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The philosophy of science Greek Philosophers

The philosophy of science is concerned with such questions as: What counts as science? How reliable are scientific theories? And what is the purpose of science? Over the years, many Quantum Diarists have struggled with such questions, none more than Byron Jennings, who has announced that he will soon retire from the blog. Many thanks to Byron for his many thought-provoking posts!

Not all philosophy is useless

By Byron Jennings | December 5, 2014
In this, the epilogue to my philosophic musing, I locate my view of the scientific method within the landscape of various philosophical traditions and also tie it into my current interest of project management.

From wavefunctions to detectors: how to think about particles

By Andrea Signori | November 9, 2014
This blog is all about particle physics and particle physicists. We can all agree, I suppose, on the notion of the particle physicist, right? There are even plenty of nice pictures up here! But do we know or are we aware of what a particle really is? This fundamental question tantalized me from the very beginning of my studies.

Foxes, hedgehogs and particle physicists

By Ken Bloom | November 11, 2012
Archilochus famously observed that “the fox knows many things, but the hedgehog knows one big thing.” This got me thinking: are particle physicists foxes or hedgehogs?

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New books for the physics fan

Saturday, December 13th, 2014

This article appeared in symmetry on Dec. 9, 2014.

These recently published popular science books will help you catch up on particle physics news, knowledge and history. Image: Artwork Sandbox Studio, Chicago, with Ana Kova

These recently published popular science books will help you catch up on particle physics news, knowledge and history. Image: Artwork Sandbox Studio, Chicago, with Ana Kova

Looking to stay current on your particle physics knowledge? Here are 10 recent popular science books you might want to check out.


1. Faraday, Maxwell and the Electromagnetic Field: How Two Men Revolutionized Physics

Nancy Forbes, Basil Mahon

Classical unified field theory came from the realization that electricity, magnetism and light all can be explained with a single electromagnetic field.

There is no modern physics without classical unified field theory—heck, there are no electronics without classical unified field theory—and there is no classical unified field theory without Michael Faraday (1791-1867) and James Clerk Maxwell (1831-1879).

The unlikely partners, born four decades apart, shared the achievement of upending a view of the world that had prevailed since Isaac Newton.

“The extraordinary idea put forward by Faraday and Maxwell was that space itself acted as a repository of energy and a transmitter of forces,” write Nancy Forbes and Basil Mahon in Faraday, Maxwell and the Electromagnetic Field: How Two Men Revolutionized Physics.

Faraday was largely self-taught and made important realizations without the benefit of a formal education in mathematics, while Maxwell was regarded as among the most brilliant mathematical physicists of his time. This double biography examines their differing lives and explains how their combined work paved the way for modern physics.


2. The Cosmic Cocktail: Three Parts Dark Matter

Katherine Freese

In The Cosmic Cocktail: Three Parts Dark Matter, physicist Katherine Freese explores the critical place dark matter occupies in our understanding of the cosmos.

It has yet to be observed directly. But, she tells us, dark matter’s day of reckoning might not be far off.

“Some new particles, unlike any from our daily experience, might be tearing through the galaxy,” she writes. “Scientists have already found hints of detection in their experiments… The nature of dark matter is one of the greatest puzzles of modern science, and it is a puzzle we are on the verge of solving.”

Freese, now the new director of the Nordic Institute for Theoretical Physics in Stockholm, admits to spending a disproportionate amount of time on the dance floor of nearby Studio 54 when she should have been focused on her doctoral studies at Columbia University. But she also tells a compelling history of the search for dark matter, from the cantankerous Fritz Zwicky’s early predictions in the 1930s to hopes for an appearance when the Large Hadron Collider fires up again in 2015.


3. The Large Hadron Collider: The Extraordinary Story of the Higgs Boson and Other Stuff that will Blow Your Mind

Don Lincoln

“My goal was to give readers an inside account of the hunt and discovery,” says Fermilab scientist Don Lincoln, a member of CERN’s CMS experiment, of his latest book, The Large Hadron Collider: The Extraordinary Story of the Higgs Boson and Other Stuff that will Blow Your Mind. “Almost all of the similar books have been written by non-physicists and theorists. I went to all the meetings, so I have a unique perspective.”

In the book, Lincoln describes the process of the discovery of the Higgs boson—and explains that it is not the end of the story.

Even though the widely celebrated appearance of the Higgs particle confirmed theorists’ predictions, Lincoln maintains that the relatively light mass of the Higgs raises enough questions to keep physicists awake at night.

“The measurement is quite inconsistent with the Standard Model and the quantum corrections,” he says. “This absolutely screams that there is something still to be found and this could be supersymmetry, extra dimensions, composite Higgs bosons or some other kind of new physics. In short, we know there is something big we’re missing.”


4. The Most Wanted Particle: The Inside Story of the Hunt for the Higgs, the Heart of the Future of Physics

Jon Butterworth

“I wanted it to give readers a sense of what it really feels like to work in a big experiment at such an amazing time and what it meant,” says University College London physicist Jon Butterworth of his book The Most Wanted Particle: The Inside Story of the Hunt for the Higgs, the Heart of the Future of Physics. “This meant occasionally the physics had to go a bit deeper than the common analogies, but also there is a lot of non-physics story which hopefully captures the real-time excitement.”

Butterworth, who works on the ATLAS experiment at CERN, uses a personalized approach to convey a sense of scene. In one chapter, he describes explaining the Higgs discovery to British TV reporter Tom Clarke while the two shoot pool.

He also uses his current hometown in England to describe his workplace at CERN, comparing the size of the Large Hadron Collider tunnel to the size of the London Underground.

The book, released in the UK in May under the title Smashing Physics: Inside the World’s Biggest Experiment, will be released in the US in January 2015.


5. The Perfect Wave: With Neutrinos at the Boundary of Space and Time

Heinrich Pas

Heinrich Pas, a theorist at the Technical University of Dortmund in Germany, studies neutrinos, particles that seem to defy the rules but may hold answers to the deepest questions of the universe.

In The Perfect Wave: With Neutrinos at the Boundary of Space and Time, Pas explains how powerful processes in the cosmos—from the fusion that lights the sun to the magnificent explosions of supernovae—are bound up in the workings of the mysterious particles.

“It is a story of an elementary particle that, just like the Silver Surfer in the superhero cartoons, surfs to the boundaries of knowledge, of the universe and of time itself,” Pas writes. “A story that captivates you as it sucks you into a maelstrom like an oversized wonderland. Jump on your board and hold tight.”


6. The Science of Interstellar

Kip Thorne

Kip S. Thorne, the Feynman Professor of Theoretical Physics Emeritus at Caltech, served as the executive producer for scientific credibility (and flexibility) on the space epic Interstellar. He explains that work in the book The Science of Interstellar.

