Wednesday, October 31, 2012

CMS proton-lead ridge: color glass condensate?

Two years ago, I reported the observation of surprising two-particle correlations at the CMS in proton-proton collisions, something that was previously observed at Brookhaven's RHIC in their 2005 gold-gold collisions.

You know that the LHC sometimes collides lead nuclei against lead nuclei instead of proton-proton pairs but a few weeks ago, it tried something something new, the proton-lead (asymmetric) collisions. Physics World tells us about a (not too) surprising result of this hybrid crashing game:
Unexpected 'ridge' seen in CMS collision data again
I wrote it's not "too" surprising because it's been seen in gold-gold, proton-proton, and lead-lead collisions, so why it should be missing in the proton-lead collisions? But there's something new about this story I haven't written about yet, and that's the reason for this new blog entry.

It's a new cute and plausible theoretical explanation of the ridge: color glass condensate.

Physics World refers to this fresh paper by a North Carolina-Brookhaven tandem:
Evidence for BFKL and saturation dynamics from di-hadron spectra at the LHC
BFKL stands for the Balitsky-Fadin-Kuraev-Lipatov equation which performs a resummation of \(\alpha_S \ln(x)\) terms that appear at each rung of the QCD ladder (here we really talk about Feynman diagrams that look like a ladder with rungs composed of QCD propagators: no kidding). For small \(x\), the right description is in terms of the so-called color glass condensate. You may try to read a funny 2006 paper by Frank Wilczek on the Origin of Mass; full text PDF as a basic background.

The color glass condensate is an interesting new state of matter – composed of quarks – that was proposed in 2000 and that is mathematically analogous to "spin glasses". There are several other cool ways to organize the quarks and their interactions that are intensely studied by some of the creative QCD folks. If I remember well, I learned about another one from a colloquium by Frank Wilczek in Massachusetts – it's called the color-flavor locking.

You know, this blog entry wasn't supposed to be too narrowly focused on the color glass condensate. It's about all similar amusing emergent concepts in QCD and the color-flavor locking is really pretty. ;-) What is it?

Before the color-flavor locking, its fathers have proposed another "behavioral mode" of the quarks in QCD, namely the following condensate (vacuum expectation value):\[

\langle q_i^\alpha C \gamma^5 q_j^\beta\rangle \propto \epsilon_{ij} \epsilon^{\alpha\beta 3}.

\] Here \(i,j\), the Latin indices, represent the flavor (up/down) while the Greek indices, \(\alpha,\beta\), encode the color. You see that the formula above picks the "third color" as a special one so \(SU(3)_c\) is broken to \(SU(2)_c\) while the chiral "flavor" \(SU(2)_L\times SU(2)_R\) group is unbroken.

But in the 1998 paper I have already mentioned, they proposed an even more natural condensate invariant under a "diagonal" \(SU(3)\), namely\[

\langle q^\alpha_{Lia} q^\beta_{Ljb} \epsilon^{ab} \rangle &=
-\langle q^{\alpha\dot a}_{Ri} q^{\beta\dot b}_{Rj} \epsilon_{\dot a\dot b} \rangle=\\
&= \kappa_1 \delta_i^\alpha \delta_j^\beta +\kappa_2 \delta_j^\alpha\delta_i^\beta.

\] Note that there is no privileged "third color" in this formula. It uses the fact that there are three "pretty light" quark flavors, namely up/down/strange, and three colors. Of course, these two numbers "3" have nothing to do with each other – which is also why the Kronecker delta relating the Greek and Latin indices above is mixing apples and oranges. But because \(3=3\), it is actually possible to choose a random identification and use it to mix apples with oranges – and flavors with colors.

Originally, you had 1st, 2nd, 3rd flavor and 1st, 2nd, 3rd color. These two numbering systems had nothing to do with each other. That's also why you could have rotated the triplets by two independent \(SU(3)\) groups. If Nature ever makes you happy and creates a condensate given by the formula above, the numbering systems for the colors and flavors are identified by the "apple-orange" or "Latin-Greek" Kronecker delta symbols and only an overall \(SU(3)\) group which rotates the colors as well as flavors so that the Kronecker delta symbol is preserved – and these transformations that transform both "versions" of the \(SU(3)\) group in the same way are known as the diagonal group – are preserved symmetries.

Note that the equation above, claiming that the vacuum expectation value has a particular form, is a conjecture. They conjectured that in some situations, it may be true or at least approximately true. But if quarks ever organize themselves so that it is true, it has physical consequences. Alford, Rajagopal, and Wilczek decided that such a condensate would lead to various new "gaps" as well as new Nambu-Goldstone bosons that may be imagined as bound states of two quarks.

Because these "behavioral patterns for quarks" – color glass condensate or color-flavor locking – differ from the usual perturbative QCD that should hold at very short distances, as well as common descriptions of protons, neutrons, and other hadrons, they are considered extreme, perhaps even more extreme than the quark-gluon plasma. But they may have rather non-extreme properties and may be realized in various situations.

From the viewpoint of fundamental high-energy physics, experiments trying to find such exotic forms of nuclear matter are not searching for new physics. Even if those phases exist, they're just manifestations of the good old QCD. However, they are so new and "hard to rigorously calculate" manifestations of QCD that it's pretty interesting, anyway. QCD with colored and flavored quarks has quite some potential for various qualitatively different types of behavior so it's desirable for theorists to propose some initially weird conjectures about what the quarks and gluons could be doing, and for experimenters to check whether the predictions are ever realized.

The ridge seems to agree with some predictions from the color glass condensate and it's pretty interesting. But don't get carried away. Two years ago, I wrote that the ridge could be a sign of the quark-gluon plasma or the dual QCD string. People don't understand these signatures too accurately so the path from the observations to the right interpretations remains somewhat wiggly and shady.

CP violation

BTW, tonight, there will be a new paper on CP violation in D meson decays that will report a 3.5-sigma deviation from the Standard Model in a particular "difference of asymmetries" quantity seen in the 2011 LHCb data.

Was Sandy systemically caused by CO2?

Anthony Watts wrote down a nice table describing which folks believe or at least pretend to believe that CO2 "caused" Hurricane Sandy and which people don't. If I simplify it a little bit, activists, liars, and crackpots such as Al Gore support the link while scientists don't. I am kind of pleased to see that for the first time, most of the media seem to agree that the people promoting the hurricane-CO2 link are hacks.

I was intrigued by a member of the former group, hardcore leftist activist Mr George Lakoff, who wrote the following text for the Huffington Post:
Global Warming Systemically Caused Hurricane Sandy
He introduces a new problematic term: "systemic causation". He believes that fossil fuels "systemically caused" Hurricane Sandy (and other weather events we don't like). The description makes it look like the construct "A systemically caused B" means "A increased the odds of B" – note that my alternative wording is equally long, much more accurate, and not requiring any new contrived phrases.

Except that Mr Lakoff believes that AIDS is only "systemically caused", not "directly caused", by the HIV virus. That's pretty interesting. Either he is an HIV denier or his definition of "systemic causation" is internally inconsistent. But let's ask two questions: Was Sandy "systemically caused" by CO2 emissions? And forgetting about the answer and focusing on genuine "systemic causes" of bad events in general, is it legitimate for the society to outlaw them?

My answer to both questions is No, although the latter question deserves a subtler discussion.

Unless you believe in astrology and similar things, you will surely agree that it's not in the power of CO2 or any other indirect hypothetical causes to adjust some "highly internal" and "seemingly random" characteristics of tropical storms such as the population of the city that the storms target. ;-)

So the fact that Sandy managed to flood some tunnels in the New York subway system, among dozens of related achievements, is pretty clearly a coincidence that can't be explained by any well-defined long-term "cause", not even a "systemic cause". Most hurricanes avoid New York, some hurricanes get there, and only the proportion may be measured or theoretically calculated. In other words, when we talk about unknown future hurricanes, we may only predict their ability to target New York or other great cities probabilistically. And we may only estimate the probability that the most important hurricane of 2013 will make landing at most 4 days before the Halloween.

(The same comment, "only probabilistic predictions are possible", obviously applies to earthquakes in Italy, too.)

Needless to say, exactly the same words apply to Katrina and New Orleans in 2005. Katrina was a big story – much bigger than Sandy (surely by the number of casualties) – because it hit a large and relatively vulnerable city of New Orleans. Sandy was a relatively big story because it affected the "greatest city in the world" although not as much as Katrina did harm New Orleans. Let's agree that the targeting is a matter of chance.

But if you subtract all the "special characteristics" of Sandy that are related to its random path, there is almost nothing left. In fact, by the Accumulated Cyclone Energy (ACE), Sandy isn't even the largest storm of the 2012 Atlantic hurricane season. It's not even the second one. It's not even the third one: Sandy is just the fourth largest Atlantic tropical storm of 2012. That shouldn't be shocking because it has made it to the Category 2 and only marginally and for a short time.

One may look for various detailed properties of Sandy – its trajectory, its area, its pre-Halloween timing, and so on. But I think it's clear that trying to attribute some "message" (I would say "divine message") to any of these detailed properties is a sign of medieval superstitions. People who try to interpret these properties as divine signals may use a quasi-scientific vocabulary but the vocabulary isn't the essence. The essence is the logic behind their thoughts and beliefs and it is equally unscientific as any other generic medieval superstitions.

