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!”

On Science Communication

In a parallel universe I might work as a science communicator.

Having completed my PhD in applied physics I wrote a bunch of job applications, one of them being a bit eccentric: I applied at the Austrian national public service broadcaster. (Adding a factoid: According to Wikipedia Austria was the last country in continental Europe after Albania to allow nationwide private television broadcasting).

Fortunately I deleted all those applications that would me make me blush today. In my application letters I referred to the physicist’s infamous skills in analytical thinking, mathematical modeling and optimization of technical processes. Skills that could be applied to basically anything – from inventing novel tractor beam generators for space ships to automatically analyzing emoticons in Facebook messages.

If I would have been required to add a social-media-style tagline in these dark ages of letters on paper and snail mail I probably would have tagged myself as combining anything, in particular experimental and theoretical physics and, above all, communicating science to different audiences. If memory serves I used the latter argument in my pitch to the broadcaster.

I do remember the last sentence of that pivotal application letter:

I could also imagine working in front of a camera.

Yes, I really did write that – based on a ‘media exposure’ of having appeared on local TV for some seconds.

This story was open-ended: I did not receive a reply until three months later, and at that time I was already employed as a materials scientist in R&D.

In case job-seeking graduate students are reading this: It was imperative that I added some more substantial arguments to my letters, that is: hands-on experience – maintaining UV excimer lasers, knowing how to handle liquid helium, decoding the output of X-ray diffractometers, explaining accounting errors to auditors of research grant managing agencies. Don’t rely on the analytical skills pitch for heaven’s sake.

I pushed that anecdote deep down into the netherworlds of my subconsciousness. Together with some colleagues I ritually burnt items reminiscent of university research and of that gruelling job hunt, such as my laboratory journals and print-outs of job applications. This spiritual event was eventually featured on a German proto-blog website and made the German equivalent of ritual burning the top search term for quite a while.

However, today I believe that the cheeky pitch to the broadcaster had anticipated my working as a covert science communicator:

Fast-forward about 20 years and I am designing and implementing Public Key Infrastructures at corporations. (Probably in vain, according to the recent reports about NSA activities). In such projects I covered anything from giving the first concise summary to the CIO (Could you explain what PKI is – in just two Powerpoint slides?) to spending nights in the data center – migrating to the new system together with other security nerds, fueled by pizza and caffeine.

The part I enjoyed most in these projects was the lecture-style introduction (the deep dive in IT training lingo) to the fundamentals of cryptography. Actually these workshops were the nucleus of a lecture I gave at a university later. I aimed at combining anything: Mathematical algorithms and anecdotes (notes from the field) about IT departments who locked themselves out of the high-security systems, stunning history of cryptography and boring  EU legislation, vendor-agnostic standards and the very details of specific products.

Usually the feedback was quite good though once the comment in the student survey read:

Her lectures are like a formula one race without pitstops.

This was a lecture given in English, so it is most likely worse when I talk in German. I guess, Austrian Broadcasting would have forced me to take a training in professional speaking.

As a Subversive Element I indulged in throwing in some slides about quantum cryptography – often this was considered the most interesting part of the presentation, second to my quantum physics stand-up edutainment in coffee breaks. The downside of that said edutainment were questions like:

And … you turned down *that* for designing PKIs?

I digress – find the end of that story here.

I guess I am obsessed with combining consulting and education. Note that I am referring to consulting in terms of working hands-on with a client, accountable for 1000 users being able to logon (or not) to their computers –  not your typical management consultant’s churning out sleek Powerpoint slides and leaving silently before you need to get your hands dirty (Paraphrasing clients’ judgements of ‘predecessors’ in projects I had to fix).

It is easy to spot educational aspects in consulting related to IT security or renewable energy. There are people who want to know how stuff really works, in particular if that helps to make yourself less dependent on utilities or on Russian gas pipelines, or to avoid being stalked by the NSA.

But now I have just started a new series of posts on Quantum Field Theory. Why on earth do I believe that this is useful or entertaining? Considering in particular that I don’t plan to cover leading edge research: I will not comment on hot new articles in Nature about stringy Theories of Everything.

