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So I’ve decided to a break from the standard progression of pedagogical posts about theoretical particle physics and comment on the recent Higgs Boson discovery. First, you can view the actual talk where the results were presented here. This video consists of two talks, one by Joe Incandela presenting the results of the CMS collaboration and the other by Fabiola Gianotti presenting the results of the ATLAS collaboration, of which my wife Elisa is a member! Some of my favorite parts are:

  • 27:00-29:00: The audible gasp in the room is for the bump on the plot. This is a plot of the number of times a particular reaction occurs, as a function of energy of the products of the reaction. An enhancement in such a plot corresponds to the presence of an intermediate unstable particle produced in the reaction.
  • 36:00-38:00: The crowd goes wild when the significance of the signal is reported to be 5.0σ, this means it not just ‘evidence’ but ‘discovery’. These words are given precise statistical meaning in the particle physics community. The attainment of ‘discovery’ was (at least to me) a huge surprise.
  • 50:00-52:00:The end of the first talk. Some nice closing remarks.
  • 1:18:00-1:20:00: A bump is seen by ATLAS which is AT THE SAME ENERGY as CMS. This is big news.
  • 1:33:00-1:37:00: The statistical significance of the ATLAS bump is quantified. Double whammy! This is 5.0σ as well! Discovery by two independent experiments. Both the crowd and the speaker can barely contain themselves.
  • 1:40:00-1:46:00: The end of the second talk. I think some nice remarks. Also, teary eyed Peter Higgs (the old guy with the bushy eyebrows) steals the show. Finally some remarks by Rolf-Dieter Heuer, the head of CERN.
  • 1:52:00-1:54:00: Remarks by former head of CERN Sir Christopher L. Smith. It took a while to find this damn particle!
  • 1:55:00 – 1:59:00: Remarks by the theorists who theorized the Higgs. My favorite parts: Peter Higgs glad he lived to see the day and François Englert missing Robert Brout, who did not live to see the day.

Anyway, while the video is as long as two Downton Abbey episodes, I can’t promise it’s any more exciting.

WARNING: If particle physics were a newspaper, the remainder of this post would belong in the ‘Opinion’ section. I’ve spent the last week or so thinking a bit about the implications of this discovery with some of my colleagues, and I’m afraid I don’t share the unbridled optimism of the press releases and articles. Undoubtedly, this discovery was historic as it (in some sense) completes the Standard Model and represents many years of dedicated blah blah blah. However, so far all indications suggest that this Higgs boson is ‘standard’ and is precisely the one that has been described in particle physics textbooks for that last ~20 years!

Of course, it is of the utmost importance that the properties of this new particle are measured precisely, as they certainly will be in the upcoming years at the LHC. However, if this particle IS the garden variety Higgs, there are few suggestions that anything else is waiting to be discovered. For example, the Higgs mass is linked to the energy at which the Standard Model ceases to make sense as a self-consistent theory. This relation has been updated recently by (amoung others) some of my CERN colleages, Degrassi, et al., in light of the recent discovery.

Fig. 1: The energy at which the Standard Model breaks down plotted against the mass of the Higgs boson. For the ~125GeV Higgs that has been discovered, the breakdown energy is waaay beyond the reach of the LHC and possibly any future earth-based particle accelerator. Figure taken from here.

The main idea from this paper which I want to emphasize is illustrated in Fig. 1. As has been known for a while, but I think not really mentioned so much, a ~125GeV Higgs boson implies that the Standard Model as we know it could be a perfectly complete theory up to VERY large energies, like ~1,000,000,000,000 GeV or so. For comparison, the maximum planned energy of the LHC is 14,000 GeV! Of course this doesn’t mean that there aren’t any new phenomena within the reach of the LHC, just that there doesn’t have to be any.

One of the few reasons for optimism is the phenomenon of ‘Dark Matter’ in astrophysics. It seems that a reasonable explanation of  the astrophysical observations is the existence of a new particle that only interacts weakly with the Standard Model particles. Such a lazy particle is affectionately termed a Weakly-Interacting Massive Particle, or WIMP for short. If one makes some simplifying assumptions about the expansion history of the universe and the Dark Matter production mechanism, a rough estimate for its energy scale is somewhat close the energies being probed at the LHC. This is the so-called ‘WIMP miracle’, and may mean that Dark Matter can be produced at CERN.

In conclusion, the Higgs boson discovery was certainly a monumental occasion for particle physics. However, there is a very real possibility that it could be the ONLY major discovery at CERN. If that’s the case, the graphic at the top of the post, which is taken from Hip-Hop is Dead (a Nas album), may also describe accelerator based particle physics as well as the academic careers of many students and postdocs currently working in the field.  Of course, in hip-hop the emergence of several new genres and communities has (in my mind) proven Nas wrong, I just hope the same thing happens in particle physics.