In the film, astronaut Cooper (Matthew McConaughey) takes leaps and bounds over, under, around and through black holes and wormholes on his quest to find a refugee planet for the population of Earth, whose food supply is devastated by global blight.

Thorne writes that “[s]ome of the science is known to be true, some of it is an educated guess, and some is speculation.”

But he takes all of it seriously; Thorne and his colleagues even wrote a scientific paper based on their computer simulations of the movie’s black hole.


7. The Singular Universe and the Reality of Time

Roberto Mangabeira Unger, Lee Smolin

Physicist Lee Smolin of Canada’s Perimeter Institute for Theoretical Physics, author of the controversial book The Trouble With Physics, collaborated with philosopher and politician Roberto Mangabeira Unger on the new book The Singular Universe and the Reality of Time.

In it, Smolin and Unger argue against the idea of the multiverse and declare that it is time to view the cosmos as being governed by laws that are evolving rather than laws that are immutable. They contend that, “everything changes sooner or later, including change itself. The laws of nature are not exempt from this impermanence.”


8. Time in Powers of Ten: Natural Phenomena and their Timescales

Gerard ‘t Hooft, Stefan Vandoren

In Time in Powers of Ten: Natural Phenomena and their Timescales, Nobel Laureate Gerard ‘t Hooft and theorist Stefan Vandoren, both of Utrecht University in the Netherlands, step back and forth in time from the minutest fractions of a second to the age of the universe and beyond. Observations range from the orbits and rotations of planets and stars, down to the decay times of atoms and elementary particles and back to geological time scales.

“The smallest matter mankind has studied moves considerably faster than the quickest computing processes of the most expeditious machine; while on the other side of the timescale we see planets, stars and entire galaxies of unimaginably old age, some billions of years,” ‘t Hooft and Vandoren write. “Scientists believe they know almost exactly how old the universe is, but even its seemingly eternal lifetime does not constitute a limit for physicists’ research.”


9. Travelling to Infinity: The True Story Behind the Theory of Everything

Jane Hawking

In Travelling to Infinity: The True Story Behind the Theory of Everything, readers are introduced to a young, floppy-haired Stephen Hawking through the eyes of his first wife, Jane Hawking (née Wilde). Hawking published versions of this book in both 1999 and 2007, and the book was reissued this year to accompany the film adaptation, The Theory of Everything.

In the book, Jane describes an early impression of Stephen from a New Year’s party in 1963: “Clearly here was someone, like me, who tended to stumble through life and managed to see the funny side of situations. Someone who, like me, was fairly shy, yet not averse to sharing his opinions, someone who unlike me had developed a sense of his own worth and had the effrontery to convey it.”

Here is a love story in which love is not enough. Hawking leaves and marries one of the nurses who tended him. Jane marries an old family friend. The two have reconciled and are on amicable terms—a good thing when the person writing your life story is your former spouse.


10. What If? Serious Scientific Answers to Absurd Hypothetical Questions

Randall Munroe

Randall Munroe’s stick-figure web comic strip, xkcd, comes with a warning: “This comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors).”

There are no dumb questions, only humorous and provocative answers from Munroe, a former NASA roboticist, in his book What If? Serious Scientific Answers to Absurd Hypothetical Questions. For example:

“Q – What would happen if the Earth and all terrestrial objects stopped spinning, but the atmosphere retained its velocity?

“A – Nearly everyone would die. THEN things would get interesting…”

In “What If?” what seems like the end is often just the beginning.

Mike Perricone


How to make a neutrino beam

Friday, December 12th, 2014

This article appeared in Fermilab Today on Dec. 11, 2014.

Ingredients for a neutrino beam: speedy protons, target, magnetic horn, decay pipe, absorbers. Image adapted from Fermilab

Ingredients for a neutrino beam: speedy protons, target, magnetic horn, decay pipe, absorbers. Image adapted from Fermilab

Fermilab is in the middle of expanding its neutrino program and is developing new detectors to study these ghostly particles. With its exquisite particle accelerator complex, Fermilab is capable of creating very intense beams of neutrinos.

Our neutrino recipe starts with a tank of hydrogen. The hydrogen atoms are fed an extra electron to make them negatively charged, allowing them to be accelerated. Once the charged atoms are accelerated, all of the electrons are ripped off, leaving a beam of positive protons. The protons are extracted into either the Booster Neutrino Beamline (BNB) or are further accelerated and extracted into the Neutrino Main Injector beamline (NuMI). Fermilab is the only laboratory with two neutrino beams. Our two beams have different energies, which allows us to study different properties of the neutrinos.

In the BNB, these protons smash into a target to break up the strong bonds of the quarks inside the proton. These collisions are so violent that they produce new quarks from their excess energy. These quarks immediately form together again into lighter composite short-lived particles called pions and kaons.

Since the pions and kaons emerge at different directions and speeds, they need to be herded together. As a bugle tunes your breath into musical notes, a horn of a different type rounds up and focuses the pions and kaons. The BNB horn looks roughly like the bell of a six-foot long bugle. It produces an electric field that in turn generates a funnel-like magnetic field, which directs all of the dispersed pions and kaons of positive electric charge straight ahead. Particles with negative charges get defocused. By switching the direction of the electric field, we can focus the negatively charged particles while defocusing the positive charges.

The focused particles in the BNB beam travel through a 50-meter long tunnel. This is where the magic happens. In this empty tunnel, the pions and kaons decay in flight into neutrinos, electrons and muons. At the end of the decay tunnel is a wall of steel and concrete to stop and absorb any particle that is not a neutrino. Because neutrinos interact so rarely, they easily whiz through the absorbers and on towards the experiments. And that’s the basic formula to make a beam of neutrinos!

A single neutrino beamline can support many experiments because the neutrinos interact too rarely to get “used up.” The BNB feeds neutrinos to MicroBooNE, and most of them go on through to the other side towards the MiniBooNE detector. Similarly, most of those go on through the other side as well and continue traveling to infinity and beyond. Detectors located in this beam measure neutrino oscillations and their interactions.

The NuMI beamline is designed similarly, but uses a different target material, two focusing horns, and a 675-meter decay pipe. The spacing between the two NuMI horns is adjustable, allowing fine-tuning of the neutrino beam energy. The NuMI beamline has higher-energy neutrinos than the BNB and thus studies different properties of neutrino oscillations.

The NuMI beamline feeds neutrinos to the MINERvA experiment and on through to the MINOS near detector. The NuMI beamline then continues about 450 miles through Earth on toward the MINOS far detector in the Soudan mine in Minnesota. By the time the beam reaches the far detector, it is about 20 miles in diameter! By having a near and far detector, we are able to observe neutrino flavor oscillations by measuring how much of the beam is electron neutrino flavor and muon neutrino flavor at each of these two detectors.

