On Learning

Some years ago I was busy with projects that required a lot of travelling but I also needed to stay up-to-date with latest product features and technologies. When a new operating system was released a colleague asked how I could do that – without having time for attending trainings. Without giving that too much thought, and having my personal test lab in mind, I replied:

I think I always try to solve some problem!

tl;dr – you can skip the rest as this has summed it all up.

About one year ago I ‘promised’ to write about education, based on my experiences as a student and as a lecturer or trainer. I haven’t done so far – as I am not sure if my simplistic theory can be generalized.

There are two very different modes of learning that I enjoy and consider effective:

  1. Trying to solve some arbitrary problem that matters to me (or a client) and starting to explore the space of knowledge from that angle.
  2. Indulging in so-called theory seemingly total unrelated to any practical problem to be solved.

Mode 2 was what I tried to convey in my post about the positive effects of reading theoretical physics textbooks in the morning. The same goes for cryptography.

I neither need advanced theoretical physics when doing calculations for heat pump systems, nor do I need the underlying math and computer science when tweaking digital certificates. When I close the theory books, I am in mode 1.

In the last weeks that mode 1 made me follow a rather steep learning curve with respect to database servers and SQL scripts. I am sure I have made any possible stupid mistake when exploring all the options. I successfully killed performance by too much nested sub-queries and it took me some time to recognize that the referral to the row before is not as straight-forward as in a spreadsheet program. One could argue that a class on database programming might have been more effective here, and I cannot prove otherwise. But most important for me was: I finally achieved what I wanted and it was pure joy all the way. I am a happy dilettante perhaps.

I might read a theoretical book on data structures and algorithms someday and let it merge with my DIY tinkering experience in my subconsciousness – as this how I think those two modes work together.

As for class-room learning and training, or generally learning with or from others, I like those ways best that cater to my two modes:

I believe that highly theoretical subjects are suited best for traditional class-room settings. You cannot google the foundations of some discipline as such foundations are not a collection of facts (each of them to be googled) but a network of interweaving concepts – you have to work with some textbook or learn from somebody who lays out that network before you in a way that allows for grasping the structure – the big picture and the details. This type of initial training also prepares you for future theoretical self-study. I still praise lectures in theoretical physics and math I attended 25 years ago to the skies.

And then there is the lecturer speaking to mode 2: The seasoned expert who talks ‘noted from the field’. The most enjoyable lecture in my degree completed last year was a geothermal energy class – given by a university professor who was also the owner of an engineering consultancy doing such projects. He introduced the theory in passing but he talked about the pitfalls that you would not expect from learning about best practices and standards.

I look back on my formal education(s) with delight as most of the lectures, labs, or projects were appealing to either mode 1 or mode 2. In contrast to most colleagues I loved the math-y theory. In projects on the other hand I had ample freedom to play with stuff – devices, software, technology – and to hone practical skills, fortunately without much supervision. In retrospect, the universities’ most important role with respect to the latter was to provide the infrastructure. By infrastructure I mean expensive equipment – such as the pulsed UV lasers I once played with, or contacts to external ‘clients’ that you would not have had a chance to get in touch otherwise. Two years ago I did the simulations part of a students’ group project, which was ‘ordered’ by the operator of a wind farm. I brought the programming skills to the table – as this was not an IT degree program –  but I was able to apply them to a new context and learn about the details of wind power.

In IT security I have always enjoyed the informal exchange of stories from the trenches with other experienced professionals – this includes participation in related forums. Besides it fosters the community spirit, and there is no need to do content-less ‘networking’ of any other sort. I have just a few days of formal education in IT.

But I believe that your mileage may vary. I applied my preferences to my teaching, that is: explaining theory in – probably too much – depth and then jumping onto any odd question asked by somebody and trying something out immediately. I was literally oscillating between the flipchart and the computer with my virtual machines – I had been compared to a particle in quantum mechanics whose exact location is unknown because of that. I am hardly able to keep to my own agenda even if I had been given any freedom whatsoever to design a lecture or training and to write every slide from scratch. And I look back in horror on delivering trainings (as an employed consultant) based on standardized slides not to be changed. I think I was not the best teacher for students and clients who expected well organized trainings – but I know that experts enjoyed our jam sessions formerly called workshops.

