Learning Physics, Metaphors, and Quantum Fields

In my series on Quantum Field Theory I wanted to document my own learning endeavors but it has turned into a meta-contemplation on the ‘explain-ability’ of theoretical physics.

Initially I had been motivated by a comment David Tong made in his introductory lecture: Comparing different QFT books he states that Steven Weinberg‘s books are hard reads because at the time of writing Weinberg was probably the person knowing more than anyone else in the world on Quantum Field Theory. On the contrary Weinberg’s book on General Relativity is accessible which Tong attributes to Weinberg’s learning GR himself when he was writing that textbook.

Probably I figured nothing can go awry if I don’t know too much myself. Of course you should know what you are talking about – avoiding to mask ignorance by vague phrases such as scientists proved, experts said, in a very complicated process XY has been done.

Yet my lengthy posts on phase space didn’t score too high on the accessibility scale. Science writer Jennifer Ouelette blames readers’ confusion on writers not knowing their target audience:

This is quite possibly the most difficult task of all. You might be surprised at how many scientists and science writers get the level of discourse wrong when attempting to write “popular science.” Brian Greene’s The Elegant Universe was an undeniably important book, and it started off quite promising, with one of the best explications of relativity my layperson’s brain has yet encountered. But the minute he got into the specifics of string theory — his area of expertise — the level of discourse shot into the stratosphere. The prose became littered with jargon and densely packed technical details. Even highly science-literate general readers found the latter half of the book rough going.

Actually, I have experienced this effect myself as a reader of popular physics books. I haven’t read The Elegant Universe, but Lisa Randall’s Warped Passages or her Knocking on Heaven’s Door are in my opinion similar with respect to an exponential learning curve.

Authors go to great lengths in explaining the mysteries of ordinary quantum mechanics: the double-slit experiment, Schrödinger’s cat, the wave-particle dualism, probably a version of Schrödinger’s equation motivated by analogies to hydrodynamics.

Curved space

An icon of a science metaphor – curved space (Wikimedia, NASA).

Then tons of different fundamental particles get introduced – hard to keep track of if you don’t a print-out of the standard model in particle physics at hand, but still doable. But suddenly you find yourself in a universe you lost touch with. Re-reading such books again now I find full-blown lectures on QFT compressed into single sentences. The compression rate here is much higher than for the petty QM explanations.

I have a theory:

The comprehensibility of a popular physics text is inversely proportional to the compression factor of the math used (even if math is not explicitly referenced).

In PI in the Sky John Barrow mulls on succinct laws of nature in terms of the unreasonable effectiveness of mathematics. An aside: Yet Barrow is as critical as Nassim Taleb with respect to the allure of Platonicity’What is most remarkable about the success of mathematics  in [particle physics and cosmology] is that they are most remote from human experience (Quote from PI in the Sky).

Important concepts in QM can be explained in high school math. My old high school physics textbook contained a calculation of the zero point energy of a Fermi gas of electrons in metals.

Equations in advanced theoretical physics might still appear simple, still using symbols taken from the Latin or Greek alphabet. But unfortunately these letters denote mathematical objects that are not simple numbers – this is highly efficient compressed notation. These objects are the proverbial mathematical machinery(*) that act on other objects. Sounds like the vague phrases I scathed before, doesn’t it? These operators are rather like a software programs using the thing to the right of this machine as an input – but that’s already too much of a metaphor as the ‘input’ is not a number either.
(*) I used the also common term mathematical crank in earlier posts which I avoid now due to obvious reasons.

You can create rather precise metaphors for differential operators in classical physics, using references to soft rolling hills and things changing in time or (three-dimensional) space. You might be able to introduce the curly small d’s in partial derivatives when applying these concepts to three-dimensional space. More than three-dimensions can be explained resorting by the beetle-on-balloon or ant-in-the-hose metaphors.

But if it gets more advanced than that I frankly run out of metaphors I am comfortable with. You ought to explain some purely mathematical concepts before you continue to discuss physics.

I think comprehension of those popular texts on advanced topics works this way:

  • You can understand anything perfectly if you have once developed a feeling for the underlying math. For example you can appreciate descriptions of physical macroscopic objects moving under the influence of gravity, such as in celestial mechanics. Even if you have forgotten the details of your high school calculus lectures you might remember some facts on acceleration and speed you need to study when cramming for your driver license test.
  • When authors start to introduce new theoretical concepts there is a grey area of understanding – allowing for stretching your current grasp of math a bit. So it might be possible to understand a gradient vector as a slope of a three-dimensional hill even if you never studied vector calculus.
  • Suddenly you are not sure if the content presented is related to anything you have a clue of or if metaphors rather lead you astray. This is where new mathematical concepts have been introduced silently.