The fact that Sandy went to New Jersey is a coincidence – one that could be predicted a few days in advance but one that has no implications for any knowledge or mechanisms that are relevant outside the end of October 2012. The fact that Sandy hit before the Halloween or before the U.S. presidential election is another coincidence. It's totally scientifically implausible to assign "causes" or "systemic causes" to such microscopic accidental characteristics of a tropical storm. Such links are equivalent to astrology and other superstitions. There isn't any conceivable natural mechanism that could impose such causal links – and there's even no conceivable mechanism or explanation that could significantly increase the chance that a hurricane is more Sandy-like if the CO2 concentration is higher. I am convinced that everyone who has been given basic scientific education – or who has a basic scientific intuition even in the absence of any formal education – must know that.

So we are back to the usual questions whether the carbon dioxide in the atmosphere may be increasing or decreasing the number or hurricanes or their average or maximum intensity. I think that the data speak a clear language: no such dependence, whether positive or negative, may be extracted from the data that seem to be fully explainable by "noise", essentially "white noise". In the future, the datasets will become more extensive and perhaps more accurate and people may see a signal we don't see today. That's why it makes sense to ask whether we may predict what they will see. I think (based on arguments I have been repeatedly explained by Richard Lindzen in particular) that if they will ever see such an impact, it should be a negative impact – fewer hurricanes or weaker hurricanes. It's because storminess and other activity is driven by temperature (and other) gradients and in a hypothetical warmer world, the equator-pole temperature difference should be smaller because the poles should warm up faster. The gradients should decrease and because the gradients power the cyclone activity (and other things, including temperature variations in general), the cyclone activity should go down.

That's my prediction but I don't know how strong the effect is. It's probably very weak and it may remain invisible for centuries and perhaps forever because "global warming caused by CO2" will most likely never have an observable effect that would go beyond a modest shift of the global mean temperature.

Even when you look at the 2012 Atlantic hurricane season which became another heavily overhyped one, you will see that the Accumulated Cyclone Energy is just 121 so far, just marginally higher than the historical average around 105. The ACEs for individual seasons are never constant. They belong to some statistical distribution. It's inevitable that sometimes, the ACE ends up being above the average, sometimes (in many years after 2005), it ends up being below the average. There's nothing shocking about either outcome: it's a law of physics that such things are not constant although left-wing, egalitarian activists often have a problem with this totally basic concept underlying all of science.

Standing doctrine vs systemic causes

Despite all the hype, there's no evidence that something is changing about the statistical distributions that encode the number, strength, and geographical location of tropical storms and there's surely no evidence that this unobserved change of the distributions has some particular reasons such as a changing CO2 concentration. We've spent way too much time with this stuff. If someone isn't able to see that my conclusion is the only one that is empirically defensible, he or she probably suffers from some hopeless mania of superstitions and it's probably impossible to rationally talk to such a person.

But I want to continue with my second topic, namely the right of "systemic causes" to lead to bans. Are bans justified by "systemic causes" i.e. causes that only affect undesirable effects probabilistically desirable and compatible with some legal principles of civilized countries based on the rule of law? I would say that the answer is mostly No and if it's Yes, it shouldn't be "complete bans" and the legislation behind some "incentives" shouldn't be dogmatic but it should be based on a careful cost-and-benefit analysis.

What do I mean?

In 2006, I informed about a Massachusetts vs EPA lawsuit that ultimately ended by the unbelievable verdict that CO2 was a pollutant that EPA has the duty to regulate. So far, thank God, this pernicious verdict hasn't been fully exploited but it's a time bomb that may still explode sometime in the future.

In 2006, I discussed an important legal technicality, the standing doctrine:
It says that the plaintiff in front of the federal courts must show that her injury is "concrete and particularized" as well as "actual or imminent". The founding fathers wrote these wise sentences exactly in order to make things like suppression of the freedom of speech or suppression of life and the work of companies with the help of hypothetical accusations impossible.
Using Mr Lakoff's new terms, a person who thinks he has been affected by a "systemic cause" has no standing in the federal courts! Indeed, it's very important that only "direct causes" may be used as arguments against a "culprit". Mr Lakoff's suggestion that we should suddenly start to fight against "systemic causes", i.e. against all kinds of acts and events that have been hypothesized to increase the chance of some undesirable "systemic consequences", is therefore extremely dangerous for the life in the U.S. and elsewhere. Such a program would have a huge potential to restrict the very basic freedoms of the citizens and corporations – well, indeed, this may be the very goal of Mr Lakoff and his comrades.

Our laws are actually already full of various regulations that are meant to suppress "systemic causes", i.e. processes that may increase the chances of undesired consequences. The laws protecting people against passive smoking may be picked as an example.

Science hasn't resolved the question whether passive smoking increases the odds of various bad diseases. There are many theoretical reasons to think that the answer should be Yes. There are also some "maverick" reasons that the influence could actually be going in the opposite direction – explanations emulating the proverb "whatever doesn't kill you makes you stronger" (those things are believed by some in the case of weak radioactivity in particular). I personally think that the former – passive smoking is somewhat unhealthy – is more likely to be true.

However, this uncertainty is often presented using big words and the possible consequences are often presented as far-reaching ones. But this is a complete distortion of what the scientific research has already found out. We don't have reliable data showing that second-hand smoking increases the probability e.g. of lung cancer; and we don't have reliable data showing that second-hand smoking decreases the probability of e.g. lung cancer.

But we actually do have lots of evidence to say that if any of these two influences exists, it's very small! This conclusion of many studies that asked this very question is often being obscured, overlooked, and censored. But it's damn real. If \(p_\text{no smoke}\) is the probability to "catch" lung cancer if you are exposed to no cigarette smoke at all, the probability for second-hand smokers is related to it by something like\[

p_\text{second-hand smokers} = (1.1\pm 0.2) p_\text{no smoke}.

\] This is a number comparable to the results of various surveys. It's not the number from any particular survey but this result is as compatible with them as any single survey from the list of actual surveys and I think it's good to offer you my own number so that you won't overestimate the importance of any particular paper in literature. There is some error margin and the results are compatible with the hypothesis that there's no influence. And they are compatible with the hypothesis that the second-hand smoke slightly increases the risk or slightly decreases the risk (for the latter, the compatibility may be worse).

But the experiments are not compatible with the hypothesis that passive smokers have a doubled risk (or, on the contrary, halved risk) of lung cancer, for example!

That's an important point that will lead you, if you're rational, to realize that the change of lung cancer risks isn't a rational reason to avoid second-hand smoke! There may be other reasons but this simply ain't one of them because if the influence exists, it is weaker than the "noise". Because of genetic and other differences, you may have a 4 times higher risk or 3 times lower risk to develop lung cancer than your friend. You don't really know what the chance is but whether you change the risk by 10% isn't a real issue and if you're unpleasant to your environment because of this small correction to the noise, you may be rightfully viewed as an intolerant jerk. This may increase the chances that someone will kill you so you may be actually shortening your life by being nasty to smokers around you.

But such "systemic causes" that increase the chances of something bad do exist. I could surely find better examples than the second-hand smoke. The society wants to thrive "statistically" so it may invent various policies that "encourage" the systemic causes of good things and "discourage" systemic causes of bad things. But it's important that such legislation shouldn't be dogmatic, black-and-white, and mindless.

Various processes we have may have "good systemic consequences" (good events whose probability is increased by the cause) as well as "bad systemic consequences" (the bad events whose chance is increased by the cause). Both of them must be taken into account. I think that if the "good systemic consequences" prevail – e.g. if we measure them in dollars – it's utterly irrational and counterproductive to legally discourage such "systemic causes".

Needless to say, even if Sandy were fully caused by CO2 emissions in 2012 – in reality, not even 1% of it is "caused" by any carbon dioxide, whether one emitted in 2012 or any other previous year – it would still fail to imply that it's irrational to regulate CO2 using this Sandy justification. The damages caused by Sandy are of order $20 billion. Imagine that this happens every year. However, the damages caused by a full ban (or near-complete ban) on CO2 would be several trillion dollars a year just for the U.S. So even if you believed the totally indefensible hypothesis that "CO2 is the systemic cause behind most Sandy-like hurricanes", it would still be indefensible to introduce laws that (almost) outlaw the carbon dioxide. The actual cost-and-benefits analysis implies that the ban would be at least 3 orders of magnitude more costly than the "damages" it tries to mitigate.

In some cases, we may find out that it's plausible that some acts contribute as "systemic causes" to some undesired consequences. In those cases, it could make sense to create laws that would force the "perpetrators" of the acts identified as "systemic causes" to pay for a corresponding fraction of the damages of the consequences that were "partly or statistically" caused by the acts.

Let me give you an example. Imagine that there's some breakthrough or change and evidence accumulates that 10% of hurricanes like Sandy are caused by the CO2 emissions. If this were true – and I don't believe that the current science suggests anything of the sort but just imagine that it will do so in the future – then it would make sense to introduce legislation that would force the CO2 emitters to pay 10% of the damages caused by future hurricanes similar to Sandy. (Without a new law, prosecution must remain impossible. A judge simply can't prosecute someone for some previously unencountered "systemic causes" because the "guilt" can't be reliably demonstrated so any "guilty" verdict would conflict with the presumption of innocence!)

