What Learning about Feynman’s Path Integrals Was Good for

I have gone to great lengths on this blog in order to explain how and why a degree in physics prepares you for seemingly different careers, or at least does not hurt.

But it would have been so simple. I will now illustrate this – using just two incomprehensible images.

Actually, I have a hidden agenda: The top page on this blog is my review of the book Student Friendly Quantum Field Theory. Of course I am trying now to coast on its success.

But I swear that each of these images made us – the Chief Engineer and me – spontaneously and independently burst out:

This is more complicated than Feynman’s path integrals!!

So if QFT does not prepare you for plumbing and IT security – I don’t know what does.

1) Feynman diagrams spotted in pipework – especially in the way the Chief Engineer depicts it! Our assumption is that this sort of documentation is commented by your typical HVAC contractor with What does he have smoked??

Piping diagram

The Chief Engineer’s Piping Diagrams. Captions in German but not important anyway. One of his ingenious inventions is built from many of these units.

2) Feynman diagrams in certificate paths – in the way authors of Request for Comments envisage the proper usage of related internet protocols. This is the kind of paper any developer sometimes SHOULD read to cross-check implementation versus best practices but I MIGHT give because the RFC is rather PhD-thesis-like.

Certificate path validation, RFC 4158

Certificate path validation, RFC 4158. Details again not important. It is about trust, cast into cryptographic relationships. Again these structures are fractal and you would see more and more trees and branches the deeper you dig.

OK – and now a true Feynman diagram. As with the other ‘examples’ a full-blown diagram would be built from lots of these units:

Susy Zerfall chi0

Feynman diagram of a SUSY process. Details – again – not important, and I would not be able to answer questions anyway (Wikimedia, user Patrick87).

As a disclaimer in case any practicing theoretical physicist feels offended: Of course I don’t intend to say that either of these things is as complicated or requires as much training as Quantum Field Theory.

But probably there is a more serious message to be uncovered here:

Feynman diagrams are often described as depicting the reactions between elementary particles. Yet they are shortcut for very unwieldy integrals in an abstract space.

Drawings of pipework and also the certificate paths seem more tangible. But the latter is replacing cryptographic relationship by little sticks and the former is after all an abstract map of physical items using 2D symbols.

In The Shallows – What the Internet Is Doing to Our Brains Nicholas Carr reminds us of the significance of our abilities to create maps at increasing levels of abstraction:

Mankind’s first maps, scratched in the dirt with a stick or carved into a stone with another stone, were as rudimentary as the scribbles of toddlers. … As more time passed, the realism became scientific in both its precision and its abstraction. The mapmaker began to use sophisticated tools like the direction-finding compass and the angle-measuring theodolite and to rely on mathematical reckonings and formulas. Eventually, in a further intellectual leap, maps came to be used not only to represent vast regions of the earth or heavens in minute detail, but to express ideas—a plan of battle, an analysis of the spread of an epidemic, a forecast of population growth.

Building and interpreting – abstract – maps of all kinds has become a key ‘skill’ – from software enterprise architecture diagrams to all sorts of infographics shared on social media like How Douglas Adams predicted the iPad.

“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.

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!”

Hyper-Jelly – Again. Why We Need Hyperspace – Even in Politics.

All of old.
Nothing else ever.
Ever tried. Ever failed.
No matter.
Try again.
Fail again.
Fail better.

This is a quote from Worstward Ho by Samuel Beckett – a poem as impenetrable and opaque as my post on quantization. There is a version of Beckett’s poem with explanations, so I try again, too!

I stated that the description of a bunch of particles (think: gas in a box) naturally invokes the introduction of a hyperspace having twice as many dimensions as the number of those particles.

But it was obviously not obvious why we need those many dimensions. You have asked:

Why do we need additional dimensions for each particles? Can’t they live in the same space?

Why does the collection of the states of all possible systems occupy a patch in 1026 dimensional phase space. ?

These are nearly the same questions.