When I embarked on another degree program myself three years ago, I stopped doing any formal teaching myself – before I had given a lecture on Public Key Infrastructure for some years, in a master’s degree program in IT security. Having completed my degree in renewable energy last year I figured that I was done now with any formal learning. So far, I feel that I don’t miss out on anything, and I stay away from related job offerings – even if ‘prestigious’.

In summary, I believe in a combination of pure, hard theory, not to be watered down, and not necessarily to be made more playful – combined with learning most intuitively and in an unguided fashion from other masters of the field and from your own experiments. This is playful no matter how often you bang your head against the wall when trying to solve a puzzle.

Physics book from 1895

A physics book written in 1895, a farewell present by former colleagues in IT – one the greatest gifts I ever got. My subconsciousness demands this is the best way to illustrate this post. I have written a German post on this book which will most likely never be translated as the essence of this post are quotes showing the peculiar use of the German language which strikes the modern reader quite odd.

How to Introduce Special Relativity (Historical Detour)

I am just reading the volume titled Waves in my favorite series of ancient textbooks on Theoretical Physics by German physics professor Wilhelm Macke. I tried to resist the urge to write about seemingly random fields of physics, and probably weird ways of presenting them – but I can’t resist any longer.

There are different ways to introduce special relativity. Typically, the Michelson-Morely experiment is presented first, as our last attempt in a futile quest to determine to absolute speed in relation to “ether”. In order to explain these results we have to accept the fact that the speed of light is the same in any inertial frame. This is weird and non-intuitive: We probably can’t help but compare a ray of light to a bunch of bullets or a fast train – whose velocity relative to us does change with our velocity. We can outrun a train but we can’t outrun light.

Michelson–Morley experiment

The Michelson–Morley experiment: If light travels in a system – think: space ship – that moves at velocity v with respect to absolute space the resulting velocity should depend on the angle between the system’s velocity and the absolute velocity. Just in the same way as the observed relative velocity of a train becomes zero if we manage to ride besides it in a car driving at the same speed as the train. But this experiments shows – via non-detected interference of beam of alleged varying velocities – that we must not calculate relative velocities of beams of light. (Wikimedia)

Yet, not accepting it would lead to even more weird consequences: After all, the theory of electromagnetism had always been relativistically invariant. The speed of light shows up as a constant in the related equations which explain perfectly how waves of light behaves.

I think the most straight-forward way to introduce special relativity is to start from its core ideas (only) – the constant speed of light and the equivalence of frames of reference. This is the simplicity and beauty of symmetry. No need to start with trains and lightning bolts, as Matthew Rave explained so well. For the more visually inclined there is an ingenious and nearly purely graphical way, called k-calculus (that is however seldom taught AFAIK – I had stumbled upon it once in a German book on relativity).

From the first principles all the weirdness of length contraction and time dilation follows naturally.

But is there a way to understand it a bit better though?

Macke also starts from the Michelson-Morely experiment  – and he adds the fact that it can be “explained” by Lorentz’ contraction hypothesis: Allowing for direction-dependent velocities – as in “ether theory” – but adding the odd fact that rulers contract in the direction of the unobservable absolution motion makes the differences the rays of light traverse go away. It also “explains” time dilatation if you consider your typical light clock and factor in the contraction of lengths:

Light clock

The classical light clock: Light travels between two mirrors. When it hits a mirror it “ticks”. If the clock moves relatively to an observer the path to be traversed between ticks appears to be longer. Thus measurement of time is tied to measurement of spatial distances.

However, length contraction could be sort of justified by tracing it back to the electromagnetic underpinnings of stuff we use in the lab. And it is the theory of electromagnetism where the weird constant speed of light sneaks in.

Contraction can be visualized by stating that like rulers and clocks are finally made from atoms, ions or molecules, whose positions are determined by electromagnetic forces. The perfect sphere of the electrostatic potential around a point charge would be turned into an ellipsoid if the charge starts moving – hence the contraction. You could hypothesize that only “electromagnetic stuff” might be subject to contraction and there might be “mechanical stuff” that would allow for measuring true time and spatial dimensions.

Thus the new weird equations about contracting rulers and slowing time are introduced as statements about electromagnetic stuff only. We use them to calculate back and forth between lengths and times displayed on clocks that suffer from the shortcomings of electromagnetic matter. The true values for x,y,z,t are still there, but finally inaccessible as any matter is electromagnetic.