For "another Sandy" whose damages are $20 billion or so, the "club of all the world's CO2 emitters" would be ordered to pay $2 billion to the fund for the victims of "another Sandy". It would save some money to the insurers and others.

You surely see where I am going. My point is that even if science accumulated evidence that CO2 helps to strengthen similar hurricanes or increase their number, the extra fees that the CO2 emitters after a similar hurricane would have to pay would be totally negligible and they wouldn't change anything whatsoever about their business. Every year, the world's CO2 emitters would pay some extra $2 billion for an Atlantic hurricane, perhaps another billion for another weather event that would be partly blamed on them, and so on. So they could share a $5 billion fine a year.

That's totally negligible because they – and we – collectively waste hundreds of billions of dollars a year by carbon markets and similar policies to regulate the CO2 emissions.

Even if you decided that the largest hurricanes we experience are partially – significantly – "systemically caused" by CO2, the damages would still be vastly smaller than the costs of the war on CO2. The insane people who defend the policies regulating CO2 need much more than an indefensible attribution of weather events to the gas we call life: they need to invent tons of events and devastation that doesn't exist at all. They need a full, unrestricted demagogy. They are living outside the reality and their survival depends on their complete separation from the reality and from the truth.

It's very important to keep all those events and hypothetical causal relationships in the context and to assign them numbers. Even if human lives are at stake, you must talk about numbers. You either count the human lives separately or identify a human life with XY million dollars, whatever the right number is, but it's totally critical to do so and to preserve a rational thinking at every step. The failure to do so opens the door to the demagogy by unhinged medieval superstitious assholes such as the scum that wants to fight against the carbon dioxide. And once these jerks see the open door, they won't hesitate to scream that an influence that is actually very insignificant, cheap, and de facto negligible (for the mankind and for the CO2 emitters) is practically infinite and Universe-threatening and enough for them to demand everything, ban anything they want, and become de facto dictators of the society.

We mustn't allow anything of the sort. We must preserve the rational and quantitative reasoning. If we manage to do so, we will inevitably protect our legal systems and habits from counterproductive policies such as the carbon regulation – and from many other bad rules that refer to "externalities" and similar things that are actually negligible if looked at properly.

And that's the memo.

Tuesday, October 30, 2012

Different ways to interpret Feynman diagrams

Feynman diagrams are the funny pictures that Richard Feynman drew on his van:

You see that a Feynman diagram is composed of several lines that meet at vertices (at the nodes of the graph). Some of the lines are straight, some of them are wiggly: this shape of each line distinguishes the particle type. For example, straight lines are often reserved for fermions while wiggly lines are reserved for photons or other gauge bosons.

Some lines (I mean line intervals) are external – one of their two endpoints is free, unattached to anything. These are the external physical particles that must obey the mass on-shell condition \(p_\mu p^\mu = m^2\) and that specify the problem we're solving (i.e. what's the probability that some particular collection of particles with some momenta and polarization will scatter and produce another or the same collection particles with other momenta and polarizations). Other lines (I mean line intervals) are internal and they are unconstrained. You must sum over all possible ways to connect the predetermined external lines by allowed vertices and allowed internal lines. If you associate a momentum with these internal lines, also known as "propagators", it doesn't have to obey the mass on-shell condition. We say that the particle is "virtual". An explanation why its \(E\) may differ from \(\sqrt{p^2+m^2}\) is that the virtual particle only exists temporarily and the energy can't be accurately measured or imposed because of the inequality \(\Delta E\cdot\Delta t\geq\hbar/2\).

Because the virtual particles are not external, they define neither the initial state nor the final state. Still, they "temporarily appear" during the process, e.g. scattering, and they influence what's happening. In fact, they're needed for almost every interaction. Also, the Feynman diagrams have vertices at which several lines meet, where they terminate. The vertices describe the real events in the spacetime in which the particles merge, splht, or otherwise interact. However, we're doing quantum mechanics so none of these points in spacetime are uniquely or objectively determined. In fact, all the choices contribute to the calculable results – the total probability amplitudes.

A Feynman diagram is a compact picture that may be drawn by most kids in the kindergarten. However, each Feynman diagram – assuming we know the context and conventions – may also be uniquely translated to an integral, a contribution to the complex "probability amplitude" whose total value is used to calculate the probability of any process in quantum field theory. The laws used to translate the kindergarten picture to a particular integral or a related mathematical expression are known as the "Feynman rules".

How do we derive them?

I will discuss three seemingly very different methods:
  • Dyson's series, an operator-based method
  • Feynman's sum over histories i.e. configurations of fields
  • Feynman's sum over histories i.e. trajectories of first-quantized particles
Richard Feynman originally derived his Feynman diagrams by the second method. As his name in the description of a method indicates, Freeman Dyson rederived the Feynman rules for the Feynman diagrams using the first method – and it was an important moment from a marketing viewpoint because this is how Freeman Dyson made Feynman diagrams extremely popular and essentially omnipresent.

The third method was added for the sake of conceptual completeness and it is the least rigorous one. However, it still gives you another way to think about the origin of Feynman diagrams – a way that is perhaps generalized in the "most straightforward way" if you try to construct Feynman diagrams for perturbative string theory.

It's important to mention that Feynman has discovered many things and methods, of course, but we shouldn't confuse them. The Feynman diagrams are the pictures on the van, tools to calculate scattering amplitudes and Green's functions. But he also invented the Feynman path integral ("sum over histories") approach to any quantum mechanical theory. It's not quite the same thing as the Feynman diagrams – it applies to any quantum theory, not just quantum field theory. However, as I have already said, he used the "sum over histories" of a quantum field theory to derive the Feynman diagrams for the first time.

Two other, conceptually differently looking ways to derive the Feynman diagrams were found later. The third method uses the "sum over histories" but applied to a "differently formulated system" than Feynman originally chose; the first method due to Dyson doesn't use the "sum over histories" at all.

Quadratic terms in the action, higher-order terms in the action

But all the three strategies to derive the Feynman rules share certain technical principles which are "independent of the formalism":
  • The lines, both the propagators and the external lines, are associated with individual fields or particle species and with the bilinear or quadratic terms they contribute to the action (and the Lagrangian or the Hamiltonian).
  • The vertices are associated with cubic, quartic, or other higher-order terms in the action (and the Lagrangian or the Hamiltonian), assuming that it is written in a polynomial form.
Let's assume we have an action and the Lagrangian that depends on the fields \(\phi_i\) in a polynomial way:\[

\LL &= a_0 + \sum_i a_{1,i} \phi_i + \frac{1}{2!} \sum_{i,j} a_{2,ij} \phi_i\phi_j +\\
&+ \frac{1}{3!}\sum_{i,j,k} a_{3,ijk} \phi_i\phi_j\phi_k+\dots

\] which continues to higher orders, if needed, and which also contains various similar terms with the (first or higher-order) spacetime derivatives \(\partial_\mu\) of the fields \(\phi_i\) contracted in various ways. We don't consider the spacetime derivatives as something that affects the order in \(\phi\) so \(\partial_\mu \phi\partial^\mu \phi\) is still a second-order term in \(\phi\). The number of fields \(\phi_i\) – the order in \(\phi\) – that appear in the cubic or higher-order term will determine how many lines are attached to the corresponding vertex of the Feynman diagram.

The individual coefficients \(a_{n,i}\) etc. are parameters or "coupling constants" of a sort. How do we treat them?

Well, the first term, the universal constant \(a_0\), is some sort of the vacuum energy density. As long as we consider dynamics without gravity, it won't affect anything that may be observed. For example, the classical (or Heisenberg) equations of motion for the operators are unaffected because the derivative of a constant such as \(a_0\) with respect to any degree of freedom vanishes. We know that even in the Newtonian physics, the overall additive shift to energy is a matter of conventions. The potential energy is \(mgh\) where \(h\) is the height but you may interpret it as the weight above your table or above the sea level or above any other level and Newton's equations still work.

If we include gravity, the term \(a_0\) acts like a cosmological constant and it curves the spacetime. Fine. We will ignore gravity here so we will ignore \(a_0\), too.

The next terms are linear, proportional to \(a_{1,i}\). They are multiplied by one copy of a quantum field. For the Lorentz invariance to hold, it should better be a scalar field and if it is not, it must be a bosonic field and the vector indices must be contracted with those of some derivatives, e.g. as in \(\partial_\mu A^\mu\).

What do we do with the linear terms?

Well, here we can't say that they don't matter. They do depend on the fields and they do matter. But we will still erase them because of a different reason: they matter too much. If the potential energy contains a term proportional to \(\phi\) near \(\phi=0\), it means that \(\phi=0\) isn't a stationary point. The value of \(\phi\) will try to "roll down" in one of the directions to minimize the potential energy. It will either do so indefinitely, in which case the Universe is a catastrophically unstable hell, or it will ultimately reach a local minimum of the potential energy. In the latter, peaceful case, you may expand around \(\phi=\phi_{\rm min}\), i.e. around the new minimum, and if you do so, the linear terms will be absent.