The last of the big Fermilab neutrino experiments is NOvA. Its near detector is off to the side of the NuMI beam and measures neutrinos only with a specific range of direction and energy. The NOvA far detector is positioned to measure the neutrinos with the same properties at a greater distance, about 500 miles away in Ash River, Minnesota. By placing the NOvA detectors 3 degrees to the side of the beam’s center, NOvA will get to make more precise oscillation measurements for a range of neutrino energies.

As more experiments are designed with more demanding requirements, Fermilab may expect to see more neutrino beamline R&D and the construction of new beamlines.

Tia Miceli


During November CERN’s Council chose Fabiola Gianotti to be the organisation’s 16th Director-General. She will take over from the current incumbent, Rolfe Heuer, on 1 January 2016 and will serve for five years.

A spokesperson for CERN’s Council stated:

“It was Dr Gianotti’s vision for CERN’s future as a world leading accelerator laboratory, coupled with her in-depth knowledge of both CERN and the field of experimental particle physics that led us to this outcome.”

CERN’s current Director General declared that:

“Fabiola Gianotti is an excellent choice to be my successor… [I] am confident that CERN will be in very good hands.”

And Gianotti herself proclaimed:

“It is a great honour and responsibility for me to be selected as the next CERN Director-General following 15 outstanding predecessors… I will fully engage myself to maintain CERN’s excellence in all its attributes, with the help of everybody, including CERN Council, staff and users from all over the world.”

Dr. Gianotti poses with the ATLAS experiment.

Dr. Gianotti poses with the ATLAS experiment. (Image credit: Claudia Marcelloni)

Dr. Gianotti hails from Italy and holds a PhD in experimental particle physics from the University of Milan. She joined CERN in 1987 and worked on various experiments including the UA2 experiment and ALEPH, a detector for the Large Electron-Positron Collider (the LHC’s predecessor). She went on to join the ATLAS experiment, for which she was leader from March 2009 to February 2013. In July 2012 she, along with a spokesperson from the CMS experiment, announced that ATLAS and CMS had observed a ‘Higgs-like particle’. The discovery of the Higgs boson was subsequently confirmed and, as a result, Peter Higgs and Francois Englert were awarded the Nobel prize for physics in December 2013. (See here).

While she is no doubt a talented physicist, Gianotti has other strings to her bow. For example, she is an accomplished pianist who once considered a career in music. She also possesses a hint of mischief: having caused quite a stir for using the comic sans font in her slideshow presentation when announcing ATLAS’s observation of the Higgs (not serious enough apparently), she went on to announce this year that CERN would be adopting comic sans as its official font! – it was April 1st. (See this post from Rob Knoops for more.)

Being CERN’s big cheese is a tough gig but Gianotti seems qualified, experienced, able and passionate enough to be a great Director-General. It is also highly refreshing to see a female appointed to the highest profile physics office in the world.

From the Quantum Diarists, good luck Fabiola!


Not all philosophy is useless.

Friday, December 5th, 2014

In this, the epilogue to my philosophic musing, I locate my view of the scientific method within the landscape of various philosophical traditions and also tie it into my current interest of project management. As strange as it may seem, this triumvirate of the scientific method, philosophy and management meet in the philosophic tradition known as pragmatism and in the work of W. Edwards Deming (1900 – 1993), a scientist and management guru who was strongly influenced by the pragmatic philosopher C.I. Lewis (1883 – 1964), who in turn strongly influenced business practices. And I do mean strongly in both cases. The thesis of this essay is that Lewis, the pragmatic philosopher, has had influence in two directions: in business practice and in the philosophy of science. Surprisingly, my views on the scientific method are very much in this pragmatic tradition and not crackpot.

The pragmatic movement was started by Charles S. Peirce (1839 – 1914) and further developed by Williams James (1842 – 1910) and John Dewey (1859 – 1952). The basic idea of philosophic pragmatism is given by Peirce in his pragmatic maxim as: “To ascertain the meaning of an intellectual conception one should consider what practical consequences might result from the truth of that conception—and the sum of these consequences constitute the entire meaning of the conception.” Another aspect of the pragmatic approach to philosophic questions was that the scientific method was taken as given with no need for justification from the outside, i.e. the scientific method was used as the definition of knowledge.
How does this differ from the workaday approach to defining knowledge? Traditionally, going back at least to Plato (428/427 or 424/423 BCE – 348/347 BCE) knowledge has been defined as:
1) Knowledge – justified true belief
The leaves open the question of how belief is justified and since no justification is ever 100% certain, we can never be sure the belief is true. That is a definite problem. No wonder the philosophic community has spent two and a half millennia in fruitless efforts to make sense of it.

A second definition of knowledge predates this and is associated with Protagoras (c. 490 B.C. – c. 420 B.C.) and the sophists:
2) Knowledge – what you can convince people is true
Essentially, the argument is that since we cannot know that a belief is true with 100% certainty; what is important is what we can convince people of. This same basic idea shows up in the work of modern philosophers of science with the idea that scientific belief is basically a social phenomenon and what is important is what the community convinces itself is true. This was part of Thomas Kuhn’s (1922 – 1996) thesis.

While we cannot know what is true, we can know what is useful. Following the lead of scientists, the pragmatists effectively defined knowledge as:
3) Knowledge – information that helps you predict and modify the future
If we take predicting and modifying the future as the practical consequence of information, this definition of knowledge is consistent with the pragmatic maxim. The standard model of particle physics is not knowledge by the strict application of definition 1) since it is not completely true; however it is knowledge by definition 3 since it helps us predict and modify the future. The scientific method is built on definition 3). The modify clause is included in the definition since the pragmatists insisted on that aspect of knowledge. For example, C.I. Lewis said that without the ability to act there is no knowledge.

The third definition of knowledge given above does not correspond to what many people think of as knowledge so Dewy suggested using the term “warranted assertions” rather than knowledge: The validity of the standard model is a warranted assertion. Fortunately, this terminology never caught on. In contrast, James’s pragmatic idea of “truth’s cash value”, derided at the time, has caught on. In a recent book “How to Measure Anything,” on risk management, Douglas W. Hubbard spends a lot of space on what is essentially the cash value of information. In business, that is what is important. The pragmatists were, perhaps, just a bit ahead of their time. Hubbard, whether he knows it or not, is a pragmatist.
Dewey coined the term “instrumentalism” to describe the pragmatic approach. An idea or a belief is like a hand, an instrument for coping. A belief has no more metaphysical status than a fork. When your fork proves inadequate to the task of eating soup, it makes little sense to argue about whether there is something inherent in the nature of forks or something inherent in the nature of soup that accounts for the failure. You just reach for a spoon . However, most pragmatists did not consider themselves to be instrumentalists but rather used the pragmatic definition of knowledge to define what is meant by real.

