[MUSIC] Hello and welcome to the sixth module of our introductory course of subatomic physics. In this sixth module, we discuss weak interactions and the Higgs mechanism. You will notice that this module is again larger than the average module. This is due to the rich phenomenology of electro-weak interactions. We recommend that you take two weeks to digest it. Before diving into our subject, in this first video we will go into more depth on the subject of antiparticles. After following this video, you will know the difference between particles and antiparticles and the connection between antiparticles and energy-momentum reversal. How do antiparticles come into play in the standard model? It all starts with the evolution equation for fields which takes the role of an equation of motion for particles. It follows from relativistic energy-momentum conservation by operator substitution. The result is the Klein-Gordon equation for free particles. Its solutions follow a continuity equation as seen here. It describes the conversation of a four-vector current j^µ of particles. The time component is the probability density of finding a particle at location x and t. Its spatial component is the particle flow density at that same position. Imagine an infinitely small volume that surrounds the point that we are considering. The density in the volume can only change – drho/dt not equal to zero – if particles enter or exit through the boundaries of the volume, which means that divergence of j should then simultaneously be non-equal to zero. This ensures that the number of particles in the volume is conserved. It might surprise you that the probability density contains the time derivative of the wave function. We will understand this fact in a little while. The Klein-Gordon equation is manifestly covariant, it contains only scalars under Lorenz transformation. Plain waves must be a solution of the Klein-Gordon equation, and describe a free particle. We use a normalization proportional to an arbitrary constant square root of N. You may take N as the number density of particles if you wish. The current density is proportional to the four-momentum. Its temporal component is proportional to the energy. This is due to the covariance of the quantity. It requires that the probability rho d^3x is invariant. The volume element changes like d^3x -> d^3x/gamma, because of the Lorentz contraction in the direction of motion. For invariance, the probability density must then change as rho -> gamma times rho. With gamma equal E/m, rho must thus be proportional to energy. It is for this reason that the probability density and current density contain respectively a time and a space derivative of the field. Let us now consider the energy eigenvalues of relativistic free particles, inserting a plane wave solution into the Klein-Gordon equation. We find solutions with positive and with negative energy. And since the density of probability is proportional to the energy, it is no more positive definite either. This contradicts the definition of probability as a real number between zero and one. To solve this problem we must interpret the continuity equation in an innovative way, introducing the electromagnetic current density. Proportional to the electric charge Q, j^0 now becomes the local charge density and thus, may be negative as well as positive in agreement with intuition. It is obvious that solutions of positive and negative energy are distinguished by the charge of the considered particle, if one wants to keep the probability density always positive. It is thus not the probability density, 2N p^µ, which is invariant, but the electromagnetic current density, 2Q N p^µ, which obeys the rules for physically sensible probabilities. The current density is conserved by the continuity equation. With this concept, we can very well spontaneously create and annihilate particles, provided that the total charge in an infinitesimal volume remains constant. All of this is a consequence of relativistic covariance only. Which are the implications of this reinterpretation for the negative energy solutions? The answer is given by Feynman's interpretation of these states. The electromagnetic current density of a free electron with charge Q = -e, ignoring its spin, is given by the first equation. This current is that of a particle with positive energy advancing in the direction of its momentum vector. Let us say that it emerges from a vertex. For a free positron, Q = +e, the current is positive but the current density is the same as for an electron of negative energy which moves in the opposite direction, say entering into the vertex. That is to say that the particle with negative energy which comes out of a vertex is equivalent to an antiparticle with positive energy that enters intervals. In this way, we can eliminate the solutions with negative energy, and replace them by antiparticles evolving in the opposite direction. This establishes a connection between the transformation between particles and antiparticles, which we call charge conjugation, and the reversal of the direction of momentum called parity. These two transformations will be analyzed in more detail in the next video. But first, we look at the consequences of Feynman's interpretation of negative energy states. These consequences are indeed quite dramatic as shown by the example of a simple double scattering process. Heisenberg's Principle allows us to reverse the order of the two scatters in time. The intermediate state will then move backwards in time. We replace it by its anti-particle moving forward in time. The double scatter then becomes a pair creation process as shown on the right followed by a particle/anti-particle annihilation process. Both contributions begin with the same initial state and end up with the same final state. We must thus sum them at the level of amplitudes when calculating the total probability of the process. When calculating simple scatters, it is sufficient to consider particles, the corresponding anti-particle process can be obtained by reversing the direction of the particles. In the next vide, we introduce the discreet transformations of space time, parity, and time reversal and charge conjugation, which transforms the wave function of a particle into that of an anti particle and vise versa.