Yes, this explanation is messy as you mix underlying – but not accessible – direction-dependent velocities with the contraction postulate added on top. This approach misses the underlying simplicity of the symmetry in nature. It is a historical approach, probably trying to do justice to the mechanical thought experiments involving trains and clocks that Einstein had also used (and that could be traced back to his childhood spent basically in the electrical engineering company run by his father and uncle, according to this biography).

What I found fascinating though is that you get consistent equations assuming the following:

  • There are true co-ordinates we can never measure; for those Galileian Transformations remain valid, that is: Time is the same in all inertial frames and distances just differ by time times the speed of the frame of reference.
  • There are “apparent” or “electromagnetic” co-ordinates that follow Lorentz Transformations – of which length contraction and time dilations are consequences.

To make these sets of transformations consistent you have to take into account that you cannot synchronize clocks in different locations if you don’t know the true velocity of the frame of reference. Synchronization is done by placing an emitter of light right in the middle of the two clocks to be synchronized, sending signals to both clocks. This is correct only if the emitter is at rest with respect to both clocks. But we cannot determine when it is at rest because we never know the true velocity.

What you can do is to assume that one frame of reference is absolutely at rest, thus implying that (true) time is independent of spatial dimensions, and the other frame of reference moving in relation to it suffers from the problem of clock synchronization – thus in this frame true time depends on the spatial co-ordinates used in that frame.

The final result is the same when you eliminate the so-called true co-ordinates from the equations.

I don’t claim its the best way to explain special relativity – I just found it interesting, as it tries to take the just hypothetical nature of 4D spacetime as far as possible while giving results in line with experiments.

And now explaining the really important stuff – and another historical detour in its own right

Yes, I changed the layout. My old theme, Garland, had been deprecated by wordpress.com. I am nostalgic – here is a screenshot –  courtesy to visitors who will read this in 200 years.

elkement.wordpress.com with theme Garland

elkement.wordpress.com using theme Garland – from March 2012 to February 2014 – with minor modifications made to colors and stylesheet in 2013.

I had checked it with an iPhone simulator – and it wasn’t simply too big or just “not responsive”, the top menu bar boundaries of divs looked scrambled. Thus I decided the days of Garland the three-column layout are over.

Now you can read my 2.000 words posts on your mobile devices – something I guess everybody has eagerly anticipated.

And I have just moved another nearly 1.000 words of meta-philosophizing on the value of learning such stuff (theory of relativity, not WordPress) from this post to another draft.

In Praise of Textbooks with Tons of Formulas (or: The Joy of Firefighting)

I know. I am repeating myself.

Maurice Barry has not only recommended Kahneman’s Thinking, Fast and Slow to me, but he also runs an interesting series of posts on his eLearning blog.

These got mixed and entangled in my mind, and I cannot help but returning to that pet topic of mine. First, some statistically irrelevant facts of my personal observations – probably an example of narrative fallacy or mistaking correlation for causation:

As you know I had planned to reconnect to my roots as a physicist for a long time despite working crazy schedules as a so-called corporate knowledge worker. Besides making the domain subversiv.at mine and populating it with content similar to the weirdest in this blog I invented my personal therapy to deflect menacing burn-out: I started reading or better working with my old physics textbooks. Due to time constraints I sometimes had to do this very early in the morning – and I am not a lark. I have read three books on sleep research recently – I know that both my sleep duration as well as my midsleep are above average and I lived in a severely sleep-deprived state most of my adult life.

Anyway, the point was: Physics textbooks gave me some rehash of things I had forgotten and prepared me to e.g. work with the heat transfer equation again. But what was more important was: These books transformed my mind in unexpected ways. Neither entertaining science-is-cool pop-sci books nor philosophical / psychological books about life, the universe and everything could do this for me at that level. (For the records: I tried these to, and I am not shy to admit I picked some self-help books also. Dale Carnegie, no less.)

There were at least two positive effects – I try to describe them in my armchair psychologist’s language. Better interpretations welcome!

Concentrating and abstract reasoning seems to be effective in stopping or overruling the internal over-thinking machine that runs in circles if you feel trapped in your life or career. Probably people like me try to over-analyze what has to be decided intuitively anyway. Keeping the thinking engine busy lets the intuitive part do its work. Whatever it was – it was pleasant, and despite the additional strain on sleep and schedule it left me more energetic, more optimistic, and above all more motivated and passionate about that non-physics work.