Now I turn to C.I. Lewis. He is alternately regarded as the last of the classical pragmatists or the first of the neo-pragmatists. He was quite influential in his day as a professor at Harvard from 1920 to his retirement in 1953. In particular, his 1929 book “Mind and the World Order” had a big influence on epistemology and surprisingly on ISO management standards. One can see a lot of the ideas developed by Kuhn already present in the work of C.I. Lewis , for example, the role of theory in interpreting observation. Or as Deming, influenced by Lewis, expressed it: “There is no knowledge without theory.” As a theorist, I like that. At the time, this was quite radical. The logical positivists took the opposite tack and tried to eliminate theory from their epistemology. Lewis and Kuhn argued this was impossible. The idea that theory was necessary for knowledge was not new to Lewis but is also present in the works of Henri Poincaré (1854 – 1912) who was duly reference by Lewis.

Another person Lewis influenced was Willard V. O. Quine (1908 – 2000), although Quine and Lewis did not agree. Quine is perhaps best known outside the realm of pure philosophy for the Duhem-Quine thesis, namely that it is impossible to test a scientific hypothesis in isolation because an empirical test of the hypothesis requires one or more background assumptions. This was the death knell of any naïve interpretation of Sir Karl Popper’s (1902 –1994) idea that science is based on falsification. But Quine’s main opponents were the logical positivists. Popper was just collateral damage. Quine published a landmark paper in 1951: “Two Dogmas of Empiricism”. I would regard this paper as the high point in the discussion of the scientific method by a philosopher and it reasonably readable (unlike Lewis’s “The Mind and the World Order”). Beside the Duhem-Quine thesis, the other radical idea is that observation underdetermines scientific models and that simplicity and conservatism are necessary to fill the gap. This idea also goes back to Poincaré and his idea of conventionalism – much of what is regarded as fact is only convention.

To a large extent my ideas match well with the ideas in “Two Dogmas of Empiricism.” Quine summarizes it nicely as: “The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges.” and “The edge of the system must be kept squared with experience; the rest, with all its elaborate myths or fictions, has as its objective the simplicity of laws.” Amen.

Unfortunately, after the two dogmas of empiricism were brought to light, the philosophy of science regressed. In a recent discussion of simplicity in science I came across, there was neither a single mention of Quine’s work nor his correct identification of the role of simplicity – to relieve the under determination of models by observation. Philosophers found no use for his ideas and have gone back to definition 1) of knowledge. Sad

Where philosophers have dropped the ball it was picked by people in, of all places management. Two people influenced by Lewis were Walter A. Shewhart (1891 – 1967) and Edwards Deming. It is said that Shewhart read Lewis’s book fourteen times and Deming read it nine times. Considering how difficult that book is, it probably required that many readings just to comprehend it. Shewhart is regarded as the father of statistical process control, a key aspect of quality control. He also invented the control chart, a key component of statistical process control. Shewhart’s 1939 book “Statistical Method from the viewpoint of Quality Control” is a classic in the field but it devoted a large part to showing how his ideas are consistent with Lewis’s epistemology. In this book, Shewhart introduced the Shewhart cycle, which was modified by Deming (and sometimes called the Deming cycle). Under its current name Do-Plan-Check-Act (DPCA cycle) it forms the basis of the ISO management standards.


The original Shewhart cycle as given in Shewhart’s book.

What is this cycle? Here it is as captured from Shewhart’s book. This is the first place where production is seen as part of a cycle and in the included caption Shewhart explicitly relates it to the scientific method as given by Lewis. Deming added another step to the cycle, which strikes me as unnecessary; the act step. It can easily be incorporated in the specification or plan stage (as it is in Shewhart’s diagram). But Deming was influenced by Lewis who regarded knowledge without the possibility of acting as impossible, hence the act step. This idea has become ingrained in ISO management standards as the slogan “continual improvement” (Clause 10 in the standards). To see the extent Deming was guided by Lewis’s ideas just look at Deming’s 1993 book “The New Economics.” He summarizes his approach in what he calls a system of profound knowledge. This has four parts: knowledge of system, knowledge of variation, theory of knowledge and knowledge of physiology. The one that seems out of place is the third; why include theory of knowledge? Deming believed that this was necessary for running a company and he explicitly refers to Lewis’s 1929 book. Making the reading of Lewis’s book mandatory for business managers would certainly have the desirable effect of cutting down the number of managers. To be fair to Deming, he does suggest starting in about the middle of the book. We have two unbroken chain – 1) Peirce, Lewis, Shewhart, Deming, ISO management standards and 2) Pierce, Lewis, Quine, my philosophical musings . It reminds one of James Burke’s TV program “Connections”.

Popper may be the person many scientists think of to justify how they work but Quine would probably be better and Quine’s teacher, C.I. Lewis, through Deming, has provided the philosophic foundation for business management. Within the context of definition 3) for knowledge both science and business have been very successful. Your reading of this essay required both. In contradistinction, standard western philosophy based on definition 1) has largely failed; philosophers still do not know how to acquire knowledge. However, not all philosophy is useless, some of it is pragmatic.


A Summer of Quantum Strings

Tuesday, December 2nd, 2014

Salvish Goomanee, an undergraduate theoretical physics student, penned the following post about his summer research at King’s College London.

Salvish Goomanee

Salvish Goomanee

This past summer, I had a brief glimpse into what it’s like to carry out theoretical research. As an undergraduate, this was an exciting foray into what I hope will be my future! The work was done under the supervision of a research associate of the theoretical physics department at King’s College London, who was kind enough to accept my proposal. We decided to look at some perturbative amplitudes of the open bosonic string, which gave me the chance to peek into the obscure world of quantum strings. (Bear with me – I’ll explain what all that means soon enough!) I was presented with several very interesting challenges during the course of the research; it was not always easy but it was great to have the chance to experience with the various mathematical methods that permeate the fundamental formalism of string theory.

After our first meeting, which lasted for about three hours, I found myself with notes and calculations and a long list of references which seemed to be the basics for understanding perturbative string theory. I obviously was not aware of what was going on but this was the beginning of the three long months of studying the new and rigorous concepts of quantum field theory and some slightly more advance mathematics as well. The goal of the research was to understand the path integral formulation and apply it to the framework of the bosonic string.