Respect the Beard

Rather than continue the classical line of High Energy Physics pedagogy, i.e. Special Relativity, Quantum Field Theory, etc., I’d like to pause for a bit and talk about one of the neat little curiosities about quantum mechanics. To begin with, I’ll recall the main equation from our previous post on Quantum Mechanics

This equation states that if a system is in state xi at time ti, to calculate the probability that it is in state xf at some later time tf I have to sum up all possible paths the system could take between the two states, and assign to each path an “arrow” given by the complex phase on the right side of the equation. This sum results in a complex number, and if I take the magnitude (distance from zero) of this complex number and square it, I get the probability.

The neat little nugget is that if I make a simple change to this equation, namely by treating the time variable on the right as an imaginary number (what does that mean, really??), it becomes the following

where the minus sign comes out basically because i times itself is minus 1. I’ve glossed over the details a bit, but this technique is known as Wick rotation. Anyway, this equation now looks like one that was derived nearly 150 years earlier, namely that of the Partition Function in statistical mechanics.

Statistical mechanics is one of my favorite branches of physics. It uses statistical techniques to describe systems that consist of many, many constituent parts. For example, in an introductory physics class, you usually talk about balls being thrown in the air, and the speed of trains and so forth, but these concepts are useless to describe a pot of water boiling. Am I to calculate the individual positions and velocities of all the individual water molecules in the pot? NO! That would be stupid. Instead what I want to know are “thermodynamic” properties of the water, like temperature and (if you ever had a grandmother make you soggy beans in one of those pots where you clamp the lid on) pressure. Statistical mechanics is able to bridge the gap between the individual properties of the water molecules and the thermodynamic properties of the pot as a whole.

This connection is made through the partition function, which (in some sense) sums up all the ways that energy can be partitioned among all the particles in the system. This function alone is enough to obtain basically all the thermodynamic information about the system. Aren’t you excited??? This was a revolution! Using the physics concepts for the microscopic single particles, like velocity, force, energy, and so forth, I’m able to write down the partition function, which then allows me to calculate macroscopic properties like temperature and pressure.

Statistical Mechanics was founded by some pretty cool guys, among them Ludwig Boltzmann, James Clerk Maxwell, and perhaps most importantly, J. Willard Gibbs, who was one of the greatest American physicists. All these guys rocked some serious beards, with Boltzmann having the best one. As someone who also has a beard, I feel a special kinship to these guys. I’ll also mention my favorite bearded basketball player, James Harden, whose picture appears at the top of the post. His Oklahoma City Thunder begin the NBA playoffs this week. GO THUNDER!!!

Anyway, when most people talk about ‘modern’ physics they usually mean quantum mechanics, special relativity, and the like, but for me statistical mechanics was really the advent of the modern era in physics. This is because statistical mechanics made direct reference to properties of the individual particles in a system, which as we discussed before, can’t be measured in practice. The idea of using these ‘invisible’ properties to formulate a theory was a radical one, and at least for Boltzmann, the scientific establishment was not initially on beard, umm, i mean on board. The scientific philosophy of the era was something like the positivism of Ernst Mach (also amply bearded) which basically said that science should only be concerned with directly observable phenomena (NOTE: I am not a philosophy of science expert). Of course, atomic theory, nuclear physics, etc. all disagree with this way of thinking, but Mach was very opposed to Boltzmann’s ideas. A more modern view was espoused by Karl Popper (not bearded) which instead demands that a valid scientific theory be ‘falsifiable’, i.e. able to be proven wrong.

Ok, to wrap things up, there’s a very deep connection between quantum mechanics and statistical mechanics. This connection is made by examining the defining equation of quantum mechanics and treating time as a purely imaginary number. When you do this, the ‘sum over possible paths’ in quantum mechanics becomes equivalent to a ‘sum over possible states’ in statistical mechanics. This is the beginning of many connections between the strange ‘quantum’ world and the less strange concepts of statistical systems. As for the ‘meaning’ of ‘imaginary time’, I’m not sure that there is one. As far as I’m concerned, it’s just a mathematical device used to establish this equivalence. If you disagree please comment!!!

Least Action Hero

        

At long last, the previously mentioned Quantum Mechanics post!!! Quantum Mechanics is essential to our understanding of the world, and was born around the turn of the (19th-20th) century. It’s rather counter-intuitive from the point of view of our everyday experience so things like thought experiments must be used to gain intuition about what’s going on. WARNING: although quantum mechanics is a very successful theory experimentally, there are many interpretations of what is actually happening. Indeed, there is an entire field of physics (which I am NOT an expert in) devoted to understanding quantum mechanics.

