Quantum Field Theory or: It’s More Than a Marble Turned into a Wiggly Line

I had been trained as an experimental physicist which meant I was good at locating vacuum leaks, adjusting lasers and lenses, telling reasonable data from artefacts, and being the only person that ever replenished the paper feed of the X-ray diffractometer (Yes, at that time we used paper records).

Exactly because of that I took pride in the fact that I attended some non-mandatory lectures in quantum theory – in particular in order to understand the quantum mechanical underpinnings of condensed matter physics and superconductivity.

Ironically, it was most likely my focus on superconductors that made me miss an important point.

Superconductivity is one of those effects that can be described as quantum physics emerging at a macroscopic scale – there is a ‘giant wave function’ comprising many particles, similar to infamous Bose-Einstein condensation. I am indulging on sloppy terminology here. The giant wave function has been proposed by Ginzburg and Landau as a phenomenological explanation of superconductivity, and Bardeen, Cooper and Schrieffer finally formulated a full ‘microscopic’ theory. If there is a probability for one particle to be in a certain place, overlapping many of those probability waves results in a simple distribution – representing how many particles are in a particular place.

What’s the irony? In condensed matter physics you are investigating the interactions of many (many, many) particles. That’s why I was under the impression that advanced theories in quantum physics are basically the theories applied to single particles plus some way of doing statistics on them. I was not familiar with the term Quantum Field Theory, but back then or in my personal condensed matter corner of the world this was called Quantum Statistics or Second Quantization.

Until very recently, when the discussions on the Higgs boson etc. rekindled my interest in fundamentals of physics, I was indeed clueless and unable to connect ‘Quantum Statistics’ with the Standard Model in particle physics and basically anything related to single particles.

So what is the connection between understanding the behavior or many particles in a piece of metal and the high-energy experiments with colliding protons?

In popular science books the transition from the classical world to the quantum world is often depicted as the replacement of solid marbles with little wiggly lines. A ‘particle’ becomes a ‘wave’. This is associated with all kinds of philosophical discussions. I tend to state that there would not be any discussion at all if we would not use these pictures. A ‘particle’ is neither a marble or a wiggly line, it is the concept of a single particle as such that ceases to exist in advanced quantum (field) theory.

It is true that simple quantum systems can be described as particles turned waves: such as the hydrogen atom that can be described nicely using a single-particle Schrödinger wave function. A particle in a box can be represented by a quantum mechanical wave function that represents the probability to find the particle at a certain position:

In order to see why and when the particle as such ceases to exist, insights from quantum mechanics (QM) need to be combined with special relativity (SR) , the famous E=mc2 in particular.

  • Large momentum – as per QM: Based on Heisenberg’s uncertainty principle, we cannot measure position and momentum of a particle precisely. So if we try to confine the particle – e.g. lock it up in a box – the uncertainty in momentum will increase. Chances increase that the particle will exhibit large momenta.
  • Large energy  – as per SR: Large (uncertainties in) momenta mean large (uncertainties in) energies. For a massless particle (as the photon, the quantum of light) energy and mass are proportional, and these properties are connected by a simple relationship in special relativity. If velocities are low compared to the speed of light you might simply think of momentum as the product of (rest) mass and velocity, whereas kinetic energy is the product of mass and velocity squared over 2.
  • Particle creation –  as per SR: We know from the most famous formula in the world that energy is equivalent to mass. So if the uncertainty in energy increases chances increase that new particles are created.  More precisely, particles are created in pairs in order not to violate other conservation laws (e.g. electrical charge).

The uncertainty principle however also related energy to time: The shorter the time scale, the higher the energy that can be used in the creation of – virtual – particles. The full-blown theory of quantum (field) theory deals with all those cases – virtual and real particles, slow or near the speed of light, lonely ones or packs.

So if we take a closer look at a particular particle, we can identify two interesting length scales:

  • The wavelength of wave associated with a single particle as long as this interpretation makes sense. This is called the De Broglie wavelength.
  • The length scale where the particle as a single entity ceases to exist. This is called the Compton wave length.

But where do this particles ‘come from’? The answer is of course – they come from an ‘underlying field’, though this may not sound as a satisfactory explanation.

It might be unusual to think of particles are something transient or something subordinate to a mysterious field.

However, we had become used to the fact that electromagnetic radiation can be viewed as or turned into particles called photons. But if we can imagine a field becoming particles why not consider particles being sort of a manifestation of a ‘field’? David Tong calls this field a ‘sea of stuff’ in his first lecture on Quantum Field Theory (BTW I would highly recommend his lectures – notes and videos – as an introduction to QFT). So protons are just ripples in this sea of proton stuff just as photons are just ripples of the electromagnetic field.

I believe that we often take some ideas in science for granted or consider them plausible or familiar if we had been exposed to them often enough. But after all: Do we really understand or feel the electromagnetic field any better as the sea of stuff that pops out electrons, protons (or the Higgs boson maybe)? Just because we are surrounded by EM wave smog transmitting Facebook posts and TV shows?

Intuition in quantum physics – if there is any – can in my point of view only be acquired by wading through the math. There is no shortcut. Stating a particle originates from a field or vice versa is just a vague replacement of something that only equations can capture precisely.

Or might there be acceptable shortcuts?

I really do enjoy MinutePhysics – watch the explanation explanation of what matter is. In passing, the electron field is introduced as an ‘electron-ness’ – similar to ‘three-ness’ invoked when we use the number of three.


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