I start from a non-physics example this time, because I believe this might convey the motivation for introducing these dimensions better.

These dimensions are not at all related to hidden, compactified, extra large dimensions you might have read about in popular physics books on string theory and cosmology. They are not tangible dimensions in the sense we could feel them – even if we weren’t like those infamous ants living on the inflating balloon.

In Austria we recently had parliamentary elections. This is the distribution of seats in parliament now:

Seat in Austrian parliament | Nick.mon Wikimedia

which is equivalent to these numbers:

SPÖ (52)
ÖVP (47)
FPÖ (40)
Grüne (24)
Team Stronach (11)
NEOS (9)

Using grand physics-inspired language I call that ordered collection of numbers: Austria’s Political State Vector.

These are six numbers thus this is a vector in a 6-dimensional Political State Space.

Before the elections websites consolidating and analyzing different polls have been more popular than ever. (There was a website run by physicist now working in finance.)

I can only display two of 6 dimensions in a plane, so the two axes represent any of those 6 dimensions. The final political state is represented by a single point in this space – the tip of an 6D arrow:

Point in phase space (Image (c) Elkement)

After the elections we know the political state vector with certainty – that is: a probability of 1.

Before the elections different polls constituted different possible state vectors – each associated with a probability lower than 1. I indicate probabilities by different hues of red:

Distribution of points in phase space (Image (c) Elkement)Each points represents a different state the system may finally settle in. Since the polls are hopefully meaningful and voters not too irrational points are not scattered randomly in space but rather close to each other.

Now imagine millions of polls – such as citizens’ political opinions tracked every millisecond by directly wiretapping their brains. This would result in millions of points, all close to each other. Not looking too closely, this is a blurred patch or spot – a fairly confined region of space covered with points that seems to merge into a continuous distribution.

Smooth distribution of points in phase space (Image (c) Elkement)

Watching the development of this red patch over time lets us speculate on the law underlying its dynamics – deriving a trend from the dynamics of voters’ opinions.

It is like figuring out the dynamics of a  moving and transforming piece of jelly:

Back to Physics

Statistical mechanics is similar, just the numbers of dimensions are much bigger.

In order to describe what each molecule of gas in a room does, we need 6 numbers per molecules – 3 for its spatial coordinates, and 3 for its velocity.

Each particle lives in the same real space where particles wiggle and bump into each other. All those additional dimensions only emerge because we want to find a mathematical representation where each potential system state shows up as a single dot – tagged with a certain probability. As in politics!

We stuff all positions and velocities of particles into an enormous state vector – one ordered collection with about 1026 different numbers corresponds to a single dot in hyperspace.

The overall goal in statistical mechanics is to calculate something we are really interested in – such as temperature of a gas. We aim at calculating probabilities for different states!

We don’t want to look to closely: We might want to compare what happens if if we start from a configuration with all molecules concentrated in a corner of the room with another one consisting of molecules everywhere in the room. But we don’t need to know where each molecule is exactly. Joseph Nebus has given an interesting example related his numerical calculation of the behavior of a car’s shock absorbers:

But what’s interesting isn’t the exact solution of the exact problem for a particular set of starting conditions. When your car goes over a bump, you’re interested in what the behavior is: is there a sudden bounce and a slide back to normal?  Does the car wobble for a short while?  Does it wobble for a long while?  What’s the behavior?

You have asked me for giving you the equations. I will try my best and keep the intro paragraphs of this post in mind.

What do we know and what do we want to calculate?

We are interested in how that patch moves and is transformed – that is probability (the hue of red) as a function of the positions and momenta of all particles. This is a function of 1026 variables, usually called a distribution function or a density.

We know anything about the system, that is the forces at play. Knowing forces is equivalent to knowing the total energy of a system as a function of any system configuration – if you know the gravitational force a planet exerts than you know gravitational energy.