The effect of silently introduced cloaked math may even be worse as readers believe they understand but have been led astray. Theoretical physicist (and seasoned science blogger) Sabine Hossenfelder states in her post on metaphors in science:

Love: Analogies and metaphors build on existing knowledge and thus help us to understand something quickly and intuitively.

Hate: This intuition is eventually always misleading. If a metaphor were exact, it wouldn’t be a metaphor.

And while in writing, art, and humor most of us are easily able to tell when an analogy ceases to work, in science it isn’t always so obvious.

My plan has been to balance metaphors and rigor by reading textbooks in parallel with popular science books. I am mainly using Zee’s Quantum Field Theory in a Nutshell, Klauber’s Student Friendly Quantum Field Theory, and Tong’s lecture notes and videos.

Feynman penguin diagram

Feynman diagrams are often used in pop-sci texts to explain particle decay paths and interactions. Actually they are shortcuts for calculating terms in daunting integrals. The penguin is not a metaphor but a crib – a funny name for a specific class of diagrams that sort of resemble penguins.

But I also enjoyed Sean Carroll’s The Particle at the End of the Universe – my favorite QFT- / Higgs-related pop-sci book. Reading his chapters on quantum fields I felt he has boldly gone where no other physicist writing pop-sci had gone before. In many popular accounts of the Higgs boson and Higgs field we find somewhat poetic accounts of particles that communicate forces, such as the photon being the intermediary of electromagnetic forces.

Sean Carroll goes to the mathematical essence of the relationship of (rather abstract) symmetries, connection fields and forces:

The connection fields define invisible ski slopes at every point in space, leading to forces that push particles in different directions, depending on how they interact. There’s a gravitational ski slope that affects every particle in the same way, an electromagnetic ski slope that pushes positively charged particles one way and negatively charged particles in the opposite direction, a strong-interaction ski slope that is only felt by quarks and gluons, and a weak-interaction ski slope that is felt by all the fermions of the Standard Model, as well as by the Higgs boson itself. 

Indeed, in his blog Carroll writes:

So in the end, recognizing that it’s a subtle topic and the discussion might prove unsatisfying, I bit the bullet and tried my best to explain why this kind of symmetry leads directly to what we think of as a force. Part of that involved explaining what a “connection” is in this context, which I’m not sure anyone has ever tried before in a popular book. And likely nobody ever will try again!

This is the best popular account of symmetries and forces I could find so far – yet I confess: I could not make 100% sense of this before I had plowed through the respective chapters in Zee’s book. This is the right place to add a disclaimer: Of course I hold myself accountable for a possibly slow absorbing power or wrong approach of self-studying, as well as for confusing my readers. My brain is just the only one I have access to for empirical analysis right now and the whole QFT thing is an experiment. I should maybe just focus on writing about current research in an accessible way or keeping a textbook-style learner’s blog blog similar to this one.

Back to metaphors: Symmetries are usually explained by invoking rotating regular objects and crystals, but I am not sure if this image will inspire anything close to gauge symmetry in readers’ minds. Probably worse: I had recalled gauge symmetry in electrodynamics, but it was not straight-forward how to apply and generalize it to quantum fields – I needed to see some equations.

Sabine Hossenfelder says:

If you spend some time with a set of equations, pushing them back and forth, you’ll come to understand how the mathematical relationships play together. But they’re not like anything. They are what they are and have to be understood on their own terms.

Actually I had planned a post on the different routes to QFT – complementary to my post on the different ways to view classical mechanics. Unfortunately I feel the mathematically formidable path integrals would lend themselves more to metaphoric popularization – and thus more confusion.

You could either start with fields and quantize them which turn the classical fields (numbers attached to any point in space and time) into mathematical operators that actually create and destroy particles. Depending on the book you pick this is introduced as something straight-forward or as a big conceptual leap. My initial struggles with re-learning QFT concepts were actually due to the fact I had been taught the ‘dull’ approach (many years ago):

  • Simple QM deals with single particles. Mathematically, the state of those is described by the probability of a particle occupying this state. Our mathematical operators let you take the proverbial quantum leap – from one state to another. In QM lingo you destroy or create states.
  • There are many particles in condensed matter, thus we just extend our abstract space. The system is not only described by the properties of each particle, but also by the number of particles present. Special relativity might not matter.
  • Thus it is somehow natural that our machinery now destroys or annihilates particles.

