mathematics « A Quantum Diaries Survivor

The worst students, the best universities December 4, 2007

Posted by dorigo in mathematics, news, physics, science.
13 comments

A flaming title, but let me explain. Italy was recently placed at the 35th place in a ranking of countries by the level of scientific knowledge of high school students. A list led by Finland, with several eastern countries figuring very well, and Italy scoring quite unlike other european countries. Disappointing, to use an euphemism.

On the other hand, today I read very conforting news, only at first sight in contradiction with the former datum: as far scientific studies are concerned, italian universities figure very high in a ranking of excellence compiled by the Centrefor Higher Education Development (CHE) in Gutersloh. In mathematics, the University of Rome-”Tor Vergata” is among the institutes “of excellence”, a very short list. Even better is the ranking of Physics institutes: here, among 24 chosen in the study, four are italian: and among them, of course, Padova (with Firenze, Pisa, and Rome-”La Sapienza”).

The parameters which were used to rank institutes were the number of scientific publications in the period 1997-2004, the number of citations, the presence of researchers among the most cited in Europe, and the use of European funds from the “Marie Curie” program, a program to favor the mobility of researchers. Apparently, my presence in Padova did not affect negatively the outcome

The slides for Lecture 1 and 2 November 28, 2007

Posted by dorigo in internet, italian blogs, mathematics, personal, physics, science.
8 comments

After overwhelming requests, I hereby make available the slides of yesterday’s lecture on the web. Beware: they are in italian, they are probably inaccurate here and there, they have not been proofread, and finally, they are nothing special… But here you go: they are viewable online here.

Actually, since it is a single file, you also get to see -in world premiere- the slides of the second lecture, which I will give tomorrow. In case you should find some blatant mistake or nagging inaccuracy, please let me know today before 10PM UT. Thank you.

UPDATE: I was asked to provide a file readily downloadable. Get it here.

Are three colours needed in particle physics ? November 18, 2007

Posted by dorigo in mathematics, personal, physics, science.
5 comments

My non-physicist readers will excuse me if this post is above their head… I intend to explain the matter in a more readable way very soon…

In the process of writing about the need for the top quark in the Standard Model, it occurred to me yesterday that the need for a renormalizable theory of subnuclear interactions - which implies the absence of axial-vector-vector anomalies, and thus a cancellation of all electric charges of the matter fields of the theory - could be combined with the requirement of the existence of charged currents among these fields imposed by the group structure of electroweak interactions, to yield more or less a priori requirements on the number of colors in quantum chromodynamics, the theory of strong interactions which keep atomic nuclei together. Here is an excerpt:

[...]if you were some deity and you were constructing the standard model from scratch, one could argue you would care for renormalizability - it would allow you to compute stuff in a simple mathematical framework. Then, having introduced doublets of quarks and leptons -you need both to construct molecular structures-, you would be facing the choice on the number of colors of quarks (which you need in order to provide stability to nuclear matter, as well as to satisfy the antisymmetric form of the total wavefunction of baryons) and the simultaneous choice of their electric charge. These are bound to satisfy the rule $\Sigma Q = -1 + N_c \times (Q_u + Q_d)=0$ in order to cancel axial-vector-vector anomalies, as well as the requirement that $Q_u-Q_d=1$ (so that W bosons allow charged current interactions between quarks as well as leptons). You would be hard pressed to choose $N_c$ any different from three if you wanted to build integer-charge mesons and baryons from $q \bar q$ and $qqq$ states! That follows from the fact that the two relations above imply $N_c=\frac{1}{1+2Q_d}$ . [...] If you know some literature on the topic of building a consistent, alternative version of the SM with different quark charges, I would be glad to get a reference [...]

