The following year, Hermann Grassmann created the exterior algebra, the algebra of anti-commuting entities known as Grassman Numbers. Then along came William Kingdon Clifford, who united Grassman’s algebra with Hamilton’s work on quaternions to create Clifford Algebra.

So, what’s the big deal here? Think spinors. Think fermions.

For quite some time, I’ve been working on a rather lengthy and detailed article on fermions and the Spin-Statistics Theorem. (There is also a lengthy and fairly comprehensive article in the works on neutrino physics, yet another on the Action Principle, and still another on Noether’s Theorem, symmetry, and conservation laws. Working on all of these in parallel is pretty time-consuming.) The aforementioned mathematical constructs are rather critical to understanding these concepts and will have to be fleshed out before I can really present that topic.  So watch this space, and consider the little mathematical family tree outlined above as a road-map of what is to come….

For more info, see:

• CLIFHISTORY
• Introduction to Clifford Algebra
• Clifford Algebra — from Wolfram MathWorld
• Clifford Algebras
• Clifford Algebras, Clifford Groups, and a Generalization of the …
• Geometric Algebra (Clifford Algebra)
• The Most Striking Equation in Mathematics « Galileo’s Pendulum
• Why Quaternions Matter « Galileo’s Pendulum
• Everything is Geometrical: Hermann Grassmann’s Algebra « Galileo’s Pendulum
• W. K. Clifford: The Geometry of Physics « Galileo’s Pendulum
• Two Particles Enter, No Particles Leave! « Galileo’s Pendulum
• Spinning Electron Got to Go Round « Galileo’s Pendulum
• Physics Quanta: From Identical Twins to Voltron, Physics Style « Galileo’s Pendulum

Read it and weep:

From the ICARUS paper.

It doesn’t end there. Other teams, including BOREXINO, LVD, and OPERA (now that they have identified possible sources of systemic error) will also be performing measurements. But it is looking like the “Ghost of OPERA” doesn’t have a ghost of a chance. So long ftl neutrinos.

• arxiv.org/ftp/arxiv/papers/1203/1203.3433.pdf
• CERN Press Release
• This Time, ICARUS Really DOES Refute OPERA | Of Particular Significance
• Superluminal Neutrinos are so 2011 | Cosmic Variance | Discover Magazine
• ICARUS Refutes Opera’s Superluminal Neutrinos – AGAIN!
• Neutrinos not faster than light : Nature News & Comment
• symmetry breaking » Blog Archive » New neutrino measurement finds particles obeying speed limit
• Not So Fast: More Evidence Neutrinos Aren’t Defying Einstein | Wired Science | Wired.com
• Not So Fast: Independent Measurement Shows Neutrinos Don’t Exceed Speed of Light | Observations, Scientific American Blog Network
• The Fat Lady Sings for OPERA : Starts With A Bang
• Neutrinos, Wishful Thinking, and the Sloth of Science « Galileo’s Pendulum
• Neutrinos in the News : Uncertain Principles

My previous articles on this topic:

• Putting the OPERA Affair in Perspective | Whiskey…Tango…Foxtrot?
• Not a Big Shocker | Whiskey…Tango…Foxtrot?
• ICARUS Nixes OPERA’s Superluminal Neutrino Result | Whiskey…Tango…Foxtrot?
• OPERA Redux | Whiskey…Tango…Foxtrot?
• FTL Neutrinos? Not so fast…. | Whiskey…Tango…Foxtrot?

Update: (April 2, 2012)

The Fat Lady has sung for FTL neutrinos. Matt Strassler has posted an excellent post-mortem of the situation based upon the limited available information.

Supersymmetry is a family of proposed extensions to the Standard Model of Particle Physics hoped to address several shortcoming in the Standard Model, including the Hierarchy Problem (why quarks are so much more massive than electrons, which in turn are so much more massive than neutrinos, and why the various fundamental forces have vastly different strenghts), reconciling General Relativity with quantum theory (i.e. quantum gravity), and unification of electroweak interactions with the strong nuclear force. Additionally, SUSY provides candidates for dark matter, as well as providing a mechanism for electroweak symmetry breaking. In short, SUSY, if valid, offers a way to fill in a LOT of gaps.

SUSY theories first came to prominence in the early 70′s, and gained considerable traction in the 80′s. It is even incorporated into many variants of string theory (with such variants referred to as superstring theories). Much is riding on SUSY, and one of the reasons for the creation of the Large Hadron Collider at CERN is to put SUSY to the test.