I also found that my work related results – the deliverables as we say – improved. I have been the utmost perfectionist ever since and my ability to create extensive documentation in parallel to doing the equivalent of cardiac surgery to IT systems is legendary (so she says in her modest manner). Nevertheless, plowing through tensor calculus and field equations helps to hone these skills even more. For those who aren’t familiar with that biotope: The mantra of other Clint-Eastwood-like firefighters is rather: Real experts don’t provide documentation!

I would lie if I would describe troubleshooting issues with digital certificates as closely related to theoretical physics. You can make some remote connections between skills that sort of related such as cryptography is math after all, but I am not operating at that deep mathematical level most of the time. I rather believe that anything rigorous and mathy puts your mind – or better its analytical subsystem – in a advanced state. Advanced refers to the better prepration to tackle a specific class of problems. The caveat is that you lose this ability if you stop reading textbooks at 4:00 AM.

Using Kahneman’s terminology (mentioned briefly in my previous post) I consider mathy science the ultimate training for system 2 – your typically slow rational decision making engine. It takes hard work and dedication at the beginning to make system 2 work effortless in some domains. In my very first lecture at the university ever the math professor stated that mathematics will purge and accelerate your brain – and right he was.

Hence I am so skeptical about joyful learning and using that science-is-cool-look-at-that-great-geeky-video-of-blackholes-and-curved-space approach. There is no simple and easy shortcut and you absolutely, positively have to love the so-called tedious work you need to put in. You are rewarded later with that grand view from the top of the mountain. The ‘trick’ is that you don’t consider it tedious work.

Kahneman is critical of so-called intuition – effortless intuitive system 1 at work – and he gives convincing accounts of cold-hearted algorithms beating humans, e.g. in picking the best candidate for a job. However, he describes his struggles with another school of thought of psychologists who are wary of algorithms. I have scathed dumb HR-acronym-checking-bots at this blog, too. But Kahneman finally reached an agreement with algorithm haters as he acknowledged that there is a specific type of expert intuition that appears like magic to outsiders. His examples: Firefighters and nurses who feel what is wrong – and act accordingly – before they can articulate it. He still believes that picking stocks or picking job applicants is not a skill and positive results don’t correlate at with skill but are completely random.

I absolutely love the example of firefighters as I can literally relate to it. Kahneman demystifies their magic abilities though as he states that this is basically pattern recognition – you have gathered similar experience, and after many years of exposure system 1 can draw from that wealth of patterns unconsciously.

Returning to my statistically irrelevant narrative this does still not explain completely why exposure to theoretical physics should make me better at analyzing faulty security protocols. Physics textbooks make you an expert in solving physics textbook problems, this is: in recognizing patterns and provide you with ideas of that type of out-of-the-box idea you sometimes need to find a clever mathematical proof. You might get better in solving that physics puzzles people enjoy sharing on social media.

But probably the relation to troubleshooting tech problems is very simple and boils down to the fact that you love to tackle formal, technical problems again and again even if many attempts are in vain. The motivation and the challenge is in looking at the problem as a black box and trying to find a clever way to get in. Every time you fail you learn something nonetheless, and that learning is a pleasure in its own right.

DOD mobile aircraft firefighting training device

“Student Friendly Quantum Field Theory”

As other authors of science blogs have pointed out: Most popular search terms are submitted by students. So I guess it is not the general public who is interested in: the theory of gyroscopes, (theory of) microwaves, (theory of) heat pumps, (theory of) falling slinkies, or the Coriolis force.

I believe that these search terms are submitted by students in physics or engineering.

“Student Friendly Quantum Field Theory” has been the top search term for this blog since I had put the textbook of the same title by Robert Klauber on my physics resources list.

So I pay my dues now and dedicate a post to this textbook: I am reviewing the first edition 2013, as I have just missed the publication of the 2nd. In short: I think the book is a pedagogical masterpiece.

This is also an auxiliary posting in my series on QFT. I want to keep this post to a reasonable non-technical level not to scare off my typical readers too much (but I apologize for some technical terms – having the “target audience” of physics students in mind).

Quantum field theory for the gifted amateur has been searched for as well. I believe indeed that this is a book for the gifted amateur in terms of a self-studying quantum physics enthusiast, at least more so than other books on QFT.