So if we perform these basic steps, we see that without a loss of generality, we may assume that the Lagrangian only begins with the bilinear or quadratic terms. The following ones are cubic, and so on.

(We could start with a quantum field theory that has nontrivial linear terms, e.g. in the scalar field, anyway. In that case, the instability of the "vacuum" we assumed would manifest itself by a non-vanishing "one-point functions" for the relevant scalar field(s). The Feynman diagrams for these one-point functions ("scattering of a 1-particle state to a 0-particle state or vice versa") are known as "tadpoles" – tadpoles have a loop(s)/head and one external leg – because a journal editor decided that Sidney Coleman's alternative term for these diagrams, the "spermion", was even more problematic than a "tadpole".)

Bilinear terms and propagators

The method of Feynman diagrams typically assumes that we are expanding around a "free field theory". A free field theory is one that isn't interacting. What does it mathematically mean? It means that its Lagrangian is purely bilinear or quadratic. If we want to extract the "relevant" bilinear Lagrangian out of a theory that has many higher-order terms as well, we simply erase the higher-order terms.

Why is a quadratic Lagrangian defining a "free theory"? It's because by taking the variation, it implies equations of motions for the fields that are linear. And linear equations obey the superposition principle: if \(\phi_A(\vec x,t)\) and \(\phi_B(\vec x,t)\) are solutions to the equations of motion, so is \(\phi_A+\phi_B\). If \(\phi_A\) describes a wave packet moving in one direction and \(\phi_B\) describes a wave packet moving in another direction, they may intersect or overlap but the wave packets may be simply added which means that they pretend that they don't see one another: they just penetrate through their friend. This is the reason why they don't interact. Linear equations describe waves that just freely propagate and don't care about anyone else. Linear equations are derived from quadratic or bilinear actions. That's why quadratic or bilinear actions define "free field theories".

If we appropriately integrate by parts, we may bring the bilinear terms to the form\[

\LL_{\rm free}=\frac{1}{2}\sum_{ij} C_{ij} \phi_i P_{ij} \phi_j

\] where \(P_{ij}\) is some operator, for example \((\partial_\mu\partial^\mu+m^2)\delta_{ij}\). The factor \(1/2\) is a convention that is natural because if we differentiate the expression above with respect to a \(\phi_i\), we produce two identical terms due to the Leibniz rule for the derivative of the product. (That's not the case if the first \(\phi_i\) were \(\phi^*_i\) which is needed when it's complex: for complex fields, including the Dirac fields etc., the factor of \(1/2\) is naturally dropped.)

So the classical equations of motion derived for those fields look like this:\[

\sum_j P_{ij} \phi_j = 0.

\] You should imagine the Klein-Gordon equation as an example of such an equation.

Some operator, e.g. the box operator, acts on the fields and gives you zero. These are linear equations. You may often explicitly write down solutions such as plane waves, \(\phi_i = \exp(ip\cdot x)\), and all their linear superpositions are solutions as well. The coefficients of these plane waves are called creation and annihilation operators etc. You may derive what spectrum of free particles may be produced by a free field theory.

This may be done in the operator approach – the free fields are infinite-dimensional harmonic oscillators defined by their raising and lowering operators – as well as by the "sum over histories" approach – the harmonic oscillator may be solved in this way as well. The "sum over histories" approach encourages you to choose the \(\ket x\) or \(\ket{ \{\phi_i(\vec x,t)\} }\) continuous (or functionally metacontinuous) basis of the Hilbert space. By the functionally metacontinuous basis, I mean a basis that gives you a basis vector for each function or \(n\)-tuple of functions \( \{\phi_i(\vec x,t=t_0) \} \) even though these functions form a set that is not only continuous but actually infinite-dimensional.

But I want to focus on the derivation of the Feynman rules including the vertices. We don't want to spend hours with a free field theory. When we construct the Feynman rules, the free part of the action determines the particles that may be created and annihilated and that define the initial and final Hilbert space as a Fock space; and it determines the propagators.

The propagators will be determined by "simply" inverting the operator \(P_{ij}\) I used to define the bilinear action above. This inverted \(P^{-1}_{ij}\) plays the role of the propagator for a simple reason: we ultimately need to solve the linear equation of motion with some function on the right hand side. Each function may be written as a combination of continuously infinitely many (i.e. as an integral over) delta-functions so we really need to solve the equation\[

\sum_j P_{ij} \phi_j = \delta^{(4)} (x-x') \cdot k_i

\] for some coefficients \(k_i\) – which may be decomposed into Kronecker deltas \(\delta_{im}\) for individual values of \(m\). The value of \(x'\) – the spacetime event where the delta-function is localized – doesn't change anything profound about the equation due to the translational symmetry. A funny thing is that the equation above may be formally solved by multiplying it with the inverse operator:\[

\phi_i = \sum_j P^{-1}_{ij} \delta^{(4)}(x-x')\cdot k_j.

\] That's why the inverse of the operator \(P_{ij}\) – which is nonlocal (the opposite to differentiation is integration and we are generalizing this fact) appears in the Feynman rules.

So far I am presenting features of the results "informally"; we are not strictly deriving any Feynman rules and we haven't chosen one of the three methods yet.

Higher-order terms

I will postpone this point but the cubic and higher-order terms in the Lagrangian will produce the vertices of the Feynman diagrams. In the position representation, the locations of the vertices must be integrated over the whole spacetime.

In the momentum representation, the vertices are interactions that appear "everywhere" and we must instead impose the 4-momentum conservation at each vertex. In the latter approach, some momenta will continue to be undetermined even if the external particles' momenta are given. The more independent "loops" the Feynman diagram has, the more independent momenta running through the propagators must be specified. All the allowed values of the loop momenta must be integrated over.

The momentum and position approaches are related by the Fourier transform. Note that the Fourier transform of a product is a "convolution" and this is the sort of mathematical facts that translates the rules from the momentum representation to the position representation and vice versa.

Starting with the methods: Dyson series

We have already leaked what the final Feynman rules should look like so let us try to derive them. Dyson's method coincides with the tools in quantum mechanics that most courses teach you at the beginning, so it's a beginner-friendly method (although this statement depends on our culture and on those perhaps suboptimal ways how we teach quantum mechanics and quantum field theory). But it's actually not the first method by which the Feynman rules were derived; Feynman originally used the "sum over histories" applied to fields.

Dyson's method uses several useful technicalities, namely the Dirac interaction picture; time ordering; and a modified Taylor expansion for the exponential.

The Dirac interaction picture is a clever compromise between Schrödinger's picture in which the operators are independent of time and the state vector evolves according to Schrödinger's equation that depends on the Hamiltonian; and the Heisenberg picture in which the state vector is independent of time and the operators evolve according the Heisenberg equations of motion that resemble the classical equations of motion with extra hats (which are omitted on this blog because it's a quantum mechanical blog).

In the Dirac interaction picture, we divide the Hamiltonian to the "easy", bilinear part we have discussed above and this "free part" is used for the Heisenberg-like evolution equations (the operators evolve in a simple linear way as a result); and the "hard", higher-order or interacting part of the Hamiltonian which is used as "the" Hamiltonian in a Schrödinger-like equation. So we have:\[

H(t) &= H_0 + V(t), \\
i\hbar \pfrac{\phi_i(\vec x,t)}{t} &= [\phi_i(\vec x,t),H_0]\\
i\hbar \ddfrac{\ket{\psi(t)}}{t} &= V(t)\ket{\psi(t)}.

\] The operators evolve according to \(H_0\), the free part, but the wave function evolves according to \(V(t)\). Note that \(V(t)\) – and of course the whole \(H(t)\) as well – is a rather general composite operator so it also depends on time: its evolution is also determined by its commutator with \(H_0\). On the other hand, \(H_0\) itself, while an operator, is \(t\)-independent because it commutes with itself.

The operator \(H_0\) depends on the elementary fields \(\phi_i\) in a quadratic way so the commutator in the second, Heisenberg-like equation above is linear in the fields \(\phi_i\). Consequently, these equations of motion are "solvable" and the solutions may be written as some combinations of the plane waves – the usual decomposition of operators \(\phi_i(\vec x,t)\) into plane waves multiplied by coefficients that are interpreted as creation and annihilation operators.

The proof that this Dirac interaction picture is equivalent to either Heisenberg or Schrödinger picture is analogous to the proof of the equivalence of the latter two pictures themselves; one just considers "something in between them".

Getting the time-ordered exponential

At any rate, we may now ask how the initial state \(\ket\psi\) at \(t=-\infty\) evolves to the final state at \(t=+\infty\) via the Schrödinger-like equation that only contains the interacting (higher-order) \(V(t)\) part of the Hamiltonian. We may divide the evolution into infinitely many infinitesimal steps by \(\epsilon\equiv \Delta t\). The evolution in each step (the process of waiting for time \(\epsilon\)) is given by the map\[

\ket\psi \mapsto \zav{ 1+\frac{\epsilon}{i\hbar} V(t) }\ket\psi.