Well, a blog is a wonderful thing, because it sometimes fulfils your wishes! In fact, I was extremely pleased by receiving today an answer to my prayer by a distinguished professor in Stony Brook, Robert Shrock. I was even more pleased to read that the question I had posed to myself was by no means a silly one. Here is Robert’s answer:

Let me respond to your question about quark charges as a function of $N_c$ . It is certainly true that $N_c=3$ is special. The condition for the cancellation of anomalies in gauged currents in the standard model is

$N_c Y_{Q_L}+Y_{L_L}=0$

where $Y$ denotes weak isospin, and $Q_L=(u,d)_L$ and $L_L=(\nu,e)_L$ denote the left-handed quark and lepton doublets (with notation for the first generation, but the relation holds, of course, for each generation). This yields the relations (e.g., see eqs. (2.15), (2.16) in my paper R. Shrock, Phys. Rev. D53, 6465 (1996) (hep-ph/9512430)):

$q_d = q_u-1 = -\frac{1}{2}( 1 + \frac{1}{N_c}(2q_e+1))$

In general, one may consider lepton charges different from the usual ones, which leads to a general set of several classes of quark charges, as displayed in Table 1 of that paper. Note that the relation $q_u-q_d=1$ is independent of $N_c$ and only depends on the $SU(2)_L \times U(1)_Y$ gauge structure with the relation $Q=T_3+(Y/2)$ and the electroweak representation content of the fermions. In section VIII of the paper I discussed some of the phenomenological properties of theories with the various classes of quark charges. For example, for $N_c=5$ , if one required the usual lepton charges, then $q_u=3/5$ , $q_d=-2/5$ and baryons would be composed of 5 quarks. The proton-like and neutron-like baryons would be given by eqs. (8.3), (8.4), namely $p=uuudd$ and $n=ddduu$ with their usual charges, eqs. (8.5) and (8.6).

The value $N_c=3$ is also special for the idea of grand unification. If one constructs a grand unified theory based on an SO(N) group with $N=4k+2$ , $k \ge 2$ , then the condition that the SM fermions of a given generation (including an electroweak-singlet neutrino) fit exactly into a spinor representation is (see eq. (9.7))

$2^{N_c+1} = 4(N_c+1)$

which has a solution only for $N_c=3$ .

Interestingly, if one considers the $N_c$ -extended standard model and studies possible embeddings in a (supersymmetric) GUT, then imposing the conditions of unbroken electromagnetic gauge invariance, asymptotic freedom of color, and three generations of quarks and leptons forces one to choose $N_c=3$ . See my recent paper ArXiv:0704.3464, published as Phys. Rev. D76, 055010 (2007).

I am writing this now, but I am feeling guilty, because I should be reading those papers first! The fact is, I want to give them more attention than what I can deserve to them in the control room of the CDF experiment, which is where I am now, with an eye on the keyboard and another on the monitors of the beam, the data taking efficiency, the silicon bias voltages… And I wanted to give my readers access to those papers as well. So let’s all read about it and get educated on this fascinating issue - I think I will twist Robert’s arm into making a guest post on the matter here!

The Goldstone Theorem for Real Dummies November 10, 2007

Posted by dorigo in mathematics, personal, physics, science.
58 comments

I have been spending the last few days preparing part of a course in particle physics for 5th year students in Physics (the second and last year of what is called “Laurea Specialistica”, like a masters degree in the US). I must say I had forgotten how much I like to study. The last serious time I spent in the company of physics books was over two years ago, but that was a very stressful occasion with an impending exam, burdened by the high stakes of getting tenured. Besides, the prospect of explaining the standard model to students who have at least some familiarity with quantum field theory is really stimulating. I am not a theorist, so in principle I am not qualified to present the theory of particle physics in an impeccable way, but the course I will teach takes a quite phenomenological-experimental point of view, so I think I will not be able to do too much damage to those innocent souls (and the course is taught for three quarters by a more experienced colleague - I only do the last part, on Higgs and collider physics).

Of course this blog has been suffering recently from my involvement in preparing the course… So I decided I would try and kill two birds with one stone, and make an attempt at making available one tiny bit of my course today, aiming at real laypersons, here. I think of it as a challenge to myself to test the inverse Feynman’s grandmother’s conjecture: whether, that is, one can explain things to grandma if one has understood them (the original conjecture states that you haven’t really understood something unless you can explain it to grandma). And since I lost both my grandmothers, you are my guinea pig for today.

Enough chat. Now, what is Goldstone’s theorem and why should you bother ? The theorem is a crucial preliminary to understand the need for a Higgs boson in the Standard Model theory of particle physics, and that would suffice to keep you awake: but it is also a very nice illustration of how the physics of a system can be extracted from a quite abstract concoction - the lagrangian density. If you do not know what a lagrangian density is, worry not: you will not really need to understand what it is in and out, because I intend to present things in a very handwaving way. That will not prevent me from calling things with their real names!