However, also the amateur should have had a thorough education in theoretical physics. If you have mastered your typical [*] four (?) semesters in theoretical physics – classical mechanics, electrodynamics, (non-relativistic) quantum theory, and statistical mechanics you should be well prepared to understand the material in this book. If the following key words trigger some memories of equations, you meet the requirements: Lagrange formalism of classical mechanics, Poisson bracket, Maxwell’s equations in four-vector notation.
[*] I have graduated at a time when bachelor’s degrees have been unheard of here in Europe – I cannot explain the prerequisites properly in terms of modern curricula or “graduate” versus “undergraduate”.

I had some exposure to quantum field theory that is used in solid state physics, too, but I don’t believe this is a pre-requisite.

I was most interested in a thorough understanding of the basics and less so in an elegant discussion of leading-edge theories. As discussed in detail earlier I can track down exactly when I don’t understand popular physics books – and by “understanding” I mean being able to recognize the math behind a popular text. (However, in this sense pop-sci books can be definition not be “understood” by the lay audience they are written for).

I didn’t have an idea how the Higgs boson gives the particles and mass, and I could not image how the electron’s spin could be a by-product yielded by a theory – so I wanted to plow through the foundations of QFT. If you want to understand the Higgs boson and field, too, this book does not yet explain this – but I believe you need some thorough grounding as given by SFQFT if you want to tackle more advanced texts.

Student Friendly Quantum Field Theory by Robert Klauber.

Student Friendly Quantum Field Theory by Robert Klauber. I have put my personal slinky on top of the book for no particular reason.

We don’t learn much about Robert Klauber himself. The blurb says:

Bob Klauber, PhD, is retired from a career of working in industry, where he led various research projects and obtained over twenty patents. At different times during and after that career, he taught a diverse number of graduate and undergraduate level physics courses.

So the author is not a tenured professor, and I believe this might be advantageous.

Written solely for students, not for peers

Klauber does not need to show off his smartness to his peers. Yes, he has some pet peeves – such as questioning the true nature of the vacuum, often painted in popular science as a violent sea consisting of pairs of particles popping out of nowhere and vanishing again. Klauber tags some opinions of his as non-mainstream [**], and he links to a few related papers of his own – but he does so in a rather humble way. Your milage may vary  but I found it very refreshing not to find allusions to the impact and grandness of this own original work or to his connectedness in the scientific community (in terms of …when I occasionally talked to Stephen Hawking last time at That Important Conference…)
[**] I feel the need to add a disclaimer: This is not at all “outsider physics” or unorthodox in the way the term is used by professionals bombarded with questionable manuscripts by authors set to refute Einstein or Newton.

But it is not an “elegant” book either. It is not providing professionals with “new ways to see QFT as you never saw it before”; it is an anti-Feynman-y book so to speak. It is not a book I would describe in the way the publishers of the Commemorative Issue of Feynman’s Physics Lectures (1989) did:

Rereading the books, one sometimes seems to catch Feynman looking over his shoulder, not at his audience, but directly at his colleagues, saying, “Look at that! Look how I finessed that point! Wasn’t that clever?”

Nothing is Trivial, Easy and Obvious – and brevity is avoided

Student Friendly Quantum Field Theory (SFQFT) is dedicated to tackling the subject from the perspective of the learning student primarily and only. Klauber goes to great lengths to anticipate questions that might be on the reader’s mind and often refers to his own learning experience – and he always perfectly nails it. He explicitly utters his contempt for declaring things trivial or straight-forward.

Klauber has put considerable efforts into developing the perfect way(s) of presenting the material. Read a summary of his pedagogical strategy here. He avoids conciseness and brevity and he wonders why these seem to be held in such high regard – in education. This also explains why a book of more than 500 pages covers basics only. The same ideas are expounded in different forms:

  • Summary upfront, “big picture”.
  • Through derivations. In case of renormalization, he also gives sort of a “detailed overview” version in a single chapter before the theory unfolds in several chapters. The structure of the book is fractal so to speak: There are whole chapters dedicated to an overview – such as Bird’s Eye View given in Ch. 1 or the summary chapter on renormalization, and each chapter and section contains their own summaries, too.
  • So-called Wholeness Charts, tabular representations of steps in derivations. I found also the charts in the first chapters extremely useful that allow for comparing non-relativistic QM and QFT, and between “particle QM” and field theory – I owe Klauber for finally clearing up my personal confusions – since I haven’t noticed before that I had been trained in non-relativistic field theories. The summary of major steps in the development of the theory for different kinds of particles are laid out in three columns of a table covering several pages, one for each type of particle.
  • Another Summary the the end.