In short, bosonic string theory is a type of string theory that deals only with bosons which are integer spin particles. To understand that a bit better, let’s leave string theory for a brief moment and consider the construction plan for a new building. Architects and engineers will work out a much smaller model first (which I am sure we have all seen!), where it would be much easier to have control upon and therefore have a better understanding how things will look like at much larger scales. Physicists will do the same thing as calculations for the bosonic string can be performed at energy scales that are relatively low. Therefore amplitudes calculated via small perturbations can be pictured as new particles. Quantum field theory is extensively employed during such processes. (Quantum field theory is basically the merging of special relativity and the laws of quantum mechanics. String theory is actually a quantum field theory that incorporates gravity which the Standard Model of particle physics does not!)

So, here I was, an undergraduate who just finished the second year exams, facing all this new stuff that I had to study and make sense of. It was challenging and somehow a bit frightening. You reach a point where you ask yourself why you should continue doing these things when it was not making much sense, but giving up was not an option! The good thing was that I would be able to finally apply the few little things I learnt from relativity (field equations, cosmological solutions and tensors, etc). Being familiar with the theory of geometrization of space-time proved to be a very good support as the notation used in perturbative string theory was then much easier to understand. It’s actually extensively employed everywhere in theoretical physics. Finally we were then able to see (and I was able to contemplate!) the various predictions, good and bad, of string theory.

This experience was something which I definitely enjoyed. I guess what made it better was the great summer we had here in London. Obviously during the hottest days we could not (and did not want to) really work inside, so we discussed our ideas our outside. One day we even spent more than four hours talking; this was one of the rare moments where an undergraduate gets to know in a bit more detail the kind of research their supervisors and lectures carry out and the level at which it is done. In the end, I was able to produce a report in which I discussed the stuff we went through. We are now looking at some other applications of the quantum strings in greater detail. All in all, it was a wonderful summer, and I’m thrilled that we’re still collaborating!


I’ve had an extremely busy few weeks! We are preparing the next LUX paper, and it’s been a hectic learning curve for me. Most importantly, I now know to never expect anything to be ready on time! It’s really exciting to know that soon I will actually have my name on a published paper – if amongst 100+ others, but that’s how it goes in experimental particle physics. It will still be a proud moment for me; I might start to feel like I’m actually useful. I managed to squeeze in a couple of trips since my last post, one of which was to a talk entitled “Insights: what makes the perfect song?” at London’s Royal Opera House. This was an Institute of Physics event, and the speaker was none other than Professor Brian Cox.

I can almost guarantee if you are British you will have heard of Brian Cox. He is a particle physicist at the University of Manchester, and a member of ATLAS, one of the general purpose LHC detectors. His floppy hair and Mancunian accent are a familiar sight and sound on the BBC; he is always expressing his love for physics and the universe in documentaries such as “Wonders of the Universe”, “Wonders of the Solar System” and most recently “Human Universe”.
I actually first saw Brian on an episode of Horizon (a British science documentary) back in 2008 entitled “What on Earth is wrong with Gravity?”. I was only a few months into AS levels in Physics, Maths, Chemistry and Biology and actually had not yet chosen to go on to do my degree in physics; I was keen at this point to go down either a medical or biochemistry route. The Horizon episode focussed on the uncomfortable discord between quantum mechanics and general relativity, and fascinated me. I was reminded how much I loved physics (my spirit had been somewhat broken by early physics AS-level classes on materials where we learnt about stress, strain and brittle fractures). I believe Horizon and Brian had at least a small influence in my decision to go on and study physics at university. I have found Cox to be like marmite amongst physicists, either loved or hated. Hate is dominant among men (jealous perhaps?) who believe he is not deserving of his professorship status, and is only a face for the media.
I managed to convince my Brian Cox-hating boyfriend to attend the Institute of Physics talk with me. He also studied physics but holds no love for it (he is selling his soul to law!); it was the music aspect that convinced him to come. Although the evening was enjoyable, we were both fairly disappointed in the lack of actual science content. The panel comprised of Christine Rice, an opera singer with a physics degree, Philip Ball, writer of “The Music Instinct”, and Maria Witek, a researcher in neuroscience.

L-R:  Christine Rice, Philip Ball, Maria Wiltek and Brian Cox at Insights: what makes the perfect song?

Despite their commendable qualifications, there were no convincing arguments as to why certain notes sound good together for example, and why our brain reacts the way it does to different things in music. We did see some statistics on the frequency of different musical intervals, saw how syncopated beats make you want to dance more (see Don’t Stop Till You Get Enough – Michael Jackson) and learnt about the Gestalt principles of harmonic progression. We really enjoyed Joe Stilgoe, a jazz pianist’s whose jaunty performances were definitely the highlight. The discussion was thought-provoking but I had been hoping for something on of the waveforms of musical harmonies and melodies, of the modes of vibration on a string, what happens in the brain when we hear a song we love, etc.

I believe the night had been more intended for Royal Opera house frequenters, of which we definitely aren’t (call me uncultured but opera singing hurts my ears), rather than physicists. As the woman sitting beside us excitedly told us, “I’m retired, I spend all the time I can learning. I just love opera and I’m here all the time, so I come to anything like this.”, perhaps this event was aimed more at the likes of her, whose attitude to being retired I found admirable (although the first thing she said to us was “I’m only here for him!” meaning Brian Cox…).

Pictured: evidence of Brian Cox doing real physics work, alongside UCL head of physics and Guardian blogger Jon Butterworth

Pictured: evidence of Brian Cox doing real physics work, alongside UCL head of physics and Guardian blogger Jon Butterworth

Sometimes, after spending weeks on end staring at lifeless code, I forget how much I love physics. The enthusiasm of a great science communicator can remind me, and Brian Cox is one of the best. He has done great things for public attitudes towards physics, whether or not he does his fair share as a professor of experimental particle physics! (I have, by the way, heard from an ATLAS member that in fact he has published some excellent papers with respectable people, such as UCL’s very own Jon Butterworth…).

I could see Lady Gaga in this... planet dress at Royal Observatory Greenwich

I could see Lady Gaga in this… a steampunk planet dress at Royal Observatory Greenwich

My second excursion was to the Royal Observatory in Greenwich. The main purpose of my visit (again, dragging along my tolerant boyfriend) was to see a show called “The Dark Universe”. Admittedly, we got a bit confused, and spent most of our trip looking at old parts of telescopes and clocks, as well as a bizarre steampunk exhibition, before we found where the actual astronomy stuff was. I would say the observatory is well worth a visit – but be warned, it’s up an incredibly steep hill. It always amazes me what the astronomers of the past managed without computers. The 30-minute show was in the Planetarium, so we were seated reclining under a dome (a set-up my boyfriend decided should be implemented in all cinemas!). Neil deGrasse Tyson narrated – another great science communicator, and I was thoroughly impressed. The visuals were stunning and the physics accurate, but still understandable to non-scientists. LUX didn’t feature in the section on dark matter (AMS – the Alpha Magnetic Spectrometer, mounted on the International Space Station, provides much better visuals!) but there was a mention of detectors deep underground, which satisfied me. The film focused on the astounding fact that we do not understand a whopping 95.1% of our universe, and I feel really lucky to be one of those people attempting to help reduce that by 26.8% by uncovering the nature of dark matter.