So let us consider a lagrangian density for a real scalar field. What is a scalar field ? Take air temperature, for instance. It is a real number defined in any point of space, a number depending on space coordinates. In quantum physics, however, a scalar field represents a particle capable of moving and interacting with its peers: doing the things that particles do, that is.

Ok but, what is then a lagrangian density ? The lagrangian density is some mathematical scribbling that enshrines the physics of our scalar field. It is defined as L=T-V, where T is the kinetic energy of the field (the particle), and V its potential energy. Think of a ball thrown in the air: Once you’ve kicked it up, it has speed -and that is a form of energy, called kinetic energy- and height -and that, too, is a form of energy: potential energy. You know what potential energy is: it is the reason why you avoid walking under a baby grand piano being lifted to the third floor. So L is just the kinetic energy of our ball subtracted of its potential energy. Despite the simple definition above, the lagrangian may take complicated forms. It is an expression which, handled the right way, can sometimes be squeezed to extract the dynamics of our particle. I will not tell you how today, but deal L with the respect it deserves, since unlike you and me, L knows everything about the scalar particle: its past, present, and future motion. Here is our lagrangian for the real scalar field $\phi$ :

Quite a far cry from “T-V”. ain’t it ? But worry not. In the expression for L above $\phi$ is our scalar field, and the curly symbol before it represents a derivative - by putting it there we express the fact that we consider the variation of the field with respect to its position in space. By multiplying together the spacetime derivatives of the field, we are in effect writing the kinetic energy it possesses. The second term is instead the potential in which our particle sits: it depends on two real parameters, $\mu^2$ and $\lambda$. Regardless of their value, we observe that L exhibits a manifest symmetry with respect to the substitution of the field with its opposite value, $\phi(x) \to -\phi(x)$ . A symmetry of the lagrangian is something worth noting: it usually reflects the presence in the theory of a conserved quantity, something that the operation of symmetry does not change, that is.

Now let us investigate more our potential V: as a first example we give both parameters $\mu^2$ , $\lambda$ a positive value. We then know the form of the potential from calculus you should have had some time in the past at school: a quartic curve, with a minimum where the field is equal to zero. It is shown in the plot below, on the left diagram.

Imagine a particle sitting at the point $\phi(x)=0$ . It is at the minimum value of the potential. If you move it about the minimum, it produces small oscillations - perturbations of its physical state - and we can compute the physics of the field using something called “perturbation series”: the potential difference introduced by the perturbation is small, so its effect is a small modification to the motion of the particle. By grouping together modifications of the same order of magnitude and summing them we can determine the dynamics. Crucially, the particle has a positive mass, corresponding to the resistence it opposes to any attempts at displacing it from the point at $\phi=0$ : you may well call it inertia.

Much more interesting is the case arising if we instead take $\mu^2<0$ . We then get the form of potential shown on the right diagram in the figure above. The potential term with a negative value of $\mu^2$ is at odds with what you would have learned by browsing the first few chapters of a quantum field theory book: it appears to represent a particle with imaginary mass. It is easy to see why it is so: it gives a “negative resistance” to any attempts of moving it from the origin, where the field is zero. The potential decreases in both directions, so it is energetically favorable for our field to roll down to one of the two saddle points. These lie at the value $\phi = \pm v = \pm \sqrt{-\mu^2/\lambda}$ , as is easy to realize by inspecting the form of V - or, if you know better, by just setting to zero the derivative of V with respect to the field.

We like to call the minimum of the potential our “vacuum”: you cannot have less energy than that. In the case of our potential with negative $\mu^2$ , the vacuum does not correspond to zero value of the field! Rather, the field takes the value $v$ . There is something wrong here: imaginary mass, vacuum containing non-zero fields… Surely, we can mend the situation by redefining our scalar field: we shift it to the minimum at +v by calling $\phi(x)=v+\eta(x)$ . The physics cannot depend on the shift of the scalar field by a constant value v, and now the lagrangian takes a different form:

In terms of the shifted field $\eta(x)$ , there is nothing wrong with the lagrangian any more: the vacuum has zero value for the field -it is a real vacuum!-, and the field has a mass term of the right sign: $-1/2 m^2 \eta^2 = -\lambda v^2 \eta^2$ so the mass of the scalar is now $m = \sqrt{2\lambda v^2} = \sqrt{-2\mu^2}$ , a positive value (forget the terms with cubic and quartic in the field: they describe self-interactions, not the mass).