However, I think it is fair to say (please comment if you disagree) that for all currently observable phenomena, different interpretations of quantum mechanics are indistinguishable. Therefore, I’ll present the point of view I think is easiest to understand. We’ve got to start with `classical’ physics, that is all of physics before quantum mechanics. This is the physics of dropping balls and crashing cars and predicting when trains will arrive. Our intuition about how this works should be pretty good, as it describes the macroscopic world in which we live.

The most succinct way I know to formulate classical physics is in terms of the ‘Least Action Principle’: Given a starting configuration of a system and an ending configuration, I want to know how the system proceeds from the starting configuration to the ending configuration. For example, if a baseball (from a really boring American sport) is in my hand now and through my office window 2 seconds later, what path did it take to get there? For each possible path, like going on a straight line between my hand and the window or taking a detour around the moon before hitting the window, I can assign a number, called the ‘action’. The path going around the moon has waaaaaaay more action than the straight line path. In fact, the path that the baseball takes is the one with the least action. End of story. For a given starting and ending point, it always takes this least action path.

Of course, I can also turn this around and specify the initial position of the baseball (my hand) and the direction and magnitude of its initial velocity (toward the window and super fast). Once I do that, the path that the baseball takes is fixed. Always. If I throw a million baseballs through a million office windows (yeah, it’s that kind of day at work), all starting from my hand and ending at the window 2 seconds later, each one will follow the same least action path.

This calls for an equation! There’s a general principle in physics that things behave smoothly, except when they don’t. When I’m assigning an action to each path, two paths which are almost identical will have almost identical actions. As I move toward the least action path, the action should decrease in a more-or-less smooth fashion, with the minimum at the the least action path (duh). If I move past this least action (or ‘classical’) path, the action will increase again (also duh). So a cartoon plot of what’s going on might look like this:

Fig.1: As the path is smoothly varied toward the classical path, the action decreases. The tangent line at the classical path has zero slope

This plot is just a fancy way to say that the least action path is the one which has the minimum action. Because of that, the tangent to the curve at the least action path is flat, i.e. it has zero slope. We can use this fact to encapsulate all of classical physics in a single tiny equation:

                                                                                  (1)

This equation says that the path the system takes is the one for which the action has a flat tangent, i.e. the slope of the tangent of the action at the classical path is zero. This also means that as I change a tiny bit from the classical path the action does not change that much. The bottom of the well in the above plot is the only place where this is true as any other point on the curve does not have a flat tangent.

Finally! We can stop talking about boring old classical physics! The main point of the preceding discussion was that classical physics is deterministic, i.e. if I specify the start and end points, or the start point and the initial velocity, I can predict with certainty what the system will do at every moment of the future. When we are doing a classical physics homework problem (booo!) the `answer’ is x(t), the state of the system at all time, after specifying the start and end points, or the start point and initial velocity.

Quantum mechanics is way different. If I were doing a quantum mechanics homework problem (<cough> nerd!) the answer would be instead be a set of numbers for each time, usually written

                                                                         (2)

Each number is the probability of finding the system in state x at time t.  In our example, given the initial and final positions of the baseball (my hand and the shattered window), at each time I have no idea where the baseball is, I only have a probability for each point in space. In fact, at any given time, there is a non-zero (but very small) probability that the baseball may be on the moon! Of course I do know that the baseball has to be somewhere at each time so I can write

                                                                                   (3)

which just says that the sum of all probabilities at a certain time has to be one, i.e. that I will always find the baseball somewhere in the universe. The thing on the right side of Eq. (2) is a bit strange. The angle brackets inside the absolute value sign are called a probability amplitude and represent a complex number. What the hell is a complex number, and why am I using it to describe something in the real world? A complex number is just a number with two parts, a real part and an imaginary part. A complex number can be specified two ways, either by giving the real and imaginary parts, or by giving the distance from zero and the angle with the real axis. See the figure. Do it.

Fig. 2: A complex number has real and imaginary parts. It can also be specified by it's length (distance from 0) and the angle it makes with the Real axis.

Formally, I can write the two representations of the same complex number as

                                                                                                                      (4)

where x,y, r, and φ are given in the figure and i is the square root of -1 (crazy).

So, the problem of calculating the probability density ρ(x,t) has been reduced to calculating the complex probability amplitude. Then we just take the length of that complex number and square it to get the probability density. There are many ways to calculate this amplitude but I think the easiest one to understand was given by Richard Feynman (in his P.H.D thesis!) which expresses the amplitude as a sum over all paths (or histories) that the system could possibly take:

                                                                                                                                                                                                                                                                               (5)

This is by far the most complicated equation we’ve encountered so far.  The left side is the probability amplitude for finding the system in the specified initial and final states at the initial and final times, respectively. Remember, if we wanted the actual probability, we would have to take the length of this complex number and square it. The right side of Eq. 5 is a sum over all possible paths that start in the initial state and end in the final state. Each path gets weighted by a ‘phase factor’ in the sum. This is the exponential factor, which is just a complex number (look at Eq. 4) with length one. This equation also contains a new fundamental constant of Nature: Planck’s constant (h).