Nothing is omitted (The ugly truth).

The downside:

Now I have understood why Dirac called this an ugly theory he refused to consider the final fundamental theory of the universe. Klauber gives you all the unwieldy algebra. I have not seen something as ugly and messy as the derivations of renormalization. The book has about 520 pages: 100 of them are dedicated to renormalization, and 85 to the calculation of cross-sections in order to compare them with experiment.

The good things:

Klauber gives you really all the derivations, not a single step is omitted. Very often equations quoted in earlier chapters are repeated for convenience of the reader. The book contains problems, but none of the derivations essential for grasping new concepts are completely outsourced to the problems sections.

Scope of the book

You can read the first chapters of the book online, and here is the Table of Contents.

Klauber suspects the addition of modern theories and applications would be confusing and I believe he is right.

He starts with the relation of QFT and non-relativistic and/or non-field-y quantum physics.  I like his penchant for the Poisson bracket in particular and the thorough distinction between wave functions and fields, and how and if there is a correspondence. Take this with a grain of salt as I had been confused a lot with an older book that referred to anything – Schrödinger wave function as well as field – as “waves”.

Klauber uses quantum electrodynamics as the example for explaining concepts. Thus he follows the historical route approximately, and he quotes Feynman who stated that he always thought about theories in terms of palpable examples.

The table of contents is rather “orthodox”.

Free fields are covered first and related equations for scalar bosons (the simplest example), fermions and vector bosons. The latter are needed as ingredients of QED – electrons and photons. I enjoyed the subtle remarks about over-emphasizing the comparison with harmonic oscillators.

Field equations for fermions (such as electrons) do not have classical counter-parts – this is where all attempts to explain by metaphor must end. I set out to write a pop-sci series on QFT and accidentally read the chapter on fermions at the same time when David Yerle posted this challenge on his blog – how to explain the electron’s spin: Now I believe there is no shortcut to understanding the electron’s spin – and as far as I recall Richard Feynman and Sean Carroll (my benchmarks in terms of providing correct popularizations) weren’t able to really explain the electron’s spin in popular terms either. There are different ways to start from but these field equations don’t have classical counterpart, and you always end up with introducing or “discovering” mathematical objects that behave in an non-intuitive way – “objects” that anti-commute without being equal to zero (There aren’t any numbers A and B that satisfy AB = -BA unless either A or B are zero).

Feynman Diagram

Picture of a Feynman diagram, inscribed by Richard P. Feynman to Wikimedia user Ancheta Wis, in Volume 3 of his Feynman Lectures on Physics (Quantum Mechanics).

Interactions are introduced via Maxwell’s equations and QED. Inspecting these equations we finally learn how symmetry and forces are related – usually cloaked as symmetry gives rise to forces in popular texts. Actually, this was one of the things I was most interested in and it was a bit hard to plow through the chapter on spinors (structures representing electrons) before getting to that point.

Symmetry is covered in two chapters – first for free fields and then for interacting fields. All that popular talk about rotating crystals etc. will rather not explain what Gauge Symmetry really is. Again I come to the conclusion that using QED (and the Lagrangian associated with Maxwell’s equation) as an example is the right thing to do, but I will need to re-read other accounts that introduce interacti0ns immediately after having explained scalar b0s0ns.

The way Feynman, Schwinger and Tomonaga dealt with infinities via renormalization is introduced after the chapter on interactions. Since this is the first time I learned about renormalization in detail it is difficult to comment on the quality. But I tend to agree with Klauber who states that students typically get lost in these extremely lengthy derivations that include many side-tracks. Klauber tries to keep it somewhat neat by giving an overview first – explaining the strategy of these iterations (answering: What the hell is going on here?) and digging deeper in the next chapters.

Applications are emphasized, so we learn about the daunting way of calculating scattering cross-sections to be compared with experiments. Caveat: Applications refer to particle physics, not to solid-state physics – but this was exactly what I, as a former condensed matter physicist, was looking for.