\] For an infinitesimal \(\epsilon\), the terms that are higher-order in \(\epsilon\) may be neglected. To exploit the formula above, we must simply perform this map infinitely many times on the initial \(\ket\psi\). Imagine that one day is very short and its length is \(\epsilon\) and use the symbol \(U_t\) for the parenthesis \(1+\epsilon V(t)/i\hbar \) above. Then the evolution over the first six days of the week will be given by\[

\ket\psi \mapsto U_{\rm Sat} U_{\rm Fri} U_{\rm Thu} U_{\rm Wed} U_{\rm Tue} U_{\rm Mon}\ket\psi.

\] Note that the Monday evolution operator acts first on the ket, so it appears on the right end of the product of evolution operators. The later day we consider, the further on the left side – further from the ket vector – it appears in the product. So the evolution from Monday to Saturday (or Sunday) is given by a product where the later operators are always placed on the left side from the earlier ones. We call such products of operators "time-ordered products".

In fact, we may define a "metaoperator" of time-ordering \({\mathcal T}\) which, if it acts on things like \(V(\text{Day1}) V(\text{Day2})\), produces the product of the operators in the right order, with the later ones standing on the left. The ordering is important because operators usually refuse to commute with each other in quantum mechanics.

Now, if you study the product of the \(U_{\rm Day}\) operators above, you will realize that the product generalizes our favorite "moderate interest rates still yield the exponential growth at the end" formula for the exponential\[

\exp(X) = \lim_{N\to \infty} \zav{ 1 + \frac XN }^N

\] where \(1/N\) may be identified with \(\epsilon\). The generalization affects two features of this formula. First, the terms \(X/N\) aren't constant, i.e. independent of \(t\), but they gradually evolve with \(t\) because they depend on \(V(t)\). Second, we mustn't forget about the time ordering. Both modifications are easily incorporated. The first one is acknowledged by writing \(X\) inside \(\exp(X)\) as the integral over time; the second one is taken into account by including the "metaoperator" of time-ordering. (I call it a "metaoperator" so that it suppresses your tendency to think that it's just an operator on the Hilbert space. It's not. It's an abstract symbol that does something with genuine operators on the Hilbert space. What it does is still linear – in the operators.)

With these modifications, we see that the evolution map is simply\[

\ket\psi\mapsto {\mathcal T} \exp\zav{ \int_{-\infty}^{+\infty}\dd t\, \zav{ \frac{V(t)}{i\hbar} } } \ket\psi

\] The time-ordered exponential is an explicit form for the evolution operator (the \(S\)-matrix) that simply evolves your Universe from minus infinity to plus infinity. In classical physics, you could rarely write such an evolution map explicitly but quantum mechanics is, in a certain sense, simpler. Linearity made it possible to "solve" the most general system by an explicit formula.

Once we have this "time-ordered exponential", we may deal with it in additional clever ways. The exponential may be Taylor-expanded, assuming that we don't forget about the time-ordering symbol in front of all the monomial terms in the Taylor expansion. The operators \(V(t)\) are polynomial in the fields and their spacetime derivatives: we allow each "elementary field" factor to either create or annihilate particles in the initial or final state (these elementary fields will become the inner end points of external lines of Feynman diagrams); or we keep the elementary fields "ready to perform internal services". In the latter case, we will need to know the correlators such as\[

\bra 0 \phi_i(\vec x,t) \phi_j(\vec x', t')\ket 0

\] which is a sort of a "response function" that may be calculated – even by the operator approaches – and which will play the role of the propagators. The remaining coefficients and tensor structures seen in \(V(t)\) will be imprinted to the Feynman rules for the vertices, the places where at least 3 lines meet.

I suppose you know these things or you will spend enough time with the derivation so that you understand many subtleties. My goal here isn't to go through one particular method in detail, however. My goal is to show you different ways how to look at the derivation of the Feynman diagrams. They seem conceptually or philosophically very different although the final predictions for the probability amplitudes are exactly equivalent.

Feynman's original method: "sum over histories" of fields

Feynman originally derived the Feynman rules by "summing over histories" of fields. The very point of the "sum over histories" approach to quantum mechanics is that we consider a classical system, the classical limit of the quantum system we want to describe, and consider all of its histories, including (and especially) those that violate the classical equations of motion. For each such a history or configuration in the spacetime, we calculate the action \(S\), and we sum i.e. integrate \(\exp(iS/\hbar)\) over all these histories, perhaps with the extra condition that the initial and final configurations agree with the specified ones (those that define the problem we want to calculate).

(See Feynman's thesis: arrival of path integrals, Why path integrals agree with the uncertainty principle, and other texts about path integrals.)

We have already mentioned that we're dividing the action, Lagrangian, or Hamiltonian to the "free part" and the "interacting part". We're doing the same thing if we use this Feynman's original method, too. To deal with the external lines, we have to describe the wave functions (or wave functionals) for the multiparticle states; this task generalizes the analogous problem with the quantum harmonic oscillator to the case of the infinite dimension and I won't discuss it in detail.

What's more important are the propagators, i.e. the internal lines, and the vertices. The propagators produce the inverse operator \(P_{ij}^{-1}\) from the Lagrangian again. These "Green's functions" have the property I have informally mentioned – they solve the "wave equation" with the Dirac delta-function on the right hand side; and they are equal to the two-point correlation functions evaluated in the vacuum.

But Feynman's path integral has a new way to derive the appearance of this inverse operator as the propagator. It boils down to the Gaussian integral\[

\int \dd^n x\,\exp(\vec x\cdot M\cdot \vec x) = \frac{\pi^{n/2}}{\sqrt{\det M}}.

\] but what is even more relevant is a modified version of this integral that has an extra linear term in the exponent aside from the bilinear piece:\[

\int \dd^n x\,\exp(\vec x\cdot M\cdot \vec x+ \vec J\cdot \vec x) = \dots

\] This more complicated integral may be solved by "completing the square" i.e. by the substitution\[

\vec x = \vec x' - \frac{1}{2} M^{-1}\cdot \vec J.

\] With this substitution, after we expand everything, the \(\vec x'\cdot \vec J\) "mixed terms" get canceled. As a replacement, we produce an extra term\[

-\frac{1}{4} \vec J\cdot M^{-1} \cdot \vec J

\] in the exponent; the coefficient \(-1/4\) arises as \(+1/4-1/2\). And because \(M\) is the matrix that is generalized by our operator \(P_{ij}\) discussed previously, we see how the inverse \(P^{-1}_{ij}\) appears sandwiched in between two vectors \(\vec J\).

The strategy to evaluate the Feynman's path integral is to imagine that this whole integral is a "perturbation" of a Gaussian integral we know how to calculate. We work with all the \(V(\vec x,t)\) interaction terms as if they were general perturbations similar to the \(\vec J\) vector above, and in this way, we reproduce all the vertices and all the propagators again.

Note that I have been even more sketchy here because this text mainly serves as a remainder that there exists a "philosophically different attitude" to the Feynman diagrams that one shouldn't overlook or dismiss just because he got used to other techniques and a different philosophy. If you want to calculate things, it's good to learn one method and ignore most of the others so that you're not distracted. But once you start to think about philosophy and generalizations, you shouldn't allow your – often random and idiosyncratic – habits to make you narrow-minded and to encourage you to overlook that there are completely different ways how to think about the same physics. These different ways to think about physics often lead to different kinds of "straightforward generalizations" that might look very unnatural or "difficult to invent" in other approaches.

In science, one must disentangle insights that are established – directly or indirectly supported by the experimental data – from arbitrary philosophical fads that you may be promoting just because you got used to them or for other not-quite-serious reasons. Of course, this broader point is the actual important punch line I am trying to convey by looking at a particular technical problem, namely methods to derive the Feynman rules.

Feynman's other method: "sum over histories" of merging and splitting particles

Once I have unmasked my real agenda, I will be even more sketchy when it comes to the third philosophical paradigm. You may "derive" the Feynman rules, at least qualitatively, from the "first-quantized approach" emulating non-relativistic quantum mechanics.

Again, in this derivation, we are "summing over histories". But they're not "histories of the fields \(\phi_i(\vec x,t)\)" as in the approach from the previous section – the original method Feynman exploited to derive the Feynman rules. Instead, we may sum over histories of ordinary mechanics, i.e. over histories of trajectories \(\vec x(t)\) for different particles in the process.

This approach, emulating non-relativistic quantum mechanics, the propagators \(D(x,y)\) arise as the probability amplitude for a particle to get from the point \(x\) of the spacetime to the point \(y\). It just happens that the form of the propagators – which have been interpreted as matrix elements of the "inverse wave operator" \(P^{-1}_{ij}\); and as two-points functions evaluated in the vacuum – may also be interpreted as the amplitude for a particle getting from one point to another.

Well, this works in some approximations and one needs to deal with antiparticles properly in order to restore the Lorentz invariance and causality (note that the sum over particles' trajectories still deals with trajectories that are superluminal almost everywhere, but the final result still obeys the restrictions and symmetries of relativity!) and it's tough. At the end, the "derivation" ends up being a heuristic one.

But morally speaking, it works. In this interpretation, a Feynman diagram encodes some histories of point-like particles that propagate in the spacetime and that merge or split at the vertices which correspond to spacetime points at which the total number of particles in the Universe may change (this step would be unusual in non-relativistic quantum mechanics, of course). The path integral over all the paths of the internal particles gives us the propagators; the vertices where the particles split or join must be accompanied by the right prefactors, index contractions, and other algebraic structures. But in some sense, it works.

