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.