And those tests are not going well. SUSY predicts the existence of supersymmetric partner particles (such as squarks, selectrons, photinos, gluinos, etc.) corresponding to the normal SM particles (quarks, electrons, photons, gluons, etc.) but at higher masses. The lighter of these superpartners should be within mass ranges accessible to the LHC. Thus far,there is no sign of them. Recent results from the LHCb collaboration have ruled out a considerable chunk of the parameter space for SUSY in their studies of the decay of B_S and B_D particles. Furthermore, results from the LHC search for the Higgs boson have put tight constraints on SUSY models as well.

SUSY may not be dead yet, but she does appear to be pretty ill….

• SUSY Still in Hiding | Not Even Wrong
• Much food for thought at major physics conference | Quantum Diaries
• Rencontres de Moriond EW 2012: Seeing less would be just as good – CERN Bulletin
• Rencontres de Moriond EW 2012: Impact of a Higgs boson on supersymmetry – CERN Bulletin
• The Incredibly Shrinking Supersymmetry
• Still Waiting for Supersymmetry | Not Even Wrong
• SUSY And The Silence Of The (Roasted) Lamb
• Where Stands Supersymmetry (as of 4/2012)? | Of Particular Significance
• Is Supersymmetry Dead?: Scientific American
• The Plot Of The Week – No SUSY In New CMS Search
• The Smell of SUSY | Not Even Wrong

These two distinguished physicists were instructed to deliver their lectures at a level suitable for an undergraduate student who had taken an introductory course in quantum mechanics. By and large, I think they succeeded, although I would not have minded a bit more mathematical detail. There were occasions upon which certain intermediate steps were glossed over, leaving room for a bit of confusion.

The book is a quick read, weighing in at only 110 pages. (The two lectures from which it was drawn were only about an hour long each.) With such brevity, it could really be considered two essays (which I shall discuss separately below. Despite that brevity, each of the two contributors weighed in on some rather heady topics, although not in great depth.

“The reason for antiparticles” by Richard P. Feynman

It is entirely fitting that Feynman should have been chosen to give the very first of the Dirac Memorial lectures, as it was Dirac who laid the groundwork for quantum electrodynamics (QED) in a 1927 paper, and it was Feynman who filled in the gaps to turn QED into a useable theory with his 1942 dissertation on the path-integral formulation of QED. Feynman’s stated goal for his lecture was to demonstrate how Dirac’s 1928 unification of quantum mechanics with special relativity (resulting in the Dirac Equation) inevitably lead to the prediction of antiparticles. Interestingly enough, he managed to do this without actually invoking the Dirac Equation itself. Feynman also provided a secondary goal for his lecture, that of demonstrating the origins of the Pauli Exclusion Principle as a direct consequence of the existence of fermions (fermionic behavior also being a direct consequence of the Dirac Equation).

I should note that a video of Feynman’s lecture is available:

The video noise at the beginning can be ignored and does not cut into the content. Unfortunately, the resolution of the video is far too low to allow the diagrams and equations on the overhead transparencies to be read. Personally, I find it somewhat enjoyable to watch the video (to get the benefit of Feynman’s charismatic delivery) while following along in the book (to see the equations and diagrams).

While Feynman’s exposition did an admirable job of explaining the origin of antiparticles, I feel that it fell somewhat short with the secondary goal regard spin-statistics. He unfortunately glossed over the mathematical reasons for half-integer spins resulting in a sign change of the wave function upon particle exchange (or, equivalently, upon a 360 degree rotational coordinate transformation), although he touched upon other arguments for this, particularly in the context of charge, parity, and temporal transformations.

As for the primary goal, though, Feynman was in fine form, bringing to his lecture the clarity that has always been a hallmark of his lectures. As an added but unstated benefit, his approach to the problem presented some insite into the origins of his famous Feynman diagrams and their mathematical underpinnings. Peppered throughout the lecture are proto-Feynman diagrams (missing only the photons, which are replaced by unspecified “disturbances” due to an imposed potential) meant to enumerate the contributions to a standard perturbative expansion in quantum mechanics, whereas proper Feynman diagrams are a method of enumerating the contributions due to various interactions to a Lagrangian in the path-integral formulation of QED. In both cases, the diagrams serve as bookkeeping devices for catching all of the leading terms of an infinite series expansion.