Even better, we can now do perturbation expansions around the new minimum, and our expansions will converge. We will be thus able to compute the dynamics for the new field $\eta$ . All is good, a true success. But there is something we had to give up in exchange: L’ is not symmetric for the operation $\eta \to -\eta$ any more!

It is important to note that the physics described by L cannot have changed as a result of a simple constant shift of the field: so we are brought to conclude that the symmetry is still there, but it is “hidden” by our choice of the vacuum at a value +v for the original field. The symmetry of L generated a degeneracy in the vacuum: two values share the minimum for V. By choosing one of the two possible vacua we have hidden the symmetry from view.

A bit harder: a complex scalar field and the Goldstone theorem

Ok, now we need to make things just a bit more complicated. We want to write a lagrangian which is symmetric under a continuous transformation law of the field, not just the simple mirroring as before. That will allow us to state Goldstone’s theorem. The simplest lagrangian we can write is the one below.

This time we have defined a field $\phi = \frac {1} {\sqrt {2}} ( \phi_1 + i \phi_2)$ : this is a complex scalar field, which is actually equivalent to two real scalar fields. Please do not worry about the imaginary symbol $i$ or the complex conjugation operator *: you will not need to even touch those with a stick. Instead, look at L. If you look at it for long enough, you realize it is an expression which remains invariant if we modify the field by a phase transformation $\phi \to \phi'=exp(i\alpha)\phi$ (with $\alpha$ the constant phase shift). That happens because every time you multiply the field by its complex conjugate the two phases annihilate, and L above only depends on such well-behaved products.

If the math I invoked above is above your head, do not worry. Suffices to accept that we have managed to put together a lagrangian which is invariant for phase transformations of the complex scalar field . The physics described by L does not depend on the value of $\alpha$ , that is. But of course it depends on the two parameters in the potential energy terms, $\lambda$ and $\mu^2$ . What happens now if we take as before the former positive and the latter negative ? Again, we get a field with an imaginary mass term, which might not really look like a good idea. Worse, we now have not just two, but a full circle of minima for the potential, lying at the values of the field satisfying $\phi_1^2+\phi_2^2=-\mu^2/\lambda$ . An infinity of choices for the vacuum! The situation is pictured in the drawing above.

Having previously worked out the simpler example of one single real scalar field, we are not impressed by the compication, since we know how to get things straight: we choose one of the vacua for a translated field by writing $\phi(x) =1/\sqrt{2} (v+\xi(x) + i\eta(x)$ .Don’t worry about yet new greek letters: we just renamed the two components of the original scalar field, after translating them to the point (v,0). In terms of the shifted fields L becomes

If we examine the latter form of L, we recognize kinetic terms (the ones with two derivatives of the field) for the scalar fields $\xi$ and $\eta$ . But while we also have a mass term (the one quadratic in the field) for $\eta$ , $\xi$ gets no mass term: that means the field is massless. The two fields correspond to orthogonal oscillations about the vacuum we have chosen: and the massless field corresponds to oscillations along the direction where the potential remains at a minimum - along the circle of minima, that is. Because of that, it encounters no resistance - no inertia, no mass.

The spontaneous breaking of the symmetry of the original lagrangian for $\phi$ has generated a mass for one of the two scalars, and a further massless scalar has appeared in the theory. This is the Goldstone theorem in a nutshell: The spontaneous breaking of a continuous symmetry of the lagrangian generates massless scalars. They correspond to fluctuations around the chosen vacuum in the direction described by the neighboring vacua.

Massless scalar particles do not belong to any reasonable theory of nature. Our world would be a quite different place if there were massless scalars around! We do not observe such particles. Indeed, there is a mathematical trick, called the Higgs mechanism, which gets rid of the massless goldstone bosons. The degrees of freedom of the theory associated to the Goldstone bosons reappear as mass terms for the weak vector bosons… But this is stuff for another lesson.