You could consider the total energy of a system that infamous formula in science fiction movies that spies copy from the computer in the secret laboratories to their USB sticks: If you know how to calculate the total energy as a function of the positions and momenta of all particles – you literally rule the world for the system under consideration.

Hyper-Planes

If we know this energy ‘world function’ we could attach a number to each point in hyperspace that indicate energy, or we could draw the hyper-planes of constant energies – equivalent of isoclines in a map.

The dimension of the hyperplane is the dimension of the hyperspace minus one, just as the familiar 2D planes floating through 3D space.

If energy changes more rapidly with varying particle positions and momenta hyper-planes get closer to each other:

Hyper-planes of constant energy in phase space (Image (c) Elkement)Incompressible Jelly

We are still in a classical world. The equations of motions of hyper-jelly are another way to restate Newton’s equations of motion. You start with writing down Force = mass x accelerating for each particle (1026 times), rearrange these equations by using those huge state vectors just introduced – and you end up with an equation describing the time evolution of the red patch.

I picked the jelly metaphor deliberately as it turns out that hyper-jelly acts as an incompressible fluid. Jelly cannot be destroyed or created. If you try to squeeze it in between two planes it will just flow faster. This really follows from Newton’s law or the conservation of energy!

Hyper-planes of constant energy and flowing distribution in phase space (Image (c) Elkement)

It might appear complicated to turn something as (seemingly) comprehensible as Newton’s law into that formalism. But that weird way of watching the time evolution of the red patch makes it actually easier to calculate what really matters.

Anything that changes in the real world – the time evolution of any quantity we can measure – is expressed via the time evolution of hyper-jelly.

The Liouville equation puts this into math.

As Richard Feynman once noted wisely (Physics Lectures, Vol.2, Ch. 25), I could put all fundamental equations into a big matrix of equations which I then call the Unwordliness, further denoted as U. Then I can unify them again as

U = 0

What I do here is not that obscure but I use some pseudo-code to obscure the most intimidating math. I do now appreciate science writers who state We use a mathematical crank that turns X into Y – despite or because they know exactly what they are talking about.

For every point in hyperspace the Liouville equation states:

(Rate of change of some interesting physical property in time) =
(Some mathematical machinery entangling spatial variations in system’s energy and spatial variations in ‘some property’)

Spatial variations in the system’s energy can be translated to the distance of those isoclines – this is exactly what Newton’s translates into! (In addition we apply the chain rule in vector calculus).

The mathematical crank is indicated using most innocent brackets, so the right-hand side reads:

{Energy function, interesting property function}

Quantization finally seems to be deceptively simple – the quantum equivalent looks very similar, with the right-hand side proportional to

[Energy function, interesting property function]

The main difference is in the brackets – square versus curly: We consider phase space so any function and changes thereof is calculated in phase space co-ordinates – positions and momenta of particles. These cannot be measured or calculated in quantum mechanics with certainty at the same time.

In a related way the exact order of operations does matter in quantum physics – whereas the classical counterparts are commutative operations. The square bracket versus the angle bracket is where these non-commutative operations are added – as additional constraints to classical theory.

I think I have reached my – current – personal limits in explaining this, while still not turning this blog into in a vector calculus lecture. Probably this stuff is usually not popularized for a reason.

My next post will focus on quantum fields again – and I try to make each post as self-consistent anyway.

From ElKement: On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace

I am still working on a more self-explanatory update to my previous physics post … trying to explain that multi-dimensional hyperspace is really a space of all potential states a single system might exhibit – a space of possibilities and not those infamous multi-dimensional world that might really be ‘out there’ according to string theorists. In the meantime, please enjoy mathematician Joseph Nebus’ additions to my post which include a down-to-earth example.

nebusresearch

I’m frightfully late on following up on this, but ElKement has another entry in the series regarding quantum field theory, this one engagingly titled “On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace”. The objective is to introduce the concept of phase space, a way of looking at physics problems that marks maybe the biggest thing one really needs to understand if one wants to be not just a physics major (or, for many parts of the field, a mathematics major) and a grad student.