I stubbornly focus on that part of science I have really grasped myself in depth – as an applied physicist slowly (re-)learning theory now. I will never reach the frontier of knowledge in contemporary physics in my lifetime. But, yes, I am guilty of sharing sensationalist physics nuggets on social media at times – and I jumped on the Negative Temperature Train last year.

My heart is in reading old text books, and in researching old patents describing inventions of the pre-digital era. If you asked me what I would save if my house is on fire I’d probably say I’d snatch the six volumes of text books in theoretical physics my former physics professor, Wilhelm Macke, has written in the 1960s. He had been the last graduate student supervised by Werner Heisenberg. Although I picked experimental physics eventually I still consider his lectures the most exceptional learning experience I ever had in life.

I have enjoyed wading through mathematical derivations ever since. Mathy physics has helped me to save money on life coaches or other therapists when I was a renowned, but nearly burnt-out ‘travelling knowledge worker’ AKA project nomad. However, I understand that advanced calculus is not everybody’s taste – you need to invest quite some time and efforts until you feel these therapeutic effects.

Yet, I aim at conveying that spirit, although I had been told repeatedly by curriculum strategists in higher education that if anything scares people off pursuing a tech or science degree – in particular, as a post-graduate degree – it is too much math, including reference to mathy terms in plain English.

However, I am motivated by a charming book:

The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

by science writer Jennifer Ouellette. According to her website, she is a recovering English major who stumbled into science writing as a struggling freelance writer… and who has been avidly exploring her inner geek ever since. How could you not love her books? Jennifer is the living proof that you can overcome math anxiety or reluctance, or even turn that into inspiration.

Richard Feynman has given a series of lectures in 1964 targeted to a lay audience, titled The Character of Physical Law.

Starting from an example in the first lecture, the gravitational field, Feynman tries expound how physics relates to mathematics in the second lecture – by the way also introducing the principle of least action as an alternative to tackle planetary motions, as discussed in the previous post.

It is also a test of your dedication as a Feynman fan as the quality of this video is low. Microsoft Research has originally brought these lectures to the internet – presenting them blended with additional background material (*) and a transcript.

You may or may not agree with Feynman’s conclusion about mathematics as the language spoken by nature:

It seems to me that it’s like: all the intellectual arguments that you can make would not in any way – or very, very little – communicate to deaf ears what the experience of music really is.

[People like] me, who’s trying to describe it to you (but is not getting it across, because it’s impossible), we’re talking to deaf ears.

This is ironic on two levels, as first of all, if anybody could get it across – it was probably Feynman. Second, I agree to him. But I will still stick to my plan and continue writing about physics, trying to indulge in the mathy aspects, but not showing off the equations in posts. Did I mention this series is an experiment?


(*) Technical note: You had to use Internet Explorer and install Microsoft Silverlight when this was launched in 2009 – now it seems to work with Firefox as well. Don’t hold be liable if it crashes your computer though!

From ElKement: Space Balls, Baywatch, and the Geekiness of Classical Mechanics

This is self-serving, but I can’t resist reblogging Joseph Nebus’ endorsement of my posts on Quantum Field Theory. Joseph is running a great blog on mathematics, and he manages to explain math in an accessible and entertaining way. I hope I will be able to do the same to theoretical physics!


Over on Elkement’s blog, Theory and Practice of Trying To Combine Just Anything, is the start of a new series about quantum field theory. Elke Stangl is trying a pretty impressive trick here in trying to describe a pretty advanced field without resorting to the piles of equations that maybe are needed to be precise, but, which also fill the page with piles of equations.

The first entry is about classical mechanics, and contrasting the familiar way that it gets introduced to people —- the whole forceequalsmasstimesacceleration bit — and an alternate description, based on what’s called the Principle of Least Action. This alternate description is as good as the familiar old Newton’s Laws in describing what’s going on, but it also makes a host of powerful new mathematical tools available. So when you get into serious physics work you tend to shift over to that model; and, if you…

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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?