Klauber uses the canonical quantization that I had tried to introduce in my series on QFT, too (though I tried to avoid the term). Nevertheless, at the end of the book a self-contained introduction to path integrals is given, too, and part of it is available online.

In summary I wholeheartedly recommend this book to any QFT newbie who is struggling with conciser texts. But I am not a professional, haven’t read all QFT books in the world and my requirements as a student are probably peculiar ones.

May the Force Field Be with You: Primer on Quantum Mechanics and Why We Need Quantum Field Theory

As Feynman explains so eloquently – and yet in a refreshingly down-to-earth way – understanding and learning physics works like this: There are no true axioms, you can start from anywhere. Your physics knowledge is like a messy landscape, built from different interconnected islands of insights. You will not memorize them all, but you need to recapture how to get from one island to another – how to connect the dots.

The beauty of theoretical physics is in jumping from dot to dot in different ways – and in pondering on the seemingly different ‘philosophical’ worldviews that different routes may provide.

This is the second post in my series about Quantum Field Theory, and I  try to give a brief overview on the concept of a field in general, and on why we need QFT to complement or replace Quantum Mechanics. I cannot avoid reiterating some that often quoted wave-particle paraphernalia in order to set the stage.

From sharp linguistic analysis we might conclude that is the notion of Field that distinguishes Quantum Field Theory from mere Quantum Theory.

I start with an example everybody uses: a so-called temperature field, which is simply: a temperature – a value, a number – attached to every point in space. An animation of monthly mean surface air temperature could be called the temporal evolution of the temperature field:

Monthly Mean Temperature

Solar energy is absorbed at the earth’s surface. In summer the net energy flow is directed from the air to the ground, in winter the energy stored in the soil is flowing to the surface again. Temperature waves are slowly propagating perpendicular to the surface of the earth.

The gradual evolution of temperature is dictated by the fact that heat flows from the hotter to the colder regions. When you deposit a lump of heat underground – Feynman once used an atomic bomb to illustrate this point – you start with a temperature field consisting of a sharp maximum, a peak, located in a region the size of the bomb. Wait for some minutes and this peak will peter out. Heat will flow outward, the temperature will rise in the outer regions and decrease in the center:

Diffluence of a bucket of heat, goverend by the Heat Transfer EquationModelling the temperature field (as I did – in relation to a specific source of heat placed underground) requires to solve the Heat Transfer Equation which is the mathy equivalent of the previous paragraph. The temperature is calculated step by step numerically: The temperature at a certain point in space determines the flow of heat nearby – the heat transferred changes the temperature – the temperature in the next minute determines the flow – and on and on.

This mundane example should tell us something about a fundamental principle – an idea that explains why fields of a more abstract variety are so important in physics: Locality.

It would not violate the principle of the conservation of energy if a bucket of heat suddenly disappeared in once place and appeared in another, separated from the first one by a light year. Intuitively we know that this is not going to happen: Any disturbance or ripple is transported by impacting something nearby.

All sorts of field equations do reflect locality, and ‘unfortunately’ this is the reason why all fundamental equations in physics require calculus. Those equations describe in a formal way how small changes in time and small variations in space do affect each other. Consider the way a sudden displacement traverses a rope:

Propagation of a waveSound waves travelling through air are governed by local field equations. So are light rays or X-rays – electromagnetic waves – travelling through empty space. The term wave is really a specific instance of the more generic field.

An electromagnetic wave can be generated by shaking an electrical charge. The disturbance is a local variation in the electrical field which gives rises to a changing magnetic field which in turn gives rise a disturbance in the electrical field …


Electromagnetic fields are more interesting than temperature fields: Temperature, after all, is not fundamental – it can be traced back to wiggling of atoms. Sound waves are equivalent to periodic changes of pressure and velocity in a gas.

Quantum Field Theory, however, should finally cover fundamental phenomena. QFT tries to explain tangible matter only in terms of ethereal fields, no less. It does not make sense to ask what these fields actually are.

I have picked light waves deliberately because those are fundamental. Due to historical reasons we are rather familiar with the wavy nature of light – such as the colorful patterns we see on or CDs whose grooves act as a diffraction grating:

Michael Faraday had introduced the concept of fields in electromagnetism, mathematically fleshed out by James C. Maxwell. Depending on the experiment (that is: on the way your prod nature to give an answer to a specifically framed question) light may behave more like a particle, a little bullet, the photon – as stipulated by Einstein.

