The Weblog of Nicholas Chapman

Visual summary of some of the core ideas of Quantum Electrodynamics. The wavepacket of a matter particle has a rotation in the complex plane. This couples with an internal rotation of the electromagnetic field, encouraging its rotation. This rotation propagates outwards in space. This EM field rotation in turn causes phase differences across other matter wavepackets, causing them to accelerate towards or away from the original particle. This is the electric force! Read on for the details. Matter Quantum electrodynamics describes the behaviour of electrically charged matter (e.g. electrons) and its interaction with the electromagnetic field. Consider electrons - they are a particle that have a property called spin. In particular they are spin-1/2 particles. Spin is a somewhat mysterious property of particles that we won't go into too much here. Anyway, spin 1/2 particles such as the electron are called fermions, and can be described with the Dirac equation. Understanding the Dirac equation is pretty tricky however - it's a weird equation which Dirac came up with to get an equation that is first order in time (meaning it has first derivatives with respect to time, but not second or higher derivatives). The Dirac equation is similar to a wave equation, and as you will see, particles described by the Dirac equation (e.g. electrons) have very wave-like behaviour. Let's see some simulations of various solutions of the Dirac equation: A wavepacket moving slowly to the right (positive momentum). The Dirac equation is an equation for a function named \( \psi \). \(\psi\) is function of position and time coordinates: \(\psi(r, t)\). It's a function from space and time coordinates to a 4-vector of complex values. Since there are 3 space coordinates and 1 time coordinate, therefore it has signature $$ \psi : \mathbb{R}^4 \to \mathbb{C}^4 $$ From a programmer's point of view, if you were to simulate a single particle on a grid using the Dirac equation, this means that at each grid cell you store 4 complex numbers. And each complex number is just 2 real numbers. Let's take a look at a stationary particle, described using a stationary gaussian wavepacket. This particle has zero momentum. A stationary particle/wavepacket. Some important things to note about the stationary particle simulation video: In the bottom middle section, you will see some phasor arrows rotating clockwise.

Phasor arrows show the complex value at a bunch of spatial points. Only the red arrows are non-zero length here: this means that \(\psi_1\) is non-zero, the other \(\psi\) components are (nearly) zero. The top-right section shows the momentum density G, from which you can see the electron spin direction (out of the screen in this case along the +z axis). Note however that the electron spin direction is *not* correlated with the direction of rotation of the complex phasor values. To reinforce that point, here's a video of a spin-down particle, e.g. one whose spin vector points in the -z direction (into the screen). Note that the momentum density curls in the opposite direction, but the phasor arrows rotate in the same clockwise direction as for the spin-up particle. A stationary particle/wavepacket with spin down (-z). Now let's look at anti-matter! These solutions are referred to as negative energy solutions: A stationary negative energy (antimatter) particle/wavepacket The crucial difference visible in the antimatter video above is that compared to normal matter, the phasor arrows spin in the opposite direction: anti-clockwise. Indeed, that's basically what antimatter is: it's a solution to the Dirac equation where the rotation direction in the complex plane goes the opposite direction from normal matter. The effect of matter on the electromagnetic field In standard quantum electrodynamics, we have a value called \( \rho \), which is the probability density of \(\psi\) at a particular position and time. For the Dirac equation, $$ \rho = \psi^\dagger \psi = |\psi_1|^2 + |\psi_2|^2 + |\psi_3|^2 + |\psi_4|^2 $$ In other words, the probability density \(\rho\) is equal to the sum of the squared lengths of the psi vectors in the complex plane. This is what is drawn in the bottom left section of the videos above. Note that \(\rho\) is greater than or equal to zero for both the matter and antimatter particles!

The probability density \(\rho\) then affects \(\Phi\), the electric potential, in the following way (in the Lorenz gauge): $$ (\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2) \Phi = q \rho $$ Where \(q\) is the electric charge of the particle, and has magnitude equal to the coupling constant between the dirac and EM field. Rearranging: $$ \begin{aligned} \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \Phi = q \rho + \nabla^2 \Phi \\ \frac{\partial^2}{\partial t^2} \Phi = c^2( q \rho + \nabla^2 \Phi) \end{aligned} $$ In words: the 'rate of change of the rate of change with respect to time' (the acceleration) of the electric potential \(\Phi\) is equal to \(c^2\) times the electric charge \(q\) times the probability density of the Dirac matter field (\( \rho \)) plus the 'curvature' \(\nabla^2\) of \(\Phi\). One key point is that a blob of particle will always try to push \(\Phi\) in the direction of \(q\): That is, if \(q\) is positive, then the particle will push \(\Phi\) in the positive direction, resulting in a positive electric potential in the EM field. If \(q\) is negative, then a particle blob will push \(\Phi\) in the negative direction, resulting in a negative electric potential. But note that \(\rho\) is always ≥ 0 for the Dirac equation, both for matter and antimatter. This means the Dirac equation cannot distinguish between the charge of a matter particle and the opposite charge of the antiparticle. This is handled in standard QED during 'second quantisation', when QED becomes a theory of operators acting on field states, and the negative sign comes from anticommutation of fermion creation and annihilation operators. Can we solve this issue without second quantisation? I think so! The key is to use the direction of rotation of the matter wave in the complex plane to source \(\Phi\), instead of just \(\rho = |\psi_1|^2 + |\psi_2|^2 + |\psi_3|^2 + |\psi_4|^2\), which does not contain this information.

