN, protons and other charged particles generated during collisions are deflected using powerful electromagnets. These magnets can yield enormous B fields of several Tesla. For comparison, Earth’s magnetic field has a strenght of ~10^-4 T, a “natural” unit which is called Gauss after the guy who measured it first in 1835.  How do these magnets work exactly? In their simplest form, they are solenoids. A solenoid is a loop of wire which produces a magnetic field when an electric current is passed through it. The magnetic field generated by such a device can be calculated beforehand by this master formula: 

B = μ * n * I,

where:

• μ is the magnetic permeability of the medium
• n is the number of coils per meter
• I is the intensity of electric current measured in Ampere.

In the void, μ = 4*3.14*10^-7 = μ_0 which is an extremely small number! While air, plastic and wood have very low μ, metals usually provide a higher value. You can write this property as relative permeability w.r.t. void μ_R = μ / μ_0 or simply μ = μ_R * m_0 (easier for straightforward calculations IMHO). Just to give you some examples of μ:

• Mu-metal: 2.5 * 10^-2 (you won’t find this at home, 75% nickel, 15% iron, plus copper and molybdenum)
• Ferrite (nickel-zinc):  8 * 10^-4
• Steel: 8.75 * 10^-4
• Nickel: 1.25 * 10^-4
• Aluminium, Copper, water: 1.25 * 10^-6
• Superconductors: 0 (due to Meissner effect)

As for I, the current you can afford to use in a “homework” setting is at most 0.8A.  A 9V battery can tipically supply 0.5Ah. The number of coils per meter depends usually on the thickness of the wire and the number of layers. We will discuss it later on. All in all, what you should get from this considerations is that the B field you can create is usually quite weak. In fact, in order to produce an enormous field the LHC makes use of superconducting magnets, whose wires have a almost naught resistance thanks to this peculiar phenomenon called superconductivity. You can drive a current of several kA (!!!) in a superconducting wire! Unfortunately, at present it is practically impossible to do this at home. However, we can learn how to make a “normal” electromagnet using some stuff you can easily buy at your preferred electric shop.

• Copper wire (or better enameled copper wire)
• Some kind of metallic core for your solenoid (a screw in our example)
• Tape
• An AC/DC transformer, possibly with variable outgoing tension 1.5 – 12V, OR a series of alkaline AA (1.5V ~2000 mAh) batteries
• A compass or a glass of water + a needle
• Optional: a LED

So, let’s get it started. First of all, you have to build up your solenoid. Armed with patience, make the first round of coils. When you arrive at the end of the metallic core, put some tape to block its end, go the beginning again and start over. Iterate this several times (at least 4). Now you can attach your solenoid to the power supply you chose. Optionally, you can put a LED in series to check if the current is flowing.

Scheme of the solenoid

Solenoid made with an iron screw

In my case, the B field I can generated is (give or take):

I = V/R = 1.2V / ~10Ohm = 0.12A

B = μ * n * I = 1.25 * 10^-4 * ( 4 layers * 10 loops / 10 cm ) * 0.006A = 6 mT

For comparison, this is the  strenght of a typical fridge magnet. According to Ohm’s first law, you

should try to keep the resistance of the circuit as low as possible, avoiding a short-circuit. This is why a LED is quite useful in this place! If you can tune you transformer output voltage, set it at minimum.

Solenoid made with a nickel core

There are several ways to see the effect of your electromagnet. For instance, you

can take it close to a compass – you’ll see the needle following your magnet. Or, you could attract some iron dust. I tried with a metallic non-magnetized needle, but unfortunately my B field wasn’t strong enough to lift it. I chose to test it in a fancy way: I put some water in a candle support, and I placed a needle floating on its surface – surface tension will keep it from sinking. Then, just act as you would with a compass. It works!