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.

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 ‘*Platonicit*y’: *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.

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!”