As an undergraduate, it’s easy to get all sorts of problems in which, to pick an example, one models a damped harmonic oscillator. A good example of this is how one models the way a car bounces up and down after it goes over a bump, when the shock absorbers are working. You as a student are given some physical properties —…

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On the Relation of Jurassic Park and Alien Jelly Flowing through Hyperspace

Yes, this is a serious physics post – no. 3 in my series on Quantum Field Theory.

I promised to explain what Quantization is. I will also argue – again – that classical mechanics is unjustly associated with pictures like this:

Steampunk wall clock (Wikimedia)

… although it is more like this:

Timelines in Back to the Future | By TheHYPO [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

This shows the timelines in Back to the Future – in case you haven’t recognized it immediately.

What I am trying to say here is – again – is so-called classical theory is as geeky, as weird, and as fascinating as quantum physics.

Experts: In case I get carried away by my metaphors – please see the bottom of this post for technical jargon and what I actually try to do here.

Get a New Perspective: Phase Space

I am using my favorite simple example: A point-shaped mass connected to an massless spring or a pendulum, oscillating forever – not subject to friction.

The speed of the mass is zero when the motion changes from ‘upward’ to ‘downward’. It is maximum when the pendulum reaches the point of minimum height. Anything oscillates: Kinetic energy is transferred to potential energy and back. Position, velocity and acceleration all follow wavy sine or cosine functions.

For purely aesthetic reasons I could also plot the velocity versus position:

Simple Harmonic Motion Orbit | By Mazemaster (Own work) [Public domain], via Wikimedia Commons

From a mathematical perspective this is similar to creating those beautiful Lissajous curves:  Connecting a signal representing position to the x input of an oscillosope and the velocity signal to the y input results in a circle or an ellipse:

Lissajous curves | User Fiducial, Wikimedia

This picture of the spring’s or pendulum’s motion is called a phase portrait in phase space. Actually we use momentum, that is: velocity times mass, but this is a technicality.

The phase portrait is a way of depicting what a physical system does or can do – in a picture that allows for quick assessment.

Non-Dull Phase Portraits

Real-life oscillating systems do not follow simple cycles. The so-called Van der Pol oscillator is a model system subject to damping. It is also non-linear because the force of friction depends on the position squared and the velocity. Non-linearity is not uncommon; also the friction an airplane or car ‘feels’ in the air is proportional to the velocity squared.

The stronger this non-linear interaction is (the parameter mu in the figure below) the more will the phase portrait deviate from the circular shape:

Van der pols equation phase portrait | By Krishnavedala (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

Searching for this image I have learned from Wikipedia that the Van der Pol oscillator is used as a model in biology – here the physical quantity considered is not a position but the action potential of a neuron (the electrical voltage across the cell’s membrane).

Thus plotting the rate of change of in a quantity we can measure plotted versus the quantity itself makes sense for diverse kinds of systems. This is not limited to natural sciences – you could also determine the phase portrait of an economic system!

Addicts of popular culture memes might have guessed already which phase portrait needs to be depicted in this post:

Reconnecting to Popular Science

Chaos Theory has become popular via the elaborations of Dr. Ian Malcolm (Jeff Goldblum) in the movie Jurassic Park. Chaotic systems exhibit phase portraits that are called Strange Attractors. An attractor is the set of points in phase space a system ‘gravitates’ to if you leave it to itself.

There is no attractor for the simple spring: This system will trace our a specific circle in phase space forever – the larger the bigger the initial push on the spring is.

The most popular strange attractor is probably the Lorentz Attractor. It  was initially associated with physical properties characteristic of temperature and the flow of air in the earth’s atmosphere, but it can be re-interpreted as a system modeling chaotic phenomena in lasers.

It might be apocryphal but I have been told that it is not the infamous flap of the butterfly’s wing that gave the related effect its name, but rather the shape of the three-dimensional attractor:

Lorenz system r28 s10 b2-6666 | By Computed in Fractint by Wikimol [Public domain], via Wikimedia Commons

We had Jurassic Park – here comes the jelly!