To add to my busy schedule this week is an offsite shift for LUX. This involves keeping an eye on the detector and the data we are taking whilst those in the USA sleep to make sure all is well. We have a vast array of sensors monitoring our detector’s condition, meaning I can see with the click of a button if something has gone wrong. This job feels like rather a lot of responsibility; I’m hoping in my next post I can say it passed without incident!


The Standard Model: What is it, how does it look, and how does it taste?


The Standard Model of Particle Physics in the cupcake representation. Credit: E. Bland

Hi All,

Today’s post is an introduction to the Standard Model of Particle Physics.  First off, saying “Standard Model of Particle Physics” is long-winded, so it is often shortened to “Standard Model” or abbreviated by “SM”. In short, the SM is presently the best description of how matter behaves, interacts, and works at very small distances and very high energies. High and small, of course, compared to our everyday experiences.

Elementary: In the SM is a collection of particles believed to be the elementary building blocks/constituents of all known matter and energy. By “elementary”, I mean that there are no smaller pieces inside these objects. To put this into perspective, humans (~2 meters) are made of cells (~10 μm); cells are made of molecules (~1 nm), or chains of atoms. Atoms (~100 pm = 1 angstrom) are made of electrons (?) that orbit a central nucleus (1~10 fm). A nucleus  is comprised of protons (1 fm) and neutrons (1 fm). Both of these are made of quarks (?) and gluons (?). However, as far as experiments have shown, SM particles are not made of smaller objects. Therefore, we call call them “elementary”, “fundamental”, and “point-like.” If there comes a day where we discover that quarks have sub-structure, then quarks will lose their “elementary” status.


Scale of the Universe. Credit: U Oregon.

Spin: Elementary particles are separated into three categories: matter, force carriers, and Higgs bosons (or fermions, gauge bosons, and the Higgs bosons). Fermions and gauge bosons have small but nonzero, intrinsic angular momentum, called spin. Angular momentum is a measure of how energetically and how quickly an object is rotating. Think of a bike wheel that never stops spinning and has only two speed: fast and half-fast. A standard unit of angular momentum at small distances is an ћ (pronounced: “h-bar”). This is like a mile or kilometer being a standard unit of distance, or a day being that for time. All gauge bosons carry the same amount of spin, 1ћ; all elementary fermions carry half as much spin, ћ/2. Particles that carry no spin are called scalars. More broadly, a boson is any particle with an integer amount of spin, i.e., 1 ћ, 2 ћ, 3 ћ, …, and a fermion is any particle with half-integer spin, i.e.,  ћ/2, 3 ћ/2, 5 ћ/2, … Spin is an example of a spacetime quantum number. Even sets of fermions make a composite boson; odd sets of fermions make composite fermions, like the proton.

Charge: There are 12 elementary fermions: the up (u), down (d), charm (c), strange (s), top (t), and bottom (b) quarks; the electron (e), muon (μ), and tau (τ) charged leptons; and the electron-neutrino (νe), muon-neutrino (νμ), and tau-neutrino (ντ) leptons. These labels/names represent another quantum number called flavor. In addition to spin, these particles also carry several different charges that cause them to be repelled or attracted when in proximity to each other, in other words to experience a force. There is the electric (or electromagnetic) charge, weak hyper charge, weak isospin charge, and strong (or color) charge. Quarks carry all charges; charged leptons carry all charges except for color; and neutrinos carry only weak charges. In fact, electric charge is a combination of hyper and isospin charges. Charges are examples of internal quantum numbers. In addition, each particle has a partner particle called an antiparticle. Particles and antiparticles have the same spacetime quantum numbers but opposite internal quantum numbers. For example: an electron is a spin-1/2 fermion with -1 electric charge; a positron (an antielectron) is a spin-1/2 fermion with +1 electric charge.

Forces: Gauge bosons are the mediators of the electromagnetic, weak, and color forces, and each force is associated with a conservation law. Fermions interact, exchange momentum, and scatter off each other by exchanging gauge bosons. For example, an electron and positron can interact by exchanging a photon. Throughout this whole process, the electric charges of the electron and positron were individually conserved.



The photon (γ) is the gauge boson for electromagnetism, and the rules of electromagnetism at small distances and high energies are called quantum electrodynamics, or QED.The gluon (g) is the gauge boson for the strong force, and its rules are called quantum chromodynamics (QCD). The strong force is responsible for holding the proton together: protons and neutrons are made up quarks that are bound together by gluons. Weak forces are responsible for certain types of radioactive decay and flavor-changing interactions. For example: an electron can radiate a W boson and become an electron-neutrino, and a top quark dominantly decays into a bottom quark and a W boson. The gauge bosons of weak isospin are the W1, W2, and W3 bosons; for weak hypercharge, this is the B boson. However, at low energies, weak charges are no longer conserved. What is conserved is the sum of isospin and hypercharge. The Ws, B, and three Higgs bosons (more on this in a bit) then combine, becoming the W+, W-, and Z bosons, collectively call the weak bosons. These are very massive particles, about 80 and 90 times more massive than the proton.

Higgs Bosons: In the SM, there are four Higgs bosons: H (sometimes call the Higgs boson), φ1, φ2, and φ3. All four Higgs are scalars (zero spin) and carry both weak isospin and weak hypercharge; two carry nonzero electric charge. A summary of all particles and how they can interact are described in this image:


Any two particles connected by a line can interact. Some bosons can interact with bosons of their own type.

Mass and Electroweak Symmetry Breaking: In the early universe, all elementary fermions and gauge bosons were massless. At some point, everything underwent a phase transition that broke the hypercharge and isospin conservation laws. During this phase transition, quarks and charged leptons acquired mass. The massless hypercharge and isospin gauge bosons along with φ1, φ2, and φ3 mixed and became the massive W+, W-, and Z bosons. Because of this, the W+, W-, and Z bosons can mediate weak interactions but, under the right conditions, behave like the scalars φ1, φ2, and φ3. This phenomenon is called electroweak symmetry breaking (EWSB). After EWSB, there is one remaining physical Higgs boson, H, which was only just discovered in 2012.