“Towards the final laws of physics” by Steven Weinberg

The second half of the book is from Steven Weinberg, who built upon Feynman’s work and extended the principles of QED to realm of weak nuclear interactions to create electroweak theory. The focus of Prof. Weinberg’s contribution was at a somewhat higher lever, focusing on what we can glean from current current physical models to predict what an eventual “theory of everything” might look like. He started off with a brief, greatly over-simplified overview of quantum mechanics (in which he obliquely managed somewhat pull off what Feynman failed to do regarding the sign change of the wave function of fermions, which for me was the high point of his contribution). He then proceeded to go into an overview of using dimensional analysis to discuss divergent terms in the Standard Model Lagrangian and how such methods are used to predict energy ranges at which gravity must operate in order to be incorporated into quantum theory. Weinberg also touched upon the importance of symmetry rules in constructing a final theory. He then closed with a quick introduction to String Theory, which, at the time of the lecture, was just beginning to strongly capture the interest of theorists (despite having been around for a decade and a half at the time).

All in all, this quick little book is a worthwhile diversion for someone with a smattering of an interest in the topic and who isn’t afraid of a handful of equations.

But wait, there’s more. LHCb is reporting more CP violations, this time with B_s mesons. Tommaso Dorigo has more on this.

Still confused about CP violations? Have a look at this excellent overview by Carl Brannen over on Tommaso’s blog.

But, roughly three hours before the first light from SN 1987A had reached us, something perhaps more significant had taken place. At three neutrino detection experiments around the world, (Japan’s Kamiokande II, America’s IMB, and Russia’s Baksan), a flood of neutrino signals was detected at 7:36 GMT. And reconstruction of the trajectories of the incoming neutrinos made it clear that they had come from SN 1987A.^2,3

I said a flood. The fact is, 24 distinct neutrino events were detected. But in the world of neutrino physics, that is huge. It is estimated that for about 13 seconds, each square foot of the Earth was bombarded by approximately 100 trillion neutrinos from the supernova. (Yes, that is “trillion” with a “t.”) So, why were so few neutrinos detected?

The simple truth of the matter is that neutrinos are VERY difficult to detect. They do not interact with other matter via electromagnetic or strong nuclear interactions, but rather only via the weak nuclear force (responsible for beta decay) and gravity. Since the weak interaction has an extremely short range (about 10^-17m – an order of magnitude smaller than protons and neutrons), the odds of a neutrino hitting something that it can interact with are infinitesimal. Neutrinos can thus pass effortlessly through an entire planet or star with little chance of hitting anything.

And it is precisely this property of neutrinos which explains why the neutrinos from SN 1987A arrived roughly three hours before light did. These neutrinos were able to zip right out from the core of the exploding star and on out into space completely unimpeded, whereas light from the explosion had to bounce around inside the exploding star for a bit, repeatedly getting captured and re-emitted by the steller plasma. Even light, with its vast speed, takes a while to get anywhere when it is constantly being intercepted.

And it is also this same property of neutrinos which makes neutrinos so difficult to detect and study. Most neutrino detectors (but not all) consist of LARGE tanks of highly purified water, frequently located deep underground to screen out other types of particles that might produce false signals. Every once in a while, a neutrino will happen to slam into and interact with one of the particles in the tank, resulting in a cone of light being emitted, which is in turn detected by light sensors lining the walls of the tank. The shape, direction, and intensity of the light cone are used to reconstruct the properties of the detected neutrino, such as its energy and trajectory. The larger the tank, the more mass there is for incoming neutrinos to interact with, thus the greater odds that such an interaction will take place within the detector. The Kamiokande II detector had 3 kilotonnes of reaction mass.

Since the discovery of SN 1987A, larger and larger neutrino detectors have been constructed to improve our ability to study neutrinos. One of the largest in the world, IceCube, uses one million kilotonnes of the Antarctic ice sheet as its detection medium.

So what knowledge has been gleaned from these colossal detectors? For one thing, the study of solar neutrinos has revealed that that they change form. Electron neutrinos, muon neutrinos, and tau neutrinos can all transform into one another. This process, known as neutrino oscillation, implies that neutrinos have some mass, albeit a tiny amount. (Until this discovery, neutrinos were thought to be completely massless, as photons are. I’m currently working on a more detailed posting about neutrinos which digs into this topic a bit more deeply, as well as touching upon sterile neutrinos and Majorana neutrinos, and the overall history of neutrino research.) The study of solar neutrinos has also provided clues for astrophysicists regarding the fusion reactions which fuel the sun.

But the greatest insights to be gleaned from neutrino astronomy are still ahead – by improving our ability to peer into the past!

Using electromagnetic phenomena (light, IR, radio waves, UV, whatever), the Cosmic Microwave Background constitutes the greatest distance that astronomers can view. This great wall represents the point in our universe’s history some 360,000 years after the Big Bang marking the start of the Recombination Epoch. Prior to this, the universe was a dense, hot plasma which was opaque to electromagnetic radiation. The Recombination Epoch marks the creation of the first neutral atoms, bringing about a reduction in the opacity of the universe.