It's this interpretation of the Feynman diagrams that has the most straightforward generalization in string theory. In string theory, we may imagine cylindrical or strip-like world sheets – histories of a single closed string or a single open string propagating in time – and they generalize the world lines. The path integral over all histories like that, between the initial closed/open string state and the final one, gives us a generalized Green's function for a single string.

And in string theory, we simply allow the topology of the world sheet to bd nontrivial – to resemble the pants diagram or the genus \(h\) surface with additional boundaries or crosscaps – and it's enough (as well as the only consistent way) to introduce interactions. While the interactions of point-like particles are given by vertices, "singular places" of the Feynman diagrams, and this singular character of the vertices is ultimately responsible for all the short-distance problems in quantum field theories, the world sheets for strings have no singular places at all. They're smooth manifolds – each open set is diffeomorphic to a subset of \(\RR^2\), especially if you work in the Euclidean signature – but if you look at a manifold globally (and only if you do so), you may determine its topology and say whether some interactions have taken place.

So this third method of interpreting the Feynman diagrams – as the sum over histories of point-like particles in the spacetime that are allowed to split and join at the vertices – which was the "most heuristic one" and the "method that was least connected to exact formulae" encoding the mathematical expressions behind the Feynman diagrams actually becomes the most straightforward, the most rigorous way to derive the analogous amplitudes in string theory.

Take the world from another point of view, interview with RPF, 36 minutes, PBS NOVA 1973. At 0:40, he also mentions that brushing your teeth is a superstition. Given my recent appreciation of the yeasts that are unaffected by the regular toothpastes, I started to think RPF had a point about this issue, too.

If you got stuck with a particular "philosophy" how to derive the Feynman rules, e.g. with Dyson's series, it could be much harder – but not impossible – to derive the mathematical expressions for multiloop string diagrams. There have been many methods due to Richard Feynman mentioned in this text but once again, the most far-reaching philosophical lesson is one that may be attributed to Richard Feynman as well:
Perhaps Feynman's most unique and towering ability was his compulsive need to do things from scratch, work out everything from first principles, understand it inside out, backwards and forwards and from as many different angles as possible.
I took the sentence from a review of a book about Feynman. It's great if you decompose things to the smallest possible blocks, rediscover them from scratch, and try to look at the pieces out of which the theoretical structure is composed from as many angles as you can. New perspectives may give you new insights, new perceptions of a deeper understanding, and new opportunities to find new laws and generalize them in ways that others couldn't think of.

And that's the memo.

P.S.: BBC and Discovery's Science Channel plan to shoot a Feynman-centered historical drama about the Challenger tragedy.

Prayer for Marta ["Let the peace remain with this land. Let anger, envy, jealousy, fear and conflicts subside, let them subside. Now when your lost control over your things will return to you, the people, it will return to you..."], an iconic politically flavored 1968 song by which the singer restarted freedom lost in 1968 during the Velvet Revolution in 1989.

P.P.S.: Ms Marta Kubišová, a top Czech pop singer in the late 1960s (Youtube videos), refused to co-operate with the pro-occupation establishment after the 1968 Soviet invasion which is why she became a harassed clerk in a vegetable shop rather than a pillar of the totalitarian entertainment similar to her ex-friend Ms Helena Vondráčková.

She just received Napoleon Bonaparte's Legion of Honor award, a well deserved one. Congratulations!

Monday, October 29, 2012

Is Hurricane Sandy unprecedented?

East Coast people's reports about their observations and courage are more than welcome

Update ex post facto: Sandy deserves the label "non-event" even more so than Irene did

The media are full of panic and strong words inspired by Hurricane Sandy, the "Frankenstorm" as some writers nicknamed it. Of course, this moniker – perhaps originally meant to indicate the Halloween time – is popular especially among those crackpots who want to argue that the storm is "man-made". Others are more modest and use the terms "mammoth, monster, superstorm" for Sandy.

A region of the New Jersey and Delaware coast is predicted to be "more like than not" to experience hurricane speeds (at least once, the 1-minute average speed must be above 74 mph). Also, see a photo gallery (27 pics) and the live broadcast from Weather Channel.

Events are being cancelled, institutions and major parts of the public transportation systems are being closed, and hundreds of thousands of people are being evacuated because of the late Category 1 hurricane that should land somewhere in New Jersey sometime on Monday evening, Eastern Daylight Saving Time, i.e. Tuesday morning European time. You may observe the predicted speed and status of the storm at the NOAA website.

I understand the sentiments behind the caution. On the other hand, I understand the skeptics much better. Many people clearly remember an isomorphic hysteria before the landing of Hurricane Irene in August 2011 which turned out to be a non-event. I remember a hurricane sometime around 2005 which was announced to land in New England – but the outcome could have been summarized by the sentence "It is raining in Boston".
Webcam: Try the rainy before-the-storm business-as-usual at The Times Square, other NYC places (sorry, the website may be overloaded)
The media are helping to preserve a certain kind of group think, a nearly religious admiration for the hurricane and its overwhelming power. I surely have the feeling that those people who refuse to evacuate their homes are considered heretics. They are not allowed to coherently describe their position in the media. But they have a rather good basis for their position, too.

Is this hurricane unprecedented? Well, it's a hurricane that comes late in the season so it may bring snow, too. (Some sources talk about an unprecedented interplay between a hurricane and some Arctic air, and so on, but they haven't really done research whether this interplay is unprecedented, either.) Also, it will manage to land in New Jersey which is pretty far from the equator and the most typical places where hurricanes land if they land at all. Also, this hurricane is targeting New Jersey, nearby places, and perhaps New York. The City is even more inhabited than New Orleans and the proper targeting is what matters for the relevance of a hurricane, indeed.

However, none of these features is really unprecedented.

Sorry, this was the Twitter #Sandy real-time timeline brought to you by TRF. I hope that we haven't lost you.

Even when you restrict your attention to hurricanes affecting New Jersey, you may find many stories in the book linked at the top or in this summary. The web page offers you 21 hurricanes between 1821 and 2011 that affected New Jersey. Irene 2011 was a hurricane that landed in New Jersey – still, most people already forgot whether it was lethal or not – but it wasn't the first hurricane with this "citizenship". On September 16th, 1903, a borderline hurricane landed near Atlantic City. Its main effect could be summarized as four inches of rainfall. Some significant hurricanes affected New Jersey in 1821, 1878, 1889, the history is quite nontrivial.

And I have focused on proud New Jersey hurricanes. If we looked elsewhere, we could enumerate tons of other hurricanes, such as the 1960 Hurricane Donna that has improved Manhattan in this way:

What about late-season hurricanes in New Jersey? Sandy must be the first one, right? Well, Hurricane Hazel landed on October 15th, 1954. And you may find much later ones. On December 2nd and 3rd, 1925, a hurricane brought 70 mph winds to Atlantic City. So calling Sandy "unprecedented" means to be ignorant about the history, sometimes very recent history. There's nothing unprecedented about hurricanes landing in New Jersey, not even about the subset that lands in October.

The journalists like to hype things because it brings them profit. Every time the end of the world arrives on the following morning, lots of newspapers are sold. But maybe they also partly want to "help the nation" and by making the stories more dramatic, they make the people more disciplined. I have some understanding for all these things. But I would be much happier to see something different as well – journalists as impartial messengers who actually describe what is happening, how people respond, and what their reasons are. A large fraction of the citizens are skeptical about the "catastrophic predictions" and many of them have decided not to evacuate their homes for reasons that are often rather sound. It would be nice if the journalists sometimes behaved as journalists in a free society – which means journalists who report and allow the readers and viewers to decide – and they were not trying to pretend that all these people are uneducated hacks and criminals and that the journalists' main holy duty is to strengthen the influence of the "authorities".

A hurricane lands somewhere in the U.S. almost every year and while it's good when the public cooperates with its representatives, it's simply unreasonable to consider all such situations "exceptional states of war" equivalent to "martial law" because we would be living under such "martial law" most of our lives (hurricanes are not the only reasons that may spark such "exceptional" measures).

I am not claiming that Sandy must turn out to be another non-event; I don't really know. The wind are strong enough so that you don't want to walk in the middle of your town. But it's rather plausible that it will be another non-event and it's wrong for the society to pretend it is not plausible, not even if this game is supposed to help a "good cause".

Sunday, October 28, 2012

Doha, Qatar will host a climate conference

What a paradox!

Qatar is a peninsula located on the opposite side of the Pilsner Gulf than Iran. Its economy is the wealthiest one not only in the Islamic world – even in comparison with Kuwait and UAE – but it's almost certainly richer than the economy of your country. The GDP per capita exceeds $100,000; no kidding. Not bad for a nation (2 million people) whose average IQ is 78. Sometimes if your ancestors are dumb enough to choose a desert as their home, you may benefit out of the stupidity because even a desert may harbor amazing wealth.