To summarize:

  1. The Standard Model of Particle physics is presently our best description of how matter behaves and interacts at very small distances and very high energies.
  2. Elementary particles are not made of any smaller particle and are divided into two categories: fermions (half-integer spin) and bosons (integer spin)
  3. Fermions make up matter (like protons and atoms), and come in 12 different varieties, or flavors.
  4. Gauge bosons mediate forces: the photon mediates electromagnetism, the gluon mediates the strong force, and the W+, W-, and Z bosons mediate the weak forces.
  5. The Higgs bosons are scalars (zero spin) and facilitate electroweak symmetry breaking. After EWSB, only one Higgs boson, H, remains.
  6. Only the photon and gluon are massless; everything else has a mass.
  7. Not everything in the SM agrees with data, but we have yet to find a better theory.


Happy Colliding

- Richard (@BraveLittleMuon)


The doorway to the registrar’s office where the final thesis check takes place

I took an entire month between defending my thesis and depositing it with the grad school. During that month, I mostly revised my thesis, but also I took care of a bunch of logistical things I had been putting off until after the defense: subletting the apartment, selling the car, engaging movers, starting to pack… and of course putting comments into the thesis from the committee. I wrote back to my (now current) new boss who said we should chat again after I “come up for air” (which is a pretty accurate way of describing it). I went grocery shopping, and for the first time in months it was fun to walk around the store imagining and planning the things I could make in my kitchen. I had spare creative energy again!

Partly I needed a full month to revise the thesis because I was making changes to the analysis within the thesis right up to the day before I defended, and I changed the wording on the concluding sentences literally 20 minutes before I presented. I didn’t have time to polish the writing because the analysis was changing so much. The professor who gave me the most detailed comments was justifiably annoyed that he didn’t have sufficient time to read the whole dissertation before the defense. It worked out in the end, because the time he needed to finish reading was a time when I didn’t want to think about my thesis in any way. I even left town and visited friends in Chicago, just to break up the routine that had become so stressful. There’s nothing quite as nice as waking up to a cooked breakfast when you’ve forgotten that cooked breakfasts are an option.

There were still thesis revisions to implement. Some major comments reflected the fact that, while some chapters had been edited within a peer group, no one had read it cover-to-cover until after the defense. The professor who had the most detailed comments wrote a 12-page email detailing his suggestions, many of which were word substitutions and thus easy to implement. Apparently I have some tics in my formal writing style.

I use slightly too many (~1.2) semicolons per page of text; this reflects my inclination to use compound sentences but also avoid parentheses in formal writing. As my high school teacher, Perryman, taught me: if you have to use parentheses you’re not being confidently declarative, and if you ever want to use nested parentheses in a formal setting, figure out what you really want to say and just say it! (subtext: or figure out why you don’t want to say it, and don’t say it. No amount of parenthesis can make a statement disappear.) Anyway, I’d rather have too many semicolons than too many parentheses; I’d rather be seen as too formal than too tentative. It’s the same argument, to me, that I’d rather wear too much black than too much pink. So, many of the semicolons stayed in despite the comments. Somehow, in the thesis haze, I didn’t think of the option of many simple single-clause sentences. Single-clause sentences are hard.

I also used the word “setup” over 100 times as a catch-all word to encompass all of the following: apparatus, configuration, software, procedure, hypothesis. I hadn’t noticed that, and I have no good reason for it, so now my thesis doesn’t use the word “setup” at all. I think. And if it does, it’s too late to change it now!

And of course there was the matter of completing the concluding paragraph so it matched the conclusion I presented in my defense seminar. That took some work. I also tried to produce some numbers to complete the description of my analysis in more detail than I needed for the defense seminar, just for archival completeness. But by the time I had fixed everything else, it was only a few hours until my deposit margin-check appointment (and also 2:30am), so I gave up on getting those numbers.

The deposit appointment was all of 5 minutes long, but marked the line between “almost done” and “DONE!!!”. The reviewing administrator realized this. She shook my hand three times in those 5 minutes. When it was done, I went outside and there were birds singing. I bought celebratory coffee and a new Wisconsin shirt. And then started packing up my apartment for the movers arriving the next morning.

During that month of re-entering society,  I had some weird conversations which reminded me how isolated I had been during the thesis. A friend who used to work in our office had started her own business, but I’d only had time to ask her about it once or perhaps twice. When we had a bit of time to catch up more, I asked how it had been during the last few months, and she replied that it had been a year. A year. It just went by and I didn’t notice, without the regular office interactions.

I’d gotten into a grove of watching a couple episodes each night of long-running TV shows with emotionally predictable episodic plot lines. Star Trek and various murder mysteries were big. The last series was “House, MD” with Hugh Laurie. By coincidence, when I defended my thesis and my stress level starting deflating, I was almost exactly at the point in the series where they ran out of mysteries from the original book it was based on, and started going more into a soap-opera style character drama. By the time I wasn’t interested in the soap opera aspects anymore, it was time to start reengaging with my real-life friends.

A few days after I moved away from Madison, when I was staying with my parents, I picked up my high school routine of reading the local paper over breakfast, starting with the comics, then local editorials. I found (or rather, my dad found) myself criticizing the writing from the point of view of a dissertator. It takes more than a few days to get out of thesis-writing mode. The little nagging conscience doesn’t go away, still telling me that the difference between ok writing and great writing is important, more so now than at any point so far in my career. For the last edits of a PhD, it might be important to criticize at that level of detail. But for a local paper, pretty much anything is useful to the community.

At lunch Saturday in a little restaurant in the medieval part of the Italian village of Assergi, I found the antidote. When I can’t read any of the articles and posters on the walls, when I can’t carry on a conversation with more than 3-word sentences, it doesn’t matter anymore if the paragraphs have a clear and concise topic sentence. I need simple text. I’m happy if I can understand the general meaning. The humility of starting over again with Italian is the antidote for the anxiety of a thesis. It’s ok to look like a fool in some ways, because I am a certified non-fool in one small part of physics.

It’s not perfect of course: there’s still a lot of anxiety inherent in living in a country without speaking the language (well enough to get by without english-speaking help). I’ll write more about the cultural transition in another post, since I have so many posts to catch up on from while I was in the thesis-hole, and this post is definitely long enough. But for now, the thesis is over.


This article appeared in Fermilab Today on Nov. 26, 2014

The plastic scintillator extrusion line, shown here, produces detector material for export to experiments around the world. Photo: Reidar Hahn

The plastic scintillator extrusion line, shown here, produces detector material for export to experiments around the world. Photo: Reidar Hahn

Small, clear pellets of polystyrene can do a lot. They can help measure cosmic muons at the Pierre Auger Observatory, search for CP violation at KEK in Japan or observe neutrino oscillation at Fermilab. But in order to do any of these they have to go through Lab 5, located in the Fermilab Village, where the Scintillation Detector Development Group, in collaboration with the Northern Illinois Center for Accelerator and Detector Design (NICADD), manufactures the exclusive source of extruded plastic scintillator.