It is the small numerical value of this constant (h = 6.62606957(29)×10−34J·s) that makes quantum mechanical effects so far removed from our everyday world. For comparison, the action (which has units of [energy] [time]) of a 0.14kg baseball traveling at 75mph ( roughly 33.5m/s, way slower than during my teenage baseball years)  for 0.1 seconds is  78.55 J·s, or about 35 orders of magnitude (factors of 10) bigger than h!

The smallness of this constant may also be used to predict Eq. 1, the defining equation of classical physics. When summing up all the phase factors corresponding to all the possible paths in Eq. (5), we have to add the arrows head-to-tail. If we have an arrow pointing along the Real axis (angle = 0 degrees) and we add to it an arrow pointing the opposite way (angle = 180 degrees), we end up with nothing. In our case, the angle corresponds to the ratio S/h, the action of a particular path divided by Planck’s constant.  If we examine a random path in the sum, say the one that goes around the moon before hitting my office window, the action will change quite a lot (compared to h) if we vary the path a tiny bit. This is because this path does not have a flat tangent in Fig. 1. In fact, for every path that we choose in this way, there is a nearby path whose arrow points in exactly the opposite direction, and the effect of the two paths cancel out as we sum them.

The only path which does not have a nearby path to cancel it is the path for which the action does not change very much when we move away from it. This is precisely the condition that we gave earlier for the classical path!!! So for a system (like the baseball and window) where the action is much larger than Planck’s constant, the only path that significantly contributes to the sum in Eq. (5) is the classical path, and we can recover Eq. (1) which determines the condition for the classical path. Furthermore, for this situation the probability that the system does not follow the classical path is certainly not zero, but it is so small that we will never see it happen. Maybe not even once if I throw baseballs at my office window all day everyday for the entire age of the universe. Perhaps I should write a research grant application for such an experiment!

Anyway, now that we’ve seen that classical mechanics is recovered as a limiting case of the Feynman path integral, we can ponder some of the implications. Of course the probabilities calculated from the path integral agree extremely well with observations, so for systems where the action is not so much larger than h we can observe effects of this sum over paths. Some famous manifestations of this are the double-slit and Stern-Gerlach experiments, but there are a ton of others. I suppose the Keanu-Reeves-Bill-and-Ted’s-Excellent-Adventure quasi-philosophical interpretation of what’s happening would be that when deciding the probability for a certain initial and final state, Nature somehow travels all possible paths (assigning each one an arrow) before summing them up to get the probability, causing the `wave particle duality’ that is so peculiar in quantum mechanics. However, this is simply one interpretation.

Of course, there are a lot of things I glossed over in this post, like the problem of measurement in quantum mechanics, some of the paradoxes, and different interpretations (here is an overview). Anyway, all I’ve done is provide a rule for calculating probabilities, which are all that can be measured. Back to throwing baseballs at office windows!!!

Hello World!

Apologies for the title! I’m new to blogging but feel the need to share some of my thoughts on particle physics and other subjects on which I am (even) less qualified.

I’ve just started working at CERN as a Theory Fellow and have mixed feelings about the `message’ that makes it from experts to the general public. Particle physics is a wonderfully rich subject that does not lend itself to headlines and twitter feeds (watch out, old man alert!!). It is my intent to make this blog not about the bleeding edge state of research in the field (for this, simply google `God Particle’ and try not to vomit) but to write more pedagogical posts about topics which I think are central to understand the main ideas about a theory (The Standard Model, yes, with capital letters) that accomplishes no small feat: it can explain all interactions between elementary particles in our world.

I remember a discussion I had over a Tartiflette (cheesy potato awesomeness) with a senior CERN theorist last year that too much of what people hear about particle physics is related to what we don’t know. Of course, this is what is most interesting to experts in the field, but in order to understand the implications of things like `The Higgs Boson’ and `Supersymmetry’, one first needs to understand the foundations. Also, I believe that some of what we do know about elementary particle physics is pretty damn elegant and deserves to be shared.

Another all-too-prevalent pet peeve of mine is `experts’ taking license to pontificate about topics outside their realm of expertise (more old man ranting). I’ll try to avoid that in this blog, or to be explicit about when I’m trying to clarify my own understanding on a subject, hopefully to incite discussion. So in short, if you want a very non-sensational, somewhat sardonic, pedagogically oriented look at particle physics (which is in what could be a very exciting time), then this blog is for you.

Most likely the next post will be somewhat of an introduction to quantum mechanics. After that who knows! Oh, and the title is a bit of a joke; despite living in France (CERN is on the border between France and Switzerland), I speak maybe 10 words of French. Basically my wife (also a particle physicist at CERN) does all the talking when we go out. I’m studying, so hopefully this will change!

 

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