The conference center above in the capital, Doha, will host the 18th session of the Conference of Parties of the UNFCCC, the loons who want to regulate carbon dioxide. That's what I call an irony. ;-) The meeting will start in one month, on November 26th, and will continue through December 7th.

You must have seen some of the unbelievable projects showing the incredible wealth of Qatar. I will return to them momentarily. Where does the wealth come from? Well, yes: it's mostly about oil. Petroleum represents over 60% of the GDP, 70% of the government revenue, and 85% of export earnings. And make no doubts about it: even the remaining part of the GDP is inflated because the products and services are sold to people who are rich because of the oil (see costs of living in Doha: 1 Qatari Riyal is $0.27) so the export fraction 85% is surely a more faithful representation of the relative importance of the petroleum.

Now, this nation hosts a conference of nutcases who want to regulate if not ban fossil fuels. Do you really believe that the folks in Qatar want to do something like that and send their economy towards another Islamic role model such as Bangladesh whose average IQ is actually higher, 82, but due to the absence of petroleum, its GDP per capita is 60 times lower than the Qatari one? Do you think it's fair that Poland whose GDP per capita is 6 times lower is being harassed because of every new coal power plant while the people who annually export oil for more money than the whole Pole's annual income are treated as generous hosts who are surely helping Pachauri et al. to fight against the evil fossil fuels, aren't they?

Maybe the Qatari folks – who have the highest per-capita CO2 production in the world – will get an exception for their good services to UNFCCC, just like Al Gore and lots of other privileged assholes who don't walk the walk.

It's just totally crazy. All the participants of the climate conference must know that it is just a preposterous theater, a sequence of crazy excuses for an exotic trip, an attempt to create policies that are so incredibly unjust and irrational that they simply can't be adopted by almost any nation. Everyone must know that the true believers who will go to Doha are much more irritated by climate skeptics who are trashing their pathological climatic religion than by those who are producing and selling megatons of fossil fuels.

Now, let me return to their not quite cheap projects.

This is Syd Mead's plan to rebuild Doha. Personal jets flying around skyscrapers resembling huge wine glasses, or whatever it is. It may look like a picture from science-fiction novels except that many very similar projects are becoming a reality on a daily basis. For example, Pearl-Qatar is a multi-billion artificial island with skyscraper homes. See Google Maps to verify it is real.

Check also the "sustainable" Amphibious floating island. They probably also have some rotating skyscrapers even though the picture below may be from Dubai (in United Arab Emirates).

Sure, all these people will be thrilled by the ideas to ban fossil fuel. Also, they like means of transportations that don't depend on fossil fuels, for example bicycles. So Doha built a 35-kilometer-long network of cycling paths:

This is a paradise of energy-efficient transportation. Well, wait a minute. This cycling path is actually fully air-conditioned by mist. Imagine that while the EU or EPA bureaucrats are terrorizing you for turning on an incandescent light bulb, the biker on the picture enjoys the ride through 35 kilometers (yes, longer than the LHC) of a thick, not isolated tunnel that is cooled down from 115 Fahrenheit degrees i.e. 46 Celsius degrees (in the summer, average high is 41 °C i.e. 106 °F) to pleasant temperatures. Relatively to an incandescent light bulb, you surely don't need much energy for such a megafridge of pleasure that was built inside a desert, do you? ;-)

Well, a desert, sorry, I didn't want to insult you. For over $5 million, one hectare of the desert is going to pretend it is green. In the Czech Republic, you may buy one hectare of a forest for $2,500-$20,000, about three orders of magnitude cheaper than in Doha.

There are of course some projects related to the production of gas and other fossil fuels, for example the modest construction above. Also, Qatar will host the 2022 World Cup so it will build about a dozen of stadiums similar to the picture below – click at it for more (most of them are powered by solar panels):

Or compare this railway station with the railway station in your town or city:

Or check some of the hotels in Qatar.

I could go on and on and on. A vast majority of all these things has been paid, is being paid, or will be paid from oil revenue. But the climate warriors are going to transform Qatar to a global symbol of the fight against fossil fuels next month, anyway. The number of psychiatrists in the world is clearly insufficient to deal with this overload of cases. ;-)

Preons probably can't exist

Don Lincoln is the star of several cute Fermilab videos in which he explains various issues in particle physics. He's also authored several related texts for Fermilab Today.

He chose a much more controversial topic, namely preons, for his fresh article in the Scientific American:
The Inner Life of Quarks
Preons are hypothetical particles smaller than leptons and quarks that leptons and quarks are made out of. But can there be such particles?

At first sight, the proposal seems natural and may be described by the word "compositeness". Atoms were not indivisible, as the Greek word indicated, but they had smaller building blocks – the nucleus and the electron. The nuclei weren't indivisible, either – they had protons and neutrons inside. The protons and neutrons weren't indivisible – they have quarks inside.

Why shouldn't this process continue? Why shouldn't there be smaller particles inside quarks? Or inside the electron and other leptons?

Many people who pose this question believe that it is a rhetorical question and they don't expect any answer. Instead, they overwhelm you with detailed speculations `bout the possible composition of quarks and leptons while they possess lots of wishful thinking when they believe that all the problems they encounter are just details that can be overcome.

(Pati and Salam introduced preons for the first time in 1974. One of the other early enough particular realizations of preons were "rishons" by Harari, Shupe, and a young Seiberg, which means "primary" in Hebrew. I guess that prazdrojs and urquells would be the Czech counterparts. The terminology describing preons has been much more diverse than the actual number of promising ideas coming from this research. The names for "almost the same thing" have included prequarks, subquarks, maons, alphons, quinks, Rishons, tweedles, helons, haplons, Y-particles, and primons.)

However, the question above is a very good, serious question and it actually has an even better answer that explains why.

Mass scales and length scales

Since the mid 1920s and realizations due to Louis de Broglie, Werner Heisenberg, and a few others, we've known about a fundamental relationship between the momentum of a particle and the wavelength of a wave that is secretly associated with it:\[

\lambda = \frac{2\pi \hbar}{p}.

\] You may use units of mature particle physicists in which \(\hbar=1\). In those units, you may omit all factors of \(\hbar\) because they're equal to one and the momentum has the same dimension as the inverse length. Note that adult physicists also tend to set \(c=1\) because the speed of light is such a natural "conversion factor" between distances and times that has been appreciated since Einstein's discovery of special relativity in 1905.

In those \(\hbar=c=1\) units, energy and momentum (and the mass) have the same units, and space and time have the same units, too. The first group is inverse to the second group. Particle physicists love to use \(1\GeV\) for the energy (and therefore also momentum and mass); the inverse \(1\GeV^{-1}\) is therefore a unit for distances and times. One gigaelectronvolt is approximately the rest mass of the proton, slightly larger than the kinetic and potential energies of the quarks inside the proton; the inverse gigaelectronvolt interpreted as a distance is relatively close to the radius of the proton.

At any rate, the de Broglie relationship above says that the greater momentum a particle has, the shorter the wave associated with it is. Similarly, the periodicity of the wave obeys\[

\Delta t = \frac{2\pi\hbar}{E}

\] where \(E\) is the energy. The phase of the wave returns to the original value after a period of time that is inversely proportional to the energy. Now, it is sort of up to you whether \(E\) is the total energy that contains the latent energies \(E=mc^2\) or whether these terms are removed. If you want a fully relativistic description and you're ready to create and annihilate particles, you obviously need to include all the terms such as \(E=mc^2\).

On the other hand, if you study a non-relativistic system, it may be OK to remove \(E=mc^2\) from the total energy and consider \(mv^2/2\) to be the leading kinetic contribution to the energy. That's how we're doing it in non-relativistic quantum mechanics. These two conventions differ by a time-dependent reparameterization of the phase of the wave function (which isn't observable),\[

\psi_\text{relativistic}(\vec x,t) = \psi_\text{non-relativistic}(\vec x,t) \cdot \exp(-i\cdot Mc^2\cdot t/ \hbar)

\] where \(M\) is the total rest mass of all the particles. The relativistic wave function's phase is just rotating around much more quickly than the non-relativistic one.

Preons don't explain any patterns

Fine. Let's return to compositeness and preons. When you conjecture that leptons and quarks have a substructure, you want this idea to lead to exciting consequences. For example, you want to explain why there are many types (flavors) of leptons and quarks out of a more economic basic list of preonic building blocks. It's not a necessary condition for preons to exist but it would be nice and sort of needed for the idea to be attractive.

This goal doesn't really work with preons. Note that it did work with quarks; that's how Gell-Mann discovered or invented quarks. There were many hadrons and the idea that all these particles were composed of quarks was actually able to explain a whole zoo of hadrons – particles related to the proton and neutron, including these two – out of a more economic list of types of quarks.

Gell-Mann's success can't really be repeated with the preons. The list of known leptons and quarks is far from "minimal" but it is not sufficiently complicated, either. Quarks have three colors under \(SU(3)_c\). And both leptons and quarks are typically \(SU(2)_W\) doublets. And both leptons and quarks come in three generations.

