Lecture 7 | New Revolutions in Particle Physics: Standard Model

The paragraph introduces the concept of how particles like photons can gain mass through a process known as spontaneous symmetry breaking, specifically via the Higgs phenomenon. It emphasizes the importance of two technical concepts: spontaneous symmetry breaking and gauge invariance. The discussion also includes an example of spontaneous symmetry breaking using a simple model of magnets, where individual magnets can point up or down, and the energy is lower when they are parallel. The text explains how the symmetry is spontaneously broken when all the magnets align in one direction due to energy considerations, and contrasts this with explicit symmetry breaking, which occurs when an external influence like a magnetic field dictates a preferred direction.

This paragraph explores the question of whether a particle with no charge but with spin can have a preferred orientation in a magnetic field. It discusses the concept of magnetic moments, using the neutron as an example. The text then delves into the propagation of spin changes and the concept of waves of reorientation. It also explains how to differentiate between spontaneous and explicit symmetry breaking in a magnetic system, highlighting the role of domain walls in the case of spontaneous symmetry breaking.

The paragraph presents a field theory example involving a real-valued field that can take on any real value. It discusses the dynamics of this field, controlled by a Lagrangian, which includes a term for kinetic energy and a potential energy term. The text explains how the potential energy can be symmetric with respect to changing the field's value, and how the field can have two states of lowest energy, representing spontaneous symmetry breaking. The discussion also touches on the concept of domain walls in this context and how they signify the breakdown of symmetry.

This section discusses the energy implications of a field transitioning between different states. It explains that a sudden change in the field state is energetically costly due to the derivative energies involved. The text then explores the conditions under which a field might prefer to remain in a state of spontaneous symmetry breaking, as opposed to a situation where the potential energy landscape does not favor any particular state, indicating unbroken symmetry. The concept of an instability in the system, where a small perturbation can significantly change the system's state, is also introduced.

The paragraph examines the behavior of a system at high temperatures, where random fluctuations could result in equal amounts of 'up' and 'down' states, on average. As the system cools, it undergoes a process where patches of aligned states form and connect, eventually leading to a state determined by random chance. This process is described as spontaneous symmetry breaking, and the paragraph distinguishes between discrete and continuous symmetries, with examples provided for each.

The text delves into the concept of continuous symmetry, where configurations can be interpolated continuously from one to another. It discusses the idea of Goldstone's theorem, which predicts the existence of massless particles, known as Goldstone bosons, when a continuous symmetry is spontaneously broken. The paragraph provides examples from field theory and condensed matter physics, such as ferromagnets and spin waves, to illustrate the concept of Goldstone bosons.

The final paragraph introduces the concept of gauge invariance, which is a type of local symmetry transformation that can vary from point to point in space. It discusses the implications of gauge invariance for the Lagrangian of a system and how introducing a gauge field can make the Lagrangian invariant under such transformations. The text touches on the idea that gauge bosons, like the photon, cannot have mass if gauge invariance is to be maintained, which leads to the Higgs phenomenon as a mechanism to give mass to particles like the Z boson.