Still here ? I would love to know if among the twentyfive readers of this post there is at least one who has made it to this last paragraph. If you are him or her, and you had no prior knowledge of quantum field theory or lagrangian formalisms, drop me a line. I’d like to know what made you think you could learn these difficult things by reading a blog post…

Guest post: Alejandro Rivero, “sBootstrap” October 16, 2007

Posted by dorigo in Blogroll, mathematics, physics, science.
34 comments

Alejandro Rivero is a theoretical physics PhD from Zaragoza University who, to his own regret, was too good with computers and too stubborn about research paths. As there was a lot more of work for computer scientists than postdocts for non commutative geometry with the orthodox Connes approach, he went into the former. But he still has more confidence on NCG than in strings. Currently he works in the BIFI, where he is involved in the project of a national infrastructure for volunteer computing.

CHEWISH PHYSICS

Coming back to the RadLab from his exile, Chew developed in the early sixties a new approach to strong interactions: the bootstrap. He observed that there was nothing in a Feynman diagram allowing to know which particles were composite and which ones were fundamental ones. Raising the flag of “nuclear democracy”, and feeling himself backed by observations of Feynman and Heisenberg, he launched a S-matrix-based program aiming to find an unique self-consistent theory of the nuclear strong force. “Nuclear democracy” was to mean that a particle can be interpreted as composite or elementary depending of its role in a particular diagram, so there were no elementary particles at all, or all the particles were equally elementary. The pursuit of further pieces of matter, subcomponents and preons and prepreons and so on, was to come to an end by asking some general properties to the scattering matrix, the particles coming ready from its pole structure.

During the first years the approach enjoyed some attention from labs everywhere, because even if you did not agree with Chew’s goals, the S-matrix was one of the very few tools available to approach to the study of strong interactions. But for the late sixties it started to be clear that the answer for strong force was more of the same: the hadrons were equal between them… because they were all composite, made of quarks. QCD became the new game, and S-matrix theory was abandoned. At the same time, one of the arguments of Chew theory, s-t duality, evolved by itself coming to beget “dual models” and then “string theory”, and then no a democracy but an unique ruler, the Planck-scale superstring.

Now, open bosonic strings, the ones you can make with two quarks, or a quark and an antiquark, lie in “Regge trajectories”: a dependence between energy and angular momentum, in a way such that all the strings of a same trajectory have a same composition: they are just excitations of its fundamental, spin 0 state. In this way, the number of strings depends of the number of different flavours of quarks you have
available to combine. And these strings are bosons. Assume each spin 0 state is not degenerate (choose the lowest energy one). Let’s postulate that for each spin 0 string there is a fermionic degree of freedom with the same electric and colour charge, but elementary. If it is so, we have saved not the “nuclear democracy” nor the bootstrap, but part of its philosophy: each elementary particle is not composite, but it is supersymmetric to a composite.

COULD THE sBOOTSTRAP WORK ? PREDICTIONS

A first test is to check for matching in degrees of freedom. For n generations of quarks and leptons, we have n charged leptons, and then 2 n degrees of freedom of charge +1 and 2n d.o.f. of charge -1. On the other side, we can combine quarks U and D and their antiquarks to form n^2 strings of charge +1 and the same of charge -1. Then 2n=n^2 implies n=2. This is already bad, because we know there are three generations. But in the neutral side it is worse: the n neutrinos give us 4 n degrees of charge 0, while in the composite side we have either 2 n^2 neutral combinations of quarks and antiquarks that should reduce to 2 n^2 -1 after taking care of the U(1) singlet (as in SU(3) you get an octet instead of a nonet).

Even if we are willing to accept the result n=2 and the extra U(1), we still need to confront quarks. And here the bartering definitely breaks. On one side, the UD sector has n^2 combinations to form charge +1/3, for which there are 2n degrees of freedom in the elementary side. Well, n^2=2n still implies n=2. But on the other side, when combining D quarks to get charge -2/3 you get n(n+1)/2 of these, while you need 2n. So from the DD sector we obtain n=3. The idea fails in the quark sector.

Thus the sBootstrap conjecture predicts 2 generations and can only work assuming an extra U(1) in the combinations for neutral strings and neglecting the possibility of partners for down quarks. Kind of failure for our idea.

Really? Let’s review… we have done an extra assumption that actually does not appear in Nature: we have assumed that all the quarks bind into strings. This is not true: the top meson has a mass greater than the W, and then it is disintegrated under electroweak force before being allowed to link into a QCD bind.

