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:

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:

Modelling 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!

Sound 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.

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 natural consequences if you do that 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.

22 Comments Add yours

  1. Peter Mander says:

    It may be stretching a point to claim that Schrödinger’s equation explains all of chemistry, but in the nuclear realm it certainly helped provide an explanation for otherwise intractable problems like alpha decay. Maybe quantum tunneling and Gamow’s theory could make a good subject for a post?

    1. elkement says:

      By ‘chemistry’ I meant mainly all phenomena that can be understood in terms of ‘single’ non-relavistic atoms.

      Superconductivity and solid-state physics has been on the back of my mind, rather than nuclear physics, when I wrote this: You can describe it phenomonologically using a simple ‘giant wave-function’ representing the ‘condensate’ (in Ginzburg-Landau theory), but you would still use Schrödinger’s ‘single-particle’ equation. But if you want to understand Cooper pairs and where the interaction really comes form, you need quantum field theory – allowing for representing interactions as particles, in this case the phonons of the lattice’s distortion mediating the interactions between electrons.

  2. M. Hatzel says:

    I am going to read the post again before I say something that requires correction. You might like this post (I have to admit, your blog enhances stuff like this for me):

    1. elkement says:

      Please say something even if it might ‘require correction’ – you can always blame it on the writer of this post ;-) who did not explain it in a comprehensible way.

      This linked post is great and it actually gives me ideas – for adding some personal encounters with ‘electromagnetic waves’ in upcoming posts. In a former life I was a laser physicist – but that experience made me more inclined to Star Wars clichés perhaps. I worked with pulsed lasers (‘dashes’ – not uncommon) and it only takes a bit of typical dust in a shaded room to ‘see the beams’ of continuous lasers. It was this view of a lab only lit by crossing laser beams in different colors that made me pick my specialization :-)

      1. M. Hatzel says:

        I’ve read it again, and I sense that this is going to take some patience on my part. The warning at the beginning, that there is no real place to begin and all we can do is connect the dots, reminds me that the big picture of all learning takes time. I feel that I have a sense of the components, that this is about the movement of fields and particles in space and time. We started with something large and common enough for all of us to know, and you led us through a way of seeing it in smaller and smaller pieces, so we could understand why. I’ll need to read it again, maybe after your next post installment.

        I am hugely impressed by all the images and links. Your efforts will not go wasted. I will keep working at this!

        1. elkement says:

          Thanks a lot for your efforts, Michelle!
          I will reward loyal readers like you with a relaxing search term poem in a “break” before the next quantum-related posting. Most likely I underestimate the overwhelming effect my attempts to lay out this vast network of interconnected ideas.

          Wikimedia is really a great repository of images. Every time I have an image in mind that should illustrate something – I find the nearly perfect equivalent in Wikimedia. I think I should create some drawings myself and release them the public domain as well in order to compensate for that. I am happy that somebody asked me yesterday for permission to use one of my (rare) original figures.

          1. M. Hatzel says:

            I will keep on with my efforts, however, I’ve started a new job and honestly, there is little brain power left over at the moment. I’ll read your search terms poem in a minute… I noticed the post when I was at the store this evening to buy a new computer for said new job. I wanted to see how the graphics compared on my choices and watched the thermal fields changing across the screens. :)

            1. elkement says:

              I can relate to this – starting something new always leaves you exhausted. I had always felt that new jobs / new projects / new places are mentally / socially (?) exhausting. Working very hard on an intellectually demanding project – but in an environment / with people I know very well – seems to be much easier.

  3. Feedback: I found this post easier to understand than the previous one. Thanks for edumacatin’ me.

    1. elkement says:

      Thanks a lot for the feedback, Steve – much appreciated!