So, if the universe prior to this era was opaque to all EM phenomena, how can astronomers ever hope to observe beyond the CMB? Currently, there are two potential options: gravitational waves, and neutrinos.

Unfortunately, we don’t really know how to detect gravitational waves just yet. For that matter, we aren’t entirely certain that they exist. They are predicted by General Relativity, which has not let us down thus far, and we have indirect observational evidence for their existence through observations of the orbital behavior of binary neutron stars. ( Russell Hulse and Joe Taylor were awarded the Nobel Prize in Physics in 1993 for this.) Currently, several major experiments are underway to attempt to detect gravitational waves directly, including LIGO, LISA and DECIGO. But, it will take some time for such efforts to bear fruit. Even once we get to the point of detecting these waves, the equipment will have to be refined to have greater sensitivity and resolution.

The other option, neutrinos, which is much further along experimentally, represents the possibility of measuring the Cosmic neutrino background, a glimpse of the universe a mere 2 seconds after the Big Bang. Pulling off such measurements will require larger and more sensitive neutrino detectors with better angular resolution. (For more on the CNB, see here and here.)

I can hardly wait to see what fruits are born by both approaches.

1. “IAUC4316: 1987A, N. Cen. 1986″. 24 February 1987.
2. Nomoto, Ken’ichi; Shigeyama, Toshikazu (1987). “Supernova 1987A: Constraints on the Theoretical Model”. In Minas Kafatos. Supernova 1987a in the Large Magellanic Cloud. Cambridge University Press. section 3.2 Shock propagation time. ISBN 0-521-35575-3
3. Improved analysis of SN1987A antineutrino events. G. Pagliaroli, F. Vissani, M.L. Costantini, A. Ianni, Astropart.Phys.31:163-176,2009.

For more information, see the following:

• Supernova 1987A | AAVSO
• Supernova 1987A in LMC
• L6S6
• Happy 25th annniversary, Supernova 1987A! | Bad Astronomy | Discover Magazine
• The Great Neutrino Tsunami | The Hammock Physicist | Johannes Koelman

BREAKING NEWS: Error Undoes Faster-Than-Light Neutrino Results – ScienceInsider

UPDATE: More authoritative sources are reporting the possibility of TWO distinct sources of experimental error having been identified:

Flaws found in faster-than-light neutrino measurement : Nature News & Comment

The latest installment of “Minute Physics” (a video series which I absolutely adore) tackles a rather Herculean task: explaining the Standard Model of Physics in a minute. It is, of course, a VERY high-level overview, and alludes to future installments designed to go into more detail.

The focus of this video is the following mathematical expression:

(Pardon the poorly aligned slash through the D.  Plain vanilla found in WordPress doesn’t seem to handle Feynman slash notation very well.)

Unfortunately, the video doesn’t really say what this is. It is the Standard Model Lagrangian, or at least an abbreviated form of it. (More detailed forms of it can be seen here and here.)

“So, what the heck is a Lagrangian?” I hear you cry.

A Lagrangian is an expression which codifies the dynamics of a physical system, and essentially consists of the kinetic energy minus the potential energy of a system. It forms the heart of the Lagrangian formulation of classical mechanics, as well as the path-integral formulation of quantum field theory.

In essence, all of physics boils down to constructing the proper Lagrangian for a physical system. Once you have the correct Lagrangian, integrate it over time to arrive at the action. Invoking the Principle of Least Action, we know that the system will evolve in such a way as to minimize the action. That is all there is to it.

Okay, I’m obviously simplifying quite a bit here. Of course, the tricky part is constructing the correct Lagrangian.

This extremely stripped-down Lagrangian shown above includes contributions (terms with an F or a D) from gauge bosons –  photon,W, Z, and gluon fields,  contributions from fermionic fields (terms with a ψ), and terms with a ϕ refer to the Higgs field.

So, how does one start with this skeleton and build a fully fleshed-out Lagrangian with which actual calculations can be performed? Well, that gets a little hairy, and the discussion will have to be spread out over time. A big hint, though: it starts with Feynman diagrams.

For excruciating detail, see the following:

• Standard Model Lagrangian – a knol by soobtoob
• www.stfc.ac.uk/ppd/resources/pdf/standardmodel09.pdf ”The Standard Model”
• www.lepp.cornell.edu/~pt267/files/notes/FlavorNotes.pdf ”Just a Taste: Lectures on Flavor Physics”