Back to the blackboard: let be n the number of generations, s the number of U quarks and r the number of D quarks. Obviously n equal or greater than r and s, and we ask both to be greater than zero. The rest of the history is section 3 of my e-print arxiv:0710.1526. Now the equations are
2 n = rs,
4n = r^2 + s^2 -1

for the lepton side, and

2n = rs,
2n = r (r+1)/2

for the quark side, and all of them solve uniquely to n=3, s=2, r=3. The sBootstrap conjecture predicts three generations AND predicts that while all the D quarks are light, only two U quarks are light.

PROSPECTIVE

There are some hints that the framework of superstring theory could be fit to keep developing on this conjecture. If we look to the colour side, the standard model with massive neutrinos has 24 fermionic degrees of freedom for each colour, including the neutral possibility. Generation-wise, it makes 8 degrees of freedom by generation; then we could have some hope of fitting them into the massless states of some sector of a superstring.

Furthermore, it is worthwhile to think of the quark-antiquark composites as oriented open strings, while the quark-quark and antiquark-antiquark are different sectors of unoriented open strings. To close an unoriented string you need to zip it against another unoriented string from the opposite sector, and then the resulting closed string carries a charge of the kind colour+anticolour, similar to a gluon. On the other side, an oriented string can close upon itself, giving a closed string uncoloured but perhaps still with electric charge… pretty much as the electroweak bosons, and it could explain the strange similarity in decay rates between the Z, W particles and the most stable mesons, of which Dorigo was kind enough to speak time ago in this blog.

A different question is, what will the LHC find, if all the superpartners have already being found? “And the prize for the experimental finding of supersymmetry goes to… hmm, Cecil Frank Powell again???“. Giving that we do not expect a QFT to have elementary fermions beyond spin 1/2, it can happen we will not find any superpartner more. And yet, we are telling that the mechanism giving mass to the stringy-bosons is the same that the mechanism giving mass to the elementary-fermions. So there should be definitely something to decrypt in the electroweak scale.

THE BALL BACK TO HEP ENERGIES

The lightness of the five quarks, and the heavy character of the top, is predicated with reference to the electroweak scale. Moreover, the superpartners are the particles bound by the QCD strong force. If we can work out some model based on superstrings, they will not be superstrings at the Planck scale but superstrings down under the TeV. It is hard to guess if a modern string theorist will be happy or sad about the sBootstrap idea.

A string-like binding fits well with other expectations of the amateur spectrologists in physicsforums.com, all of them from HEP energy data: the quotient between the mass of Z and W has been noticed, by Hans de Vries, to be very much as the one of the binding two relativistic particles with total angular momentum 1 and 1/2. Some breaking in the neutral vs charged pion was also related to the quotient between the mass of a lepton and those of a gauge boson, and some of the octet breaking or mixing was very near of relationships involving only lepton masses. Furthermore, Koide’s formula works better when mixing composite particles, and then it could be a formula to be met in the initial three supermultiplets, before further symmetry breaking and mixing in the strong sector. This fact has been another of the guiding principles towards our suggestion of a composite elementary symmetry. Of course all the amateur findings, without a backing theory, could be birthday coincidences: how many people must meet in a room to have a 50% probability of finding two persons with a common birthday? But note that some formulae are grouped in very nicely symmetric families.

In any case, the plot associated to the Z width mystery provides an approximation of the scale range: from the mass of the pion -and muon- to the point where the scaling of charged pion meets the scaling of neutral pion. The latter goes with the cube of the mass, the former with the quintic power. The lines meet at 2.6 TeV, and of course the effect of the breaking of electroweak symmetry starts to blur already before, as we climb beyond the 0.1 TeV of the Z mass.

Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” October 4, 2007

Posted by dorigo in Blogroll, internet, mathematics, physics, science.
80 comments

Marni D. Sheppeard is a theoretical physicist in New Zealand. After 20 years of stints in experimental condensed matter physics, biomedical engineering, lattice QCD, electronics manufacture, programming, teaching, mountaineering and the real world, she has finally found some time to work on her main interest, quantum gravity. You’ll also find her serving excellent lattes in central Christchurch.

IS CATEGORY THEORY USEFUL ?

In the past, physics has made great progress with the mathematics of classical symmetries. For example, motion of a body in a plane can be decomposed into straight line translations and rotations.