Like vinyl siding on a house, long thin blocks of plastic scintillator cover the surfaces of certain particle detectors. The plastic absorbs energy from collisions and releases it as measurable flashes of light. Fermilab’s Alan Bross and Anna Pla-Dalmau first partnered with local vendors to develop the concept and produce cost-effective scintillator material for the MINOS neutrino oscillation experiment. Later, with NIU’s Gerald Blazey, they built the in-house facility that has now exported high-quality extruded scintillator to experiments worldwide.

“It was clear that extruded scintillator would have a big impact on large neutrino detectors,” Bross said, “but its widespread application was not foreseen.”

Industrially manufactured polystyrene scintillators can be costly — requiring a labor-intensive process of casting purified materials individually in molds that have to be cleaned constantly. Producing the number of pieces needed for large-scale projects such as MINOS through casting would have been prohibitively expensive.

Extrusion, in contrast, presses melted plastic pellets through a die to create a continuous noodle of scintillator (typically about four centimeters wide by two centimeters tall) at a much lower cost. The first step in the production line mixes into the melted plastic two additives that enhance polystyrene’s natural scintillating property. As the material reaches the die, it receives a white, highly reflective coating that holds in scintillation light. Two cold water tanks respectively bathe and shower the scintillator strip before it is cool enough to handle. A puller controls its speed, and a robotic saw finally cuts it to length. The final product contains either a groove or a hole meant for a wavelength-shifting fiber that captures the scintillation light and sends the signal to electronics in the most useful form possible.

Bross had been working on various aspects of the scintillator cost problem since 1989, and he and Pla-Dalmau successfully extruded experiment-quality plastic scintillator with their vendors just in time to make MINOS a reality. In 2003, NICADD purchased and located at Lab 5 many of the machines needed to form an in-house production line.

“The investment made by Blazey and NICADD opened extruded scintillators to numerous experiments,” Pla-Dalmau said. “Without this contribution from NIU, who knows if this equipment would have ever been available to Fermilab and the rest of the physics community?”

Blazey agreed that collaboration was an important part of the plastic scintillator development.

“Together the two institutions had the capacity to build the resources necessary to develop state-of-the-art scintillator detector elements for numerous experiments inside and outside high-energy physics,” Blazey said. “The two institutions remain strong collaborators.”

Between their other responsibilities at Fermilab, the SDD group continues to study ways to make their scintillator more efficient. One task ahead, according to Bross, is to work modern, glass wavelength-shifting fibers into their final product.

“Incorporation of the fibers into the extrusions has always been a tedious part of the process,” he said. “We would like to change that.”

Troy Rummler


Geometry and interactions

Tuesday, November 25th, 2014

Or, how do we mathematically describe the interaction of particles?

In my previous post, I addressed some questions concerning the nature of the wavefunction, the most truthful mathematical representation of a particle. Now let us make this simple idea more complete, getting closer to the deep mathematical structure of particle physics. This post is a bit more “mathematical” than the last, and will likely make the most sense to those who have taken a calculus course. But if you bear with me, you may also come to discover that this makes particle interactions even more attractive!

The field theory approach considers wavefunctions as fields. In the same way as the temperature field \(T(x,t)\) gives the value of the temperature in a room at space \(x\) and time \(t\), the wavefunction \(\phi (x,t)\) quantifies the probability of presence of a particle at space point \(x\) and time \(t\).
Cool! But if this sounds too abstract to you, then you should remember what Max Planck said concerning the rise of quantum physics: “The increasing distance between the image of the physical world and our common-sense perception of it simply indicates that we are gradually getting closer to reality”.

Almost all current studies in particle physics focus on interactions and decays of particles. How does the concept of interaction fit into the mathematical scheme?

The mother of all the properties of particles is called the Lagrangian function. Through this object a lot of properties of the theory can be computed. Here let’s consider the Lagrangian function for a complex scalar field without mass (one of the simplest available), representing particles with electric charge and no spin:

\(L(x) = \partial_\mu \phi(x)^* \partial^\mu \phi(x) \).

Mmm… Is it just a bunch of derivatives of fields? Not really. What do we mean when we read \(\phi(x)\)? Mathematically, we are considering \(\phi\) as a vector living in a vector space “attached” to the space-time point \(x\). For the nerds of geometry, we are dealing with fiber bundles, structures that can be represented pictorially in this way:


Click on image for larger version

The important consequence is that, if \(x\) and \(y\) are two different space-time points, a field \(\phi(x)\) lives in a different vector space (fiber) with respect to \(\phi(y)\)! For this reason, we are not allowed to perform operations with them, like taking their sum or difference (it’s like comparing a pear with an apple… either sum two apples or two pears, please). This feature is highly non-trivial, because it changes the way we need to think about derivatives.

In the \(L\) function we have terms containing derivatives of the field \(\phi(x)\). Doing this, we are actually taking the difference of the value of the field at two different space-time points. But … we just outlined that we are not allowed to do it! How can we solve this issue?

If we want to compare fields pertaining to the same vector space, we need to slightly modify the notion of derivative introducing the covariant derivative \(D\):

\( D_\mu = \partial_\mu + ig A_\mu(x) \).

Here, on top of the derivative \(\partial\), there is the action of the “connection” \(A(x)\), a structure which takes care of “moving” all the fields in the same vector space, and eventually allows us to compare apples with apples and pears with pears.
So, a better way to write down the Lagrangian function is:

\(L(x) = D_\mu \phi(x)^* D^\mu \phi(x) \).

If we expand \(D\) in terms of the derivative and the connection, \(L\) reads:

\(L(x) = \partial_\mu \phi(x)^* \partial^\mu \phi(x) +ig A_\mu (\partial^\mu \phi^* \phi – \phi^* \partial^\mu \phi) + g^2 A^2 \phi^* \phi \).

Do you recognize the role of these three terms? The first one represents the propagation of the field \(\phi\). The last two are responsible for the interactions between the fields \(\phi, \phi^*\) and the \(A\) field, referred to as the “photon” in this context.


Click on image for larger version

This slightly hand-waving argument involving fields and space-time is a simple handle to understand how the interactions among particles emerge as a geometric feature of the theory.

If we consider more sophisticated fields with spin and color charges, the argument doesn’t change. We need to consider a more refined “connection” \(A\), and we could see the physical interactions among quarks and gluons (namely QCD, Quantum Chromo Dynamics) emerging just from the mathematics.

 Probably the professor of geometry in my undergrad course would call this explanation “Spaghetti Mathematics”, but I think it can give you a flavor of the mathematical subtleties involved in the theory of particle physics.