A single point-particle on a spring can move only along a line – it has a single degree of freedom. You need just a two-dimensional plane to plot its velocity over position.

Allowing for motion in three-dimensional space means we need to add additional dimensions: The motion is fully characterized by the (x,y,z) positions in 3D space plus the 3 components of velocity. Actually, this three-dimensional vector is called velocity – its size is called speed.

Thus we need already 6 dimensions in phase space to describe the motion of an idealized point-shaped particle. Now throw in an additional point-particle: We need 12 numbers to track both particles – hence 12 dimensions in phase space.

Why can’t the two particles simply use the same space?(*) Both particles still live in the same 3D space, they could also inhabit the same 6D phase space. The 12D representation has an advantage though: The whole system is represented by a single dot which make our lives easier if we contemplate different systems at once.

Now consider a system consisting of zillions of individual particles. Consider 1 cubic meter of air containing about 1025 molecules. Viewing these particles in a Newtonian, classical way means to track their individual positions and velocities. In a pre-quantum mechanical deterministic assessment of the world you know the past and the future by calculating these particles’ trajectories from their positions and velocities at a certain point of time.

Of course this is not doable and leads to practical non-determinism due to calculation errors piling up and amplifying. This is a 1025 body problem, much much much more difficult than the three-body problem.

Fortunately we don’t really need all those numbers in detail – useful properties of a gas such as the temperature constitute gross statistical averages of the individual particles’ properties. Thus we want to get a feeling how the phase portrait develops ‘on average’, not looking too meticulously at every dot.

The full-blown phase space of the system of all molecules in a cubic meter of air has about 1026 dimensions – 6 for each of the 1025 particles (Physicists don’t care about a factor of 6 versus a factor of 10). Each state of the system is sort of a snapshot what the system really does at a point of time. It is a vector in 1026 dimensional space – a looooong ordered collection of numbers, but nonetheless conceptually not different from the familiar 3D ‘arrow-vector’.

Since we are interesting in averages and probabilities we don’t watch a single point in phase space. We don’t follow a particular system.

We rather imagine an enormous number of different systems under different conditions.

Considering the gas in the cubic vessel this means: We imagine molecule 1 being at the center and very fast whereas molecule 10 is slow and in the upper right corner, and molecule 666 is in the lower left corner and has medium. Now extend this description to 1025 particles.

But we know something about all of these configurations: There is a maximum x, y and z particles can have – the phase portrait is limited by these maximum dimensions as the circle representing the spring was. The particles have all kinds of speeds in all kinds of directions, but there is a most probably speed related to temperature.

The collection of the states of all possible systems occupy a patch in 1026 dimensional phase space.

This patch gradually peters out at the edges in velocities’ directions.

Now let’s allow the vessel for growing: The patch will become bigger in spatial dimensions as particles can have any position in the larger cube. Since the temperature will decrease due to the expansion the mean velocity will decrease – assuming the cube is insulated.

The time evolution of the system (of these systems, each representing a possible system) is represented by a distribution of this hyper-dimensional patch transforming and morphing. Since we consider so many different states – otherwise probabilities don’t make sense – we don’t see the granular nature due to individual points – it’s like a piece of jelly moving and transforming:

Precisely defined initial configurations of systems configurations have a tendency to get mangled and smeared out. Note again that each point in the jelly is not equivalent to a molecule of gas but it is a point in an abstract configuration space with a huge number of dimensions. We can only make it accessible via projections into our 3D world or a 2D plane.

The analogy to jelly or honey or any fluid is more apt than it may seem

The temporal evolution in this hyperspace is indeed governed by equations that are amazingly similar to those governing an incompressible liquid – such as water. There is continuity and locality: Hyper-Jelly can’t get lost and be created. Any increase in hyper-jelly in a tiny volume of phase space can only be attributed to jelly flowing in to this volume from adjacent little volumes.

