Why do we have to combine Einstein's Relativity with Quantum Mechanics?
The result is our most successful theory yet - quantum field theory
As an undergraduate, I had a sort of unshakeable trust in both physics and the scientists who had constructed it. My lectures in mechanics, quantum physics and fluid dynamics were structured. There was always some witty yet predictable theme and story of how the courses presented the mathematical ideas. The presentation of ideas was always so confident that it came across as prophetic genius from the more remarkable men and women before me. Everything was clean. Everyone knew the answers - and I was just merely a student that had to go along for the ride.
Because of this reason, whenever I was confused, I would blame myself for not being smart or savvy enough to understand what was going on.
However, as soon as I started my master's in theoretical physics, the story changed. My confidence in the clean, elegant mechanics and quantum physics theories began to deteriorate once I started learning about some of its serious issues. I will cover some of these physical and mathematical inconsistencies in the following sections. At this time, I was also introduced to the subject of quantum field theory - a field of physics and mathematics designed to tackle some of the issues I will mention below.
Quantum field theory is a mathematical tool that says that the fundamental building blocks of nature are fields [1]. A field is just a function that allocates a number to a particular point in space and time. So, for example, the temperature is a field - at every point in space, I can give you a number. This number is measured either in Fahrenheit or Celsius. After we first construct fields, we turn them into particles with quantisation. Quantisation is the mathematical procedure of interpreting the vibrations of fields as particles in their own right and then making predictions about what happens when particles interact.
It is important to stress that quantum field theory is not a 'theory' itself per se. It is merely a framework, a scaffolding, if you will, in which physicists can input a particular Lagrangian and then do the math to see what comes out.
Quantum field theory did little to ease my pain and confusion, however. As a subject itself, I found it messy, imperfect, unstructured, and most of all, extremely difficult to grasp. Yet, despite this difficulty, it is probably one of the most successful theories ever created in the history of modern science in terms of experimental evidence. In this post, I will talk about the failures of quantum mechanics and how quantum field theory tries to resolve them. I will then explain some basic field theory to get a taste of the mathematics required to tackle a subject like this.
The problem of instantaneous influences
In this section, I will talk about a philosophical oddity in the formulation of traditional mechanics. By 'traditional mechanics', I mean physics derived from Newton. The problem is this. In the classical formulation of physics, object A can affect object B instantaneously, even if they are far apart. The way that forces are modelled in the math implies an instantaneous reaction. This phenomenon is what we call an 'instantaneous influence'. Think of it as an invisible line that transmits forces between objects at instantaneous speed.
For simplicity, let's take the example of Newtonian gravity - a theory used to predict the forces between planets. In high school, you may have learned that the gravitational force applied between two objects with masses m1 and m2, looks like
\[ \mathbf{F_ 1 } = - G \frac{ m_ 1 m_ 2 } { | \mathbf{ r} _1 - \mathbf { r } _ 2 | ^ 2} \]
The bottom part of the fraction is just the distance between the two objects at any given moment in time. Because Newtonian gravity is formulated in this way, we find that if the object with mass m2 were to move a bit, the force on the object with mass m1 would change instantly. The case where we calculate the force between two electrons, through the 'Coulomb model' of repulsion, is very similar. The force is also instantaneous.
In the example above, we have the issue that information between physical objects is passed instantaneously. The forces propagate between objects at infinite speed. Why is this an issue? In experiments, we have observed that nature has a speed limit. We know that in nature, nothing exceeds the speed of light. The fact that an object can influence another beyond the speed of light directly contradicts this principle. Physical theories which do not take into account this speed limit of light are called non-relativistic.
The same difficulty applies to quantum mechanics, which is non-relativistic. For example, suppose we had a particle stored in a box. If the box were 1 metre in length, width, and height, we could compute its position wavefunction by solving the Schrodinger equation. The probability distribution of its position depends on the size of the box. Now, if we were to adjust the size of the box ever so slightly, we would find that the corresponding probability distribution has changed instantaneously. This instantaneous change in the probability function means that the particle would have detected changes in the size of the box instantaneously.
So, to correct this, quantum field theory incorporates a concept called 'locality'. The idea is straightforward - the physical properties at a particular space in time can only be affected by things at that point. There is a 'propagation speed' between how particles interact, and forces are carried by messenger particles called virtual particles which travel at a finite speed. This propagation speed is also less than or equal to the speed of light, which is consistent with Einstein's theory of special relativity.