The applications presented in relation to this approach were all taken from solid state physics where you deal with lots of particles anyway and creating and destroying some was not a big deal. It is more exciting if virtual particles are created from the vacuum and violating the conservation of energy for a short time, in line with the uncertainty principle.

The alternative route to this one (technically called the canonical quantization) is so-called path integral formalism. Zee introduces it via an anecdote of a wise guy student (called Feynman) who pesters his teacher with questions on the classical double-slit experiment: A particle emitted from a source passes through one of two holes and a detector records spatially varying intensity based on interference. Now wise guy asks: What if we drill a third hole, a fourth hole, a fifth hole? What if we add a second screen, a third screen? The answer is that adding additional paths the particle might take the amplitudes related to these paths will also contribute to the interference pattern.

Now the final question is: What if we remove all screens – drilling infinite holes into those screens? Then all possible paths the particle can traverse from source to detector would contribute. You sum over all (potential) histories.

I guess, a reasonable pop-sci article would probably not go into further details of what it means to sum over an infinite number of paths and yet get reasonable – finite – results, or to expound why on earth this should be similar to operators destroying particles. We should add that the whole amplitude-adding business was presented as an axiom. This is weird, but this is how the world seems to work! (Paraphrasing Feynman).

Then we would insert an opaque blackbox [something about the complicated machinery – see details on path integrals if you really want to] and jump directly to things that can eventually be calculated like scattering cross-sections and predictions how particle will interact with each other in the LHC … and gossip about Noble Prize winners.

Yet it is so tempting to ponder on how the classical action (introduced here) is related to this path integral: Everything we ‘know about the world’ is stuffed into the field-theoretical counterpart of the action. The action defines the phase (‘angle’) attached to a path. (Also Feynman talks about rotating arrows!) Quantum phenomena emerge when the action becomes comparable to Planck’s constant. If the action is much bigger most of the paths are cancelled out because  If phases fluctuate wildly contributions of different amplitudes get cancelled.

“I am not gonna simplify it. If you don’t like it – that’s too bad!”

And Now for Something Completely Different: Quantum Fields!

Do I miss assignments and exams? Definitely not, and I am now – finally, really, absolutely – determined to complete another program I had set for myself about 2-3 years ago. I had not been able to pull it off in addition to being a moonlighting student.

Since about 10 years I have been recycling my physics knowledge on a regular basis. By recycling I mean: Reading text books as a hobby – sometimes practiced at 5:00 AM, before I jumped into my company car to drive to an IT customer’s site.

I have read the occasional popular physics book, too, but I found peace and tranquility in working through long-winded mathematical derivations. I dare say, this is what kept me sane. Actually, this is the therapy I would recommend to all those burnt-out managers and consultants in the corporate world – especially to the geeky ones.

Having finished that Recapture What You Had Already Known program I was shocked that I – trained as an applied physicist – had missed some major advances in theoretical physics. I could not make head or tail of how the Higgs field is giving the particles mass.

There is some glaring irony: I was not completely ignorant of theoretical physics – I took some non-mandatory classes to understand the theory superconductivity, the field I worked in as an experimental physicist. The Higgs mechanism is very similar to phase transitions like a metal become superconducting – but obviously I was not able (anymore?) to make this mental connection.

Even worse I was ignorant of the big questions in physics. What is it exactly that we don’t know today? Where is the final frontier no theorist has boldly gone before?

Thus I set out to systematically study the language theoretical physicists speak today: Quantum Field Theory (QFT) and General Relativity (GR). I managed to work through about 50% of the books, lectures notes and videos I had selected – then I tried to take combining anything too far and finally focussed on thermodynamics, solar and wind power and the smart grid for two years. By the way, my coffee reduction program came to a grinding halt as well as you have to drink coffee as a student who has to meet deadlines and submit assignments.

I am resuming the QFT / GR program now.