The collection of all possible rotations is described by a circle in the plane, marked with a reference point which represents no rotation. Other points on the circle represent rotations by an angle corresponding to the angle between the given point and the reference point. Any rotation has a reverse operation, namely a rotation by the same angle in the opposite direction, which equals the rotation that moves from the original point around the remainder of the circle. That is, a pair of rotations is represented by a pair of arrows which make up a circle. These arrows combine to form a single arrow around the whole circle. Note that this is the same physical operation as the original marked reference point. Two shorter arrows could also be combined into a single rotation.

The idea of combining arrows in this way is what Category Theory is about. By definition, a category is just a collection of points along with arrows between them which can be combined to form new arrows. But points don’t have to represent actual points in a classical space. A point might represent a whole space, and the arrows ways of mapping one space into another. By studying these arrows we have a way to look at properties of spaces that doesn’t involve drawing complicated pictures in higher dimensions, because a space is represented by a single point and arrows are only one dimensional. But the concept of category is even more general than this. Points might represent sets, and the arrows functions between sets. Now much of 20th century mathematics is built upon the properties of sets, and here there is the possibility of using functions and sets together. But there seems to be a snag: surely the collection of points and arrows in any category must form sets! And it is true that the basic axioms for a category state the existence of such sets. However, by being a little bit more sneaky we can describe the category of all sets and then ask ourselves about other categories that might replace this one.

Why on earth would we want to do this in physics? First, observe that for rotations in the plane there is only one point, the marked reference. All classical symmetries, in any dimension, correspond to categories with only one point. In the quantum world, however, we secretly play with categories with more than one point. For example, an atom emits light only at certain frequencies determined by state transitions for its electrons. A transition may be pictured as an arrow between states. Two consecutive transitions combine to give a photon of frequency equal to the sum of its component transitions. This looks similar to the planar rotations, except that each state now gives a point in the category.

Another category associated to quantum physics takes points to be spaces of states. This category has a logical aspect, similar to the category of sets but also with glaring differences. Whereas elements of sets obey the rule that they either exist or do not exist, quantum matter only takes on this feature when it is observed. The logic of quantum mechanics is built from operators (projections) that take a space of states and pick out a specific choice of state. Such an operator, as an arrow in the quantum category, has the feature that doing it twice is the same as doing it once, because once a state is chosen the second arrow will just select the state again. This type of operation appears in many places in mathematics. In category theory, a map that selects the point at the start of an arrow is such an operator, because after the point is selected the second iteration of the map just reselects the point, which is viewed as an arrow from the point to itself that does nothing. Category theory is unique in the way it combines algebra, geometry and logic.

But the important feature of a quantum particle that is really not understood is its mass. To derive mass numbers correctly would require some knowledge about Quantum Gravity, the theory of quantized mass. Only the photon has zero mass and travels at the local speed of light relative to observers made of massive particles. In the late 1960s, Sir Roger Penrose considered spacetime events as collections of light rays incident on a point from the celestial sphere. Mathematically, this amounts to thinking of the sphere as a category of sets of light rays, with its own non-classical logic. This is still a one dimensional category. Categories in higher dimensions, with higher dimensional arrows, are currently been investigated as a language for gravity.

In three dimensions a basic arrow may be replaced by a cube. We imagine that the cube contains arrows on the faces and one arrow in the interior. An analogue of a projection operator is a map that sends the whole cube to one half of its boundary. Is it possible that particle masses could be derived from the simple algebra associated to such operators? Carl Brannen has been studying projection operator algebras associated to cubes. He has shown that the electron, muon and tau masses are described by a simple three-by-three array of numbers, as are the three neutrino masses. Current work is extending this analysis to baryons and mesons.

The giant calculator joke July 30, 2007

Posted by dorigo in computers, games, humor, mathematics, personal, travel.
9 comments

Today I was back in my office and found a way to have a laugh with Devis, a colleague who recently spent a few days in the US.

Devis had been harassed before the trip by another colleague, Andrea, who wanted to save a hundred bucks on a multi-function pocket calculator. This is commonplace for italians traveling to the States: there always is a relative or a friend who knows electronics are cheaper overseas, and who will force you to spend endless afternoons in shopping malls in search for the requested item - and at times, to conceal said item while passing customs in order to save import taxes. And with the very advantageous exchange rate of dollars per euro these days, these sorts of requests have only gotten worse.