  4. Another nice, concise introduction! One f the things that always strikes me is the sheer brilliance of those who were able to synthesize the available evidence into a framework that accounts for all (at least most) of the known observations. My favourite starring point for an introduction to physics is Newton’s Laws because it is now my experience that high school students have already encountered them in the popular media before coming to my class. I ask them where they heard them and it’s generally from some youtube video or kid’s TV show. When I check the source, somewhere in the introduction the Narrator exclaims “This is Easy!” and that’s the part I always like to harp on in class. Newton’s laws are fairly easy (Maybe not the third which is nearly always pretty much a mystery for young students. After all, walls do not push back in the ordinary sense!!!) but only ONCE YOU UNDERSTAND THEM WELL.
    I liken it to swimming. Swimming is easy once you know how to swim. Through practice it becomes automatic. As for Newton’s Laws, students have to acknowledge these items before they can really understand them:
    First Law: Things tend to remain in motion. Bullshit! They come to a rest. Just slide your book across the table. See it came to a rest. Even curling stones on ice come to a rest. The brilliance of Newton was seeing this thing called friction and recognizing it as just another force.
    Second Law: We DO NOT introduce it properly in school because the students simply do not have the math background. Instead we try to rationalize the direct proportion between Force and acceleration and the inverse proportion between mass and acceleration. Kid have GREAT diffivulty with direct proportions. Ask, “If one cat can eat a saucer of milk in one minute, how long will it take a thousand cats to eat a thousand saucers?” and notice that you get the wrong answer each time. As for inverse proportions… well. Hmmmm. Not great. I generally ask them to consider pizza and people. We order a 24 slice pizza. How many pieces each if there is one person? two? three? four? six? eight? twelve? twenty four? Now graph the relationship between # slices each and # people. Look–it’s a curve called a hyperbola. Oh, and then the students really need to gi straight to the lab and do two sets of measurements. In one they vary the applied force on a cart and emasure the resulting a and in the other they hold F constant and vary m while measuring a. In both cases then then plot the independent variable vs. a and then draw conclusions. As a follow up then need to plot 1/m vs a as well.
    Third Law: My rule–NEVER NEVER use the phrase “to every action there is an equal and opposite reaction” I hate it and it totally messes up students because OBJECTS DO NOT REACT!!! I usually standardize arund F(sub)ab = -F(sub)ba.
    Whoops, ranting again.
    At any rate, Elke you kept it at the respectful level it should be. Easy to grasp but not too easy

    1. elkement says:

      Thanks, Maurice – I really appreciate your detailed comments as really you know from your experience how it feels to communicate ideas in physics! My respect for professional science writers is growing – you live in danger of making things too complicated for true lay audiences but too simplified for experts ot tech-savvy readers of pop-sci books.
      Newton’s laws are great examples indeed. Yes, ‘action and reaction’ is a very opaque concept. I think I tend to present the action-reaction as conservation of momentum. I like the typical experiment though – those little vehicles connected by springs, different masses, shooting off in opposite directions. You could demonstrate the conservation of momentum by considering the center of gravity (at rest).

  5. Ok … I think I got most of that and, like NicoLite (above), I enjoyed the MinutePhysics quite well. You’ve done a fine job of explaining this without numbers … whew! I’m always impressed with the capacity of folks like you to think ‘this way.’ My brain is hardwired in a TOTALLY different way. I’m glad we’re all different though … makes the world so much more interesting … don’t you think? I’m ready for more. D

    1. elkement says:

      *Relief* Thanks, Dave – I appreciate your feedback. Some physics colleagues liked it too, thus I seem to have managed to find some ‘pop-sci’ explanations that do not take the metaphors too far.

      As for thinking this way … I would be interested in psychologists’ opinions and research on when and how ‘quantitative’ or ‘abstract’ (or what you want to call that) thinking is established. Is there a time slot in early childhood that must not be missed? I have no idea – it came naturally to me as far as I can remember.

      However, I was not at all your typical physics / math nerd at school – I was interested in anything and very curious ever since. I can remember growing a plant from seeds as a child for the first time was as exciting as doing optics experiments with lenses from dismantled glasses, growing crystals from toxic solutions, or crafting paper polyhedrons. I even considered majoring in literature and philosophy very seriously.

    1. elkement says:

      Yes – absolutely! I find his metaphors stunning as they are so apt (and I am extremely critical about physics metaphors). I wonder how much time it takes to create such as three minutes video.

  6. David Yerle says:

    Can you believe I had not read about the Amplituhedron yet? This is amazing! I had read that there had been a lot of progress simplifying calculations for Feynman diagrams, but this is a completely different level.
    I found the explanation of fields to be pretty accurate! Does this mean you’re going to blog more about physics?

    1. elkement says:

      Thanks, David! Yes – I am going to blog more about physics, my personal QFT learning experiences in particular. I have once been made familiar to what was called ‘Second Quantization’ at that time – but only applied to many-particle systems (BCS theory of superconductivity e.g., and phonons –> I would found the cult of the phonon BTW).
      I want to understand in detail how the Higgs gives the particles mass now. Theoretically I should as I have seen popular accounts that explain the Higgs mechanics in terms of phase transitions and superconductivity – but I have not made this mental connection yet.
      Above all, it is a challenge in writing about this in a pop-sci way. I have really no clue yet how to describe quantization, operators and stuff :-)

      1. David Yerle says:

        And neither do I! Good luck with it…

      2. Joseph Nebus says:

        I know I’ve tried to describe operators to people who know a little bit of calculus, but not enough, and they’ve gone away … at least, claiming to understand what I was talking about. I’ve been hesitant to try probing whether they actually get it because I’m scared of what the answer might be.

        1. elkement says:

          Thanks, Joseph – this sounds familiar. I think this is the same effect as readers of pop-sci books on string theory claiming they “understood” it.
          I have an idea in mind how to present quantization which is probably very unusual – thus maybe even more disastrous. I think I will not be able to resist that temptation though.
          But before that I will allow for a break and analyze the search terms my visitors submitted in the past quarter :-)

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