For example, instead of Coulomb's law dictating that the attraction between two electrons are instantaneous, quantum electrodynamics enforces this speed limit with different mechanics. When there is an attraction between two electrically charged particles, there is a force carrier that mediates the force between them. This force carrier is called a virtual photon. Once again, since photons can only travel at the speed of light, the force between the two electrons propagate only at a finite speed.
To calculate the force between electrons in quantum field theory, we calculate a 'scattering amplitude'. Scattering amplitudes are probabilities associated with a collection of particles interacting with one another to produce outputs. The Feynman diagram above visualises when an electron interacts with another electron. In the interaction, the particle responsible for the force between them, the photon, is pictured with the squiggly line above.
The 'identical particle problem'
In addition, non-relativistic classical and quantum don't address a simple yet profound question about the nature of particles in general. The question is this about why particles are essentially the same everywhere. For example, it is not apparent why an electron observed in London should be the same as an electron on one of Saturn's rings - but it is. If you find an electron anywhere - it will have the same charge and mass as other identified electrons, along with other fundamental properties.
The identical particle problem implies that it might not be appropriate to view particles as the fundamental objects in nature. Rather, there must be some more fundamental electron field that permeates space in the case of electrons above. The use of fields is where quantum field theory gets its name. Field vibrate at different frequencies, which represent energy. Quantum field theory then constructs operators which represent 'particle generators' for a given frequency. When an operator is applied to a base state, a particle is modelled as popping into existence. With this, we can model particles popping in and out of existence on top of a fundamental base field.
Mathematically, particles are modelled using objects called creation and annihilation operators. To construct a particle in quantum field theory, a creation operator is applied to a 'zero' state called the ground state. Creation and annihilation operators are used to generate and destroy particles in this way.
\[ | \mathbf {{ p } _ 1 } \rangle = a _{\mathbf { p } _ 1 } ^ \dagger | \text{base state} \rangle \]
The left-hand of this equation represents a single particle with momentum \( \mathbf{ p } _ 1 \), and the right-hand side represents its creation from the ground state with the creation operator.
The success of quantum electrodynamics
Now that we've talked about some of the problems quantum field theory has solved let's discuss its successes. Quantum electrodynamics is a theory that uses quantum field theory to make predictions about electromagnetic phenomena, like the magnetic properties of fundamental particles. In quantum electrodynamics, the 'spinor field' is the base field that generates electrons. The base field that produces photons, the electromagnetic force carrier, is called the 'electromagnetic potential'.
Once again, creation and annihilation operators exist to create electrons out of this base field. Furthermore, some constraints on the mathematics and symmetry of the theory imply the existence of 'antiparticles'. The anti-particle of the electron is the positron. Carl D. Anderson discovered the positron in 1932 in a cloud chamber. It is of the same mass yet opposite charge of the electron.
One of the most scientifically accurate predictions that it has come up with is the magnetic dipole of the muon. The magnetic dipole of a particle is a quantity associated with the magnetic field a particle produces. Experimentally, it has been measured to the value below, [2]
\[ g_{\text{experiment}} = 2 + 233 \, 184 \, 600 (1680) \times 10 ^{ -11} \]
The calculation in quantum electrodynamics pins the muon diploe moment to a value extremely close to this, at
\[ g _{\text{theoretical}} = 2 + 233 \, 183 \,478 (308) \times 10 ^{ -11} \]
This degree of agreement in experiment versus theory is almost unheard of in any other field of science. In addition to the muon magnetic dipole prediction, quantum field theory has also sent itself as a tool to understand condensed matter physics. For example, quantum field theory is a tool to understand superfluid Helium's behaviour and materials with ultra-low resistivity to electric currents (called superconductors).
Wrap up
I have barely done the vast and subtle nature of quantum field theory any justice in this article. However, I hope that I've shed some light on some of the problems in the non-relativistic fields of classical mechanics and quantum mechanics that quantum field theory serves to solve.
References
[1] Landau, L. The Classical Theory of Fields, 1975 (Pergamon)
[2] Wilczek, F. American Physical Society Centenary issue of Reviews of Modern Physics, March 1999.