Devis had agreed to look for the pocket calculator (some fancy Hewlett-Packard model, apparently quite expensive), but once in the Chicago area, he had been unable to find it. So he stopped at a Wal-Mart, and had a brilliant, brilliant idea. Here is what he brought back:

At 10″ by 6″, the thing is mastodontic - indeed, it can be attached to a wall (there is even a suitable hook on the back). We laughed when we guessed the face Andrea will put up when presented with the “pocket” calculator. But then we had an idea for an even more fun joke.

The calculator has a silvery finish, and if one walks by keeping it sideways as a book, with its back showing, everybody will think it is a fancy, extra-slim laptop computer. So we already decided that at the next seminar or conference I have the occasion to follow, I will arrive late bringing the calculator with me, and walk confidently to the front row making sure everybody notices me. I will then sit down, carefully place the calculator in front of me, and meaningfully start poking on the giant rubbery keys, one finger per hand, with a concentrated look.

I bet you want to do it yourself. You have my permission. Get yours here, but don’t forget to report on the laughs.

3/fb reached! June 13, 2007

Posted by dorigo in mathematics, news, physics, science.
21 comments

The Tevatron has been painstakingly producing proton-antiproton collisions for six years now, at an accelerating pace. The goal of 3 inverse femtobarns of collisions has now been crossed, as shown in the graph below (of which you can always find an up-to-date version in http://www.fnal.gov/pub/now/tevlum.html ).

To understand what the heck is a integrated luminosity of three inverse femtobarns, just think about shooting a lot of bullets with a short gun at a dime placed ten yards away. You do not expect to hit the dime, do you ? In fact, your bullets will cover a wide area around the dime rather randomly - say an area of about a square foot wide (if you are good). In order to hit the dime you will on average need to shoot a number of bullets equal to the ratio between the area covered by the dime (about a sixth of a square inch, or one cm^2) and one square foot - or about a thousand of them.

Let us put this in a tidier form: if we shot 1000 bullets per square foot, that is exactly the same concept physicists talk about when discussing the amount of collisions they managed to make: an “integrated luminosity” L = 1000/sq ft, or about 1/cm^2 since a square foot is about 1000 cm^2. Since the dime has a cross section S of one cm^2, you expect to have made N= SL = 1 cm^2 x 1/cm^2 = 1 hit on average!

Now, physicists use the same line of reasoning to estimate the chance of producing a rare process when colliding particles. Rare processes have a very small cross section, and so to produce them we need to shoot many bullets! The dreaded “inverse femtobarn” is nothing but a measure of the number of bullets we shot per unit area. A femtobarn is the impossibly small area of 10^-39 cm^2: the area of a square whose side is a few billionths of a billionth of a meter. So three inverse femtobarns means having “illuminated” with three protons every such square of the incoming antiproton.

With three inverse femtobarns, one can produce really rare processes. The total cross section of a proton-antiproton collision is about S = 10^-25 cm^2, so with L=3/fb = 3/(10^-39 cm^2) we have actually produced N = SL = 3 x 10^14 collisions (or three hundred thousand billions)! Now, a really rare process such as Higgs boson production has a cross section of a few hundred femtobarns: with 3 inverse femtobarns of data we expect to have produced several hundreds of them!

The problem, then, is finding these few hundred Higgs bosons in the three hundred thousand billions… Hehm. It is much harder than finding the dime once it was hit by your bullet!

Not to worry. CDF and D0 are up to the task. And the Tevatron is expected to more than double the total amount of collisions it produced this far by the end of 2009… More dimes to find, more chances to get rich.

« newer posts older posts »

A Quantum Diaries Survivor

The worst students, the best universities December 4, 2007

The slides for Lecture 1 and 2 November 28, 2007

Are three colours needed in particle physics ? November 18, 2007

The Goldstone Theorem for Real Dummies November 10, 2007

The giant calculator joke July 30, 2007

3/fb reached! June 13, 2007

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A Quantum Diaries Survivor

The worst students, the best universities December 4, 2007

The slides for Lecture 1 and 2 November 28, 2007

Are three colours needed in particle physics ? November 18, 2007

Guest post: George Barouxis - “Extrinsic” Relativity November 16, 2007

The Goldstone Theorem for Real Dummies November 10, 2007

Guest post: Alejandro Rivero, “sBootstrap” October 16, 2007

Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” October 4, 2007

A couple of things September 13, 2007

The giant calculator joke July 30, 2007

3/fb reached! June 13, 2007

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