These are three ways in which there seems to be a "pattern" in the list of types of quarks and leptons; three directions in which the lists of quarks and leptons seem to be "extended". But none of them may be nicely explained by preons. First, you can't really explain why there are \(SU(2)_W\) doublets or \(SU(3)_c\) triplets. Whatever elementary particles you choose, they must ultimately carry some nonzero \(SU(2)_W\) and \(SU(3)_c\) charges – and the charges of the doublets and triplets are really the minimal ones (the simplest representations) so whatever the preons are, they can't really be simpler than quarks or leptons.

(Here I am assuming that the gauge bosons and gauge fields aren't "composite". The possibility of their compositeness is related to preons and the discussion why it's problematic would be similar to this one but it would differ in some important details. The conclusion is that composite gauge bosons are even more problematic than preons.)

Also, you won't be able to produce three families out of a "simpler list of preons". To produce exactly three families, you need something that comes in three flavors, i.e. a particle of "pure flavor" that has three subtypes and that binds to other particles to make them first- or second- or third-generation quarks or leptons. But there must still be other particles that carry the weak and strong charges so the result just can't be simpler.

The comments above were really way too optimistic. The actual problems with the "diversity of the bound states" that you get out of preons are much worse. Much like there are hundreds of hadron species, you typically predict hundreds of bound states of preons. Moreover, they should allow multiple arrangements of the preons' spins, they should be ready to be excited, and they should produce much more structured bound states. None of these things is observed and the predicted structure just doesn't seem to have anything to do with the observed, rather simple, list of quark and lepton species.

But there exists a problem with preons that is even more serious: their mass.

If it makes any sense to talk about them as new particles, they must have some intrinsic rest mass, much like quarks and leptons. What can the mass be? We may divide the possibilities to two groups. The masses may either be smaller than \(1\TeV\) or greater than \(1\TeV\). I chose this energy because it's the energy that is slightly smaller than the LHC beams and that is already "pretty nicely accessible" by the LHC collider. Maybe I should have said \(100\GeV\) but let's not be too picky.

If the new hypothetical preons are lighter than \(1\TeV\), then the new hypothetical particles are so light that the LHC collider must be producing them rather routinely. If that were so, they would add extra bumps and resonances and corrections and dilution to various charts coming from the LHC. Those graphs would be incompatible with the Standard Model that assumes that there are no preons, of course. But it's not happening. The Standard Model works even though it shouldn't work if the preons were real and light.

So we're left with the other possibility, namely that preons are heavier than \(1\TeV\) or \(100\GeV\) or whatever energy similar to the cutting-edge energies probed by the LHC these days. But that's even worse because the very purpose of preons is to explain quarks and leptons as bound states of preons – and the known quarks and leptons are much lighter than \(1\TeV\).

To get a \(100\MeV\) strange quark, to pick a random "mediocre mass" example, the rest mass of preon(s) inside the quark, several \(\TeV\), would have to be almost precisely cancelled by other contributions to the mass and energy, with the accuracy better than 1 in 10,000. Clearly, the extra terms can't be kinetic energy which is positively definite: the compensating terms would have to be types of negative (binding) potential energy.

But it's extremely unlikely for the energy to be canceled this accurately, especially if you expect that the cancellation holds for many different bound states of preons (because many quarks and leptons are light).

Note that the virial theorem tells us that in non-relativistic physics, it's normal that the kinetic energy and the potential energy are of the same order. For example, for the harmonic oscillator with the \(kx^2/2\) potential energy, the average kinetic energy and the average potential energy are the same. For the Kepler/Coulomb problem, \(V\sim - k/r\), and the kinetic energy is \((-1/2)\) times the (negative) potential energy. More generally,\[

2\langle E_{\rm kin}\rangle = -\sum_{m=1}^N \langle \vec F_m\cdot \vec r_m\rangle

\] and if the potential goes like \(V\sim k r^n\), then \[

\langle E_{\rm kin} \rangle =\frac{n}{2} \langle V\rangle.

\] If you need the potential energy to cancel, you have to assume \(n=-2\). But the attractive potentials \(-1/r^2\) are extremely unnatural in 3+1 dimensions where \(-1/r\) is the only natural solution to the Poisson-like equations you typically derive from quantum field theories. You won't be able to derive them from any meaningful theory. Moreover, relativistic corrections will destroy the agreement even if you reached one. I was assuming that the motion of preons may be represented by non-relativistic physics – because the preons are pretty heavy and at relativistic speeds, they would be superheavy. If you assume that they're heavy and relativistic (near the speed of light), you will face an even tougher task to compensate their relativistically enhanced kinetic energy.

Even if you fine-tuned some parameters to get a cancellation, it will probably not work for other preon bound states. The degree of fine-tuning needed to obtain many light bound states is probably amazingly high. And we're just imposing a few conditions – the existence of light bound states that may be called "leptons and quarks". We should also impose all other known conditions – e.g. the non-existence of all the other bound states that the preon model could predict and the right interactions of the bound states with each other and with other particles – and if we do so, we find out that our problems are worse than just a huge amount of fine-tuning. We simply won't find any working model at all even if we're eager to insert arbitrarily fine-tuned parameters.

If you think about the arguments above, you are essentially learning that you shouldn't even attempt to explain light elementary particles – those that are lighter than the energy frontier, e.g. the energy scale that is being probed by the current collider – as composites. It can never really work. Quarks and leptons are much lighter than the LHC beam energy and because no sign of compositeness (involving new point-like particles) has been found, it really means that there can't be any.

Compositeness has done everything for us

So while the idea of compositeness is responsible for many advances in the history of physics, nothing guarantees that such "easy steps" may be done indefinitely. In fact, it seems likely that there won't be another step of this sort although some bold proposals that the top quark etc. could still be composite exist and are marginally compatible with the known facts.

After all, wouldn't you find it painful if the progress in physics were reduced to repeating the same step "our particles are composed of even smaller ones" that you would repeatedly and increasingly more mechanically apply to the current list of particles? The creativity in physics would be evaporating.

There exists a sense in which quarks and leptons are composite and the counter-arguments above are circumvented. In string theory, a lepton or a quark is a string. That means that you may interpret each such elementary particle as a bound state of "many string bits", pearls or beads along the string. If the number of conjectured "smaller building blocks" becomes infinite, like it is in the case of the stringy shape of an elementary particle, the cancellation between the kinetic and potential energy may become totally natural.

Despite the inner structure of elementary particles, string theory has an explanation why there are massless (or approximately massless, in various approximations) particles in the stringy spectrum. To some extent, this masslessness is guaranteed by having the "critical spacetime dimension" \(D=10\) or \(D=26\) for the superstring and bosonic string case, respectively. Well, string theory circumvents another problem we mentioned, too. We said that the kinetic energy is positive and the sum of all such positive terms must be positive, too. However, string theory uses the important fact that the sum of all positive integers equals \(-1/12\) which provides us with a very natural opportunity to cancel infinitely many terms although all of them seem to be positive.

Comparing preons and superpartners

The LHC hasn't found traces of any new particles beyond those postulated by the Standard Model of particle physics yet. However, that doesn't mean that all proposals for new physics are in the same trouble. In particular, I think it's important to explicitly compare preons with superpartners predicted by the supersymmetry.

At some point in the discussion above, I mentioned that preons could be either lighter or heavier than \(1\TeV\). The case of "light new particles" is generally excluded by the LHC (and previous experiments) because we would have already produced these new particles if they existed and if they were light.

The case of preons heavier than \(1\TeV\) was problematic because their "already high mass" must have been accurately cancelled by some negative contributions to the total energy/mass of the bound states and the negative potential energy required to do so seemed impossible, fine-tuned, and generally hopeless.

But the case of superpartners heavier than \(1\TeV\) doesn't have any problems of this sort. No supersymmetry phenomenologist really has any "rock solid" argument of this sort that would imply that the gluino is lighter than \(1\TeV\) or heavier than \(1\TeV\). We just don't know, these new particles may be discovered at every moment, and even at several \(\TeV\) or so, they still immensely improve the situation with the fine-tuning of the Higgs mass etc.

So while preons are pretty much completely dead – because you just can't construct light particles out of heavy ones, if I oversimplify just a tiny bit – superpartners remain immensely viable and well-motivated. The superpartners may still be rather light – the lower bound on their mass are often significantly lower than the lower bounds on other particles' masses in models of new physics – but there's nothing wrong about their being much heavier, either.

Much like in many texts, it's important not to become a dogmatic advocate of some ideas you decide to "love" in the first five minutes of your research. You could fall in love with the preons. Except that if you impartially study them in much more detail, you find out that this paradigm doesn't really agree with the known features of the world of particles well and some clever enough arguments may actually exclude rather vast and almost universal classes of such models. You should never become a blinded advocate of a theory who becomes blind to arguments of a certain type, e.g. the negative ones that unmask a general disease of your pet theory.

Preons are pretty much hopeless while other models of new physics remain extremely well motivated and promising.

And that's the memo.

P.S.: There will be a Hadron Collider Physics HCP 2012 conference in Kyoto in two weeks; see some of the ATLAS talks under HCP-2012. The detectors should update some of their data from 5 to 12+ inverse femtobarns of the 2012 data which means from 10 to 17 inverse femtobarns of total data. It's just a 30% improvement in the accuracy. Expect much more in March 2013 in Moriond.

Also, Czechia celebrates the main national holiday today, the anniversary of the 1918 birth of Czechoslovakia.