In summary: Classical mechanical systems comprising many degrees of freedom – that is: many components that have freedom to move in a different way than other parts of the system – can be best viewed in the multi-dimensional space whose dimensions are (something like) positions and (something like) the related momenta.

Can it get more geeky than that in quantum theory?

Finally: Quantization

I said in the previous post that quantization of fields or waves is like turning down intensity in order to bring out the particle-like rippled nature of that wave. In the same way you could say that you add blurry waviness to idealized point-shaped particles.

Another is to consider the loss in information via Heisenberg’s Uncertainly Principle: You cannot know both the position and the momentum of a particle or a classical wave exactly at the same time. By the way, this is why we picked momenta  and not velocities to generate phase space.

You calculate positions and momenta of small little volumes that constitute that flowing and crawling patches of jelly at a point of time from positions and momenta the point of time before. That’s the essence of Newtonian mechanics (and conservation of matter) applied to fluids.

Doing numerical calculation in hydrodynamics you think of jelly as divided into small little flexible cubes – you divide it mentally using a grid, and you apply a mathematical operation that creates the new state of this digitized jelly from the old one.

Since we are still discussing a classical world we do know positions and momenta with certainty. This translates to stating (in math) that it does not matter if you do calculations involving positions first or for momenta.

There are different ways of carrying out steps in these calculations because you could do them one way of the other – they are commutative.

Calculating something in this respect is similar to asking nature for a property or measuring that quantity.

Thus when we apply a quantum viewpoint and quantize a classical system calculating momentum first and position second or doing it the other way around will yield different results.

The quantum way of handling the system of those  1025 particles looks the same as the classical equations at first glance. The difference is in the rules for carrying out calculation involving positions and momenta – so-called conjugate variables.

Thus quantization means you take the classical equations of motion and give the mathematical symbols a new meaning and impose new, restricting rules.

I probably could just have stated that without going off those tangent.

However, any system of interest in the real world is not composed of isolated particles. We live in a world of those enormous phase spaces.

In addition, working with large abstract spaces like this is at the heart of quantum field theory: We start with something spread out in space – a field with infinite degrees in freedom. Considering different state vectors in these quantum systems is considering all possible configurations of this field at every point in space!

(*) This was a question asked on G+. I edited the post to incorporate the answer.

_______________________________________

Expert information:

I have taken a detour through statistical mechanics: Introducing Liouville equations as equation of continuity in a multi-dimensional phase space. The operations mentioned – related to positions of velocities – are the replacement of time derivatives via Hamiltonians equations. I resisted the temptation to mention the hyper-planes of constant energy. Replacing the Poisson bracket in classical mechanics with the commutator in quantum mechanics turns the Liouville equation into its quantum counterpart, also called Von Neumann equation.

I know that a discussion about the true nature of temperature is opening a can of worms. We should rather describe temperature as the width of a distribution rather than the average, as a beam of molecules all travelling in the same direction at the same speed have a temperature of zero Kelvin – not an option due to zero point energy.

The Lorenz equations have been applied to the electrical fields in lasers by Haken – here is a related paper. I did not go into the difference of the phase portrait of a system showing its time evolution and the attractor which is the system’s final state. I also didn’t stress that was is a three dimensional image of the Lorenz attractor and in this case the ‘velocities’ are not depicted. You could say it is the 3D projection of the 6D phase portrait. I basically wanted to demonstrate – using catchy images, admittedly – that representations in phase space allows for a quick assessment of a system.

I also tried to introduce the notion of a state vector in classical terms, not jumping to bras and kets in the quantum world as if a state vector does not have a classical counterpart.

I have picked an example of a system undergoing a change in temperature (non-stationary – not the example you would start with in statistical thermodynamics) and swept all considerations on ergodicity and related meaningful time evolutions of systems in phase space under the rug.

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 …

Electromagneticwave3D

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.