Popular physics accounts of quantum mechanics often make me cringe. It’s not that Schrödinger’s equation, the double-slit experiment and Schrödinger’s cat are invalid examples – they have great explanatory power in elucidating the inherent strangeness of the quantum world (This sounds like a clichéd blurb, doesn’t it?). But real interesting stuff starts where many particles are involved and/or particles are very energetic. And this is just not covered in the simple picture (the non-relativistic Schrödinger equation).
I have already ranted about that in my post Quantum Field Theory or: It’s More Than a Marble Turned into a Wiggly Line.

We had interesting discussions about MOOCs (Massive Open Online Courses) and I had complained about gamification. You might argue that working towards a degree and submitting assignments that get graded is the old-school version of gamification, and you might be right. I try to avoid gamification now. Though I will watch educational videos as supplementary material I will not sign-up for any course now, become part of a group or participate in anything that provides me with deadlines and with feedback. The discerning reader might argue that my public announcement of a private study program is just the same way of holding myself accountable.

I try to rise to the challenge of posting some ‘pop-sci’, math-free articles about QFT although or because I want to understand it at a fundamental level. What QFT adds to the spooky weird nature of quantum mechanics are several layers and concepts of mathematical abstractions. You can explain the Higgs mechanism and symmetry breaking by referring to a potential well shaped like a mexican hat and a small marble moving in its brim. But what kind of space is this, and what exactly is the ball?

I find these concepts most fascinating – because it is these abstract ‘hyper’ spaces where classical physics start to get as interesting and spooky quantum physics. QFT builds on Classical Field Theory, and the latter is underrated in terms of geek factor.

My program will loosely follow these lectures by David Tong. I picked those because lecture notes are available, too. Above all, I enjoy Tong’s blackboard & chalk presentation style.

I have found the first lecture also on Youtube though I prefer the formats provided Perimeter Institute (previous link), with the snapshots of the blackboard displayed side-by-side with the videos.

Science geeks and life-long education junkies: Do you prefer blackboard or Powerpoint?

 

Why Fat Particles Radiate Less

I am just reading Knocking on Heaven’s Door by Lisa Randall which has a chapter on the impressive machinery of Large Hadron Collider. The LHC has been built to smash proton beams against each other: Protons, not electrons. Why protons? I stumbled upon the following statement:

“But accelerated particles radiate, and the lighter they are, the more they do so”.

Electrons would cause higher radiation losses and less energy would be available for the creation of new particles in collisions.

But why is this so? In order to prove this, you would go through a calculation of the electromagnetic field generated by the moving particles based on Maxwell’s equations (which are relativistic per default).

I think you can understand it qualitatively from this chain of reasoning:

If a particle is forced to move on a curved path, it is accelerated – such as planets are accelerated all the time by the gravitational force exerted by the sun.

Consider a curved part of the LHC’s trajectory – the radius is given. The acceleration of particles moving in circles is equal to v2/R with R being the radius of curvature and v the speed of the particle. So acceleration increases with increasing speed.

Charged particles lose energy via electromagnetic radiation when they are accelerated. This can be understood from conservation of energy: If a particle would be slowed down in free space (friction due to collisions with particles in the atmosphere being not an option), the energy has to go somewhere. If a particle is accelerated, some force does work on it (which is also true for an orbiting particle.
This argument has been used to prove the classical model of the atom as a miniature solar system wrong: If an electron would orbit round the core it would lose energy and finally ‘fall down’ into the core. So we need a quantum mechanics to explain the stability of atoms.

Particles are accelerated by electrical fields: the energy transferred to particles of the same electrical charge would be the same for a proton or an electron (except the sign). For particles with velocities close to the speed of light relativistic effects cannot be neglected so the energy of a particle of rest mass m and velocity v is (c = speed of light)

For smaller velocities this reduces to the sum of the ‘rest mass energy’ mc2 and the kinetic energy mv2/2.

If the energy is given a particle with higher rest mass would exhibit a smaller velocity. Thus its acceleration in a toroidal tube (~v2) would be smaller.

CERN LHC Tunnel1

LHC Tunnel, CERN (Wikimedia)


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Further reading and notes

– Note that I prefer to call the ‘rest mass’ just ‘mass’ – I would not introduce a so-called relativistic mass.
– As usual, I am recommending Feynman’s Physics Lectures. Even in Volume 1 he gives a concise introduction on the radiation of charges. Actually he states that he developed an expression for the electrical field caused by a single point charge for the purpose of this lecture only that had not been published elsewhere before.  Volume 2 comprises electrodynamics in depth.