In Compton Scattering a photon partially transfers energy when colliding with an electron: The change in the photon’s frequency corresponds with its loss in energy. Based on the angle between the trajectories of the electron and the photon energy and momentum transfer can be calculated – using the same reasoning that can be applied to colliding billiard balls.

Compton Effect

We tend to consider electrons fundamental particles. But they give proof of their wave-like properties when beams of accelerated electrons are utilized in analyzing the microstructure of materials. In transmission electron microscopy diffraction patterns are generated that allow for identification of the underlying crystal lattice:

A complete quantum description of an electron or a photon does contain both the wave and particle aspects. Diffraction patterns like this can be interpreted as highlighting the regions where the probabilities to encounter a particle are maximum.

Schrödinger has given the world that famous equation named after him that does allow for calculating those probabilities. It is his equation that let us imagine point-shaped particles as blurred wave packets:

Schrödinger’s equation explains all of chemistry: It allows for calculating the shape of electrons’ orbitals. It explains the size of the hydrogen atom and it explains why electrons can inhabit stable ‘orbits’ at all – in contrast to the older picture of the orbiting point charge that would lose energy all  the time and finally fall into the nucleus.

But this so-called quantum mechanical picture does not explain essential phenomena though:

  • Pauli’s exclusion principle explains why matter is extended in space – particles need to put into different orbitals, different little volumes in space. But It is s a rule you fill in by hand, phenomenologically!
  • Schrödinger’s equations discribes single particles as blurry probability waves, but it still makes sense to call these the equivalents of well-defined single particles. It does not make sense anymore if we take into account special relativity.

Heisenberg’s uncertainty principle – a consequence of Schrödinger’s equation – dictates that we cannot know both position and momentum or both energy and time of a particle. For a very short period of time conservation of energy can be violated which means the energy associated with ‘a particle’ is allowed to fluctuate.

As per the most famous formula in the world energy is equivalent to mass. When the energy of ‘a particle’ fluctuates wildly virtual particles – whose energy is roughly equal to the allowed fluctuations – can pop into existence intermittently.

However, in order to make quantum mechanics needed to me made compatible with special relativity it was not sufficient to tweak Schrödinger’s equation just a bit.

Relativistically correct Quantum Field Theory is rather based on the concept of an underlying field pervading space. Particles are just ripples in this ur-stuff – I owe to Frank Wilczek for that metaphor. A different field is attributed to each variety of fundamental particles.

You need to take a quantum leap… It takes some mathematical rules to move from the classical description of the world to the quantum one, sometimes called quantization. Using a very crude analogy quantization is like making a beam of light dimmer and dimmer until it reveals its granular nature – turning the wavy ray of light into a cascade of photonic bullets.

In QFT you start from a classical field that should represent particles and then apply the machinery quantization to that field (which is called second quantization although you do not quantize twice.). Amazingly, the electron’s spin and Pauli’s principle are a natural consequence if you do it right. Paul Dirac‘s achievement in crafting the first relativistically correct equation for the electron cannot be overstated.

I found these fields the most difficult concepts to digest, but probably for technical reasons:

Historically  – and this includes some of those old text books I am so fond of – candidate versions of alleged quantum mechanical wave equations have been tested to no avail, such as the Klein-Gordon equation. However this equation turned out to make sense later – when re-interpreted as a classical field equation that still needs to be quantized.

It is hard to make sense of those fields intuitively. However, there is one field we are already familiar with: Photons are ripples arising from the electromagnetic field. Maxwell’s equations describing these fields had been compatible with special relativity – they predate the theory of relativity, and the speed of light shows up as a natural constant. No tweaks required!

I will work hard to turn the math of quantization into comprehensive explanations, risking epic failure. For now I hand over to MinutePhysics for an illustration of the correspondence of particles and fields:

Disclaimer – Bonus Track:

In this series I do not attempt to cover latest research on unified field theories, quantum gravity and the like. But since I started crafting this article, writing about locality when that article on an alleged simple way to replace field theoretical calculations went viral. The principle of locality may not hold anymore when things get really interesting – in the regime of tiny local dimensions and high energy.

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…

View original post 72 more words