But to do this we need to abandon the Dirac equation and go back to a second-order equation of motion for the matter field. If we square the free Dirac equation ('free' = the equation for a particle not interacting with an EM field), we get the Klein-Gordon equation (See technical appendix 'Klein-Gordon equation from the Dirac Equation'). If we square the non-free Dirac equation (e.g. square the equation for a particle interacting with an EM field), we get the Feynman-Gell-Mann equation (FGM equation)[1]. Crucially, for the Klein-Gordon equation, \(\rho\) changes to become $$ \rho = -\frac{\hbar}{m c^2}|\psi|^2 \frac{\partial (\arg \psi)}{\partial t} \\ $$ In other words, the charge density \(\rho\) is proportional to the square of the magnitude (length of phasor vectors) of \(\psi\), multiplied by the signed rate of rotation of \(\psi\) in the complex plane. (See technical appendix 'Klein-Gordon charge density and phasor rotation direction') \(\rho\) for the 2 Klein-Gordon components that describe a Dirac-like particle becomes: $$ \rho = -\sum_{i=1}^{2} \frac{\hbar}{m c^2}|\psi_i|^2 \frac{\partial (\arg \psi_i)}{\partial t} \\ $$ i.e. we sum the individual component \(\rho\) values. With the squared Dirac equation, a particle and an antiparticle will have opposite rotation directions in the complex plane, resulting in opposite charge densities. The value of \(q\) can have the same sign for both particle and antiparticle solutions - and the resulting opposite electrical potentials will come from the opposite directions of phase rotation! The other nice thing about the Klein-Gordon equation is that it's much easier to understand, in my opinion, than the Dirac equation, because it isn't full of 4x4 matrices! I plan to write another essay on understanding the Klein-Gordon equation, stay tuned for that. Many answers to the mysteries of the universe are contained within it! With all that said, let's move on to the core intuition-building of how the Dirac matter field and the EM field interact: The intuition for the interaction of the Matter and EM fields In broad strokes: 1. Matter rotating in the complex plane encourages a corresponding rotation in the complex plane of the EM field.

2. This rotation in the EM field spreads outwards like a wave. 3. This rotation in the EM field in turn encourages a corresponding rotation in the matter field. To explain this in more detail, first let me introduce the metaphor of a clutch. The simplest kind of clutch is basically two metal discs connected to axles. When they are 'engaged' (pushed together), rotation of one of them causes, via friction, corresponding rotation in the other disc. If you have ever driven a manual car, then you will have been manually controlling a clutch. When the clutch is engaged, the engine drive shaft engages to the shaft that runs to the wheels, causing the wheels to turn. A clutch with the two clutch discs separated. The shaft from the engine does not turn the shaft to the wheels. A clutch with the two clutch discs touching. Friction between the discs causes the wheel shaft to turn also. See How a clutch works. (3D Animation) for a short introduction to clutches. In QED, matter has a rotation in the complex plane. Some of this rotation comes from momentum, some of this rotation comes from mass. This rotation couples (engages) with the rotation of the EM field, which also can be thought of as having a rotation in the complex plane. It's not a 1:1 correspondence of rotation, indeed the matter rotation is generally much faster. It's like the clutch between the rotations is only weakly engaged. The rotation of the matter field 'encourages' rotation of the EM field. If we identify \(\Phi\) as the rate of rotation in the complex plane of the EM field, then the equation we saw before $$ \frac{\partial^2}{\partial t^2} \Phi = c^2( q \rho + \nabla^2 \Phi) $$ Starts to make sense - it's basically saying to increase the rotation speed of the EM field if the rotation of the matter field is positive, and to decrease it if negative. Or to be a little more precise: increase the EM field rotation speed in the clockwise direction if the matter field is rotating in the clockwise direction, and to increase the EM field rotation speed in the anticlockwise direction if the matter field is rotating in the anticlockwise direction. (