Learning Outcomes:

When one does not know the principles or the fundamental equations that govern the phenomena one observes, one often obtains clues to them by looking at the symmetries or selection rules they exhibit. Historically, Maxwell constructed his unified theory of electromagnetism by assuming the existence of the displacement current, which was required by current conservation. Discovery of special relativity was inspired by Lorentz invariance of Maxwell’s equations. History also shows that disclosed symmetries, more often than not, turn out to be approximate. Isospin and
flavor SU(3), which led to the discovery of the quark model, are such examples. Today what people believe to be strictly conserved are based on gauge symmetries. Energy-momentum, angular momentum, and charge conservations are in this category. The conservation of baryon and lepton numbers, although phenomenological valid, are not considered as strict laws. Their mixing is an essential feature of the grand unified theories (GUTs). Symmetries and conservation laws are intimately connected. Whether based on gauge symmetry or not, apparent symmetries and conservation laws can be powerful tools for discovering new physics. Symmetry means one cannot tell the difference before and after a certain transformation. For the case of continuous space-time translation, this means there is no special position or absolute coordinate system in setting space-time coordinates for describing a physical phenomenon. The same experiments done at two different places (say in Tokyo and in New York) or done at different times (today or tomorrow) should produce the same result. In other words, translational invariance means the existence of certain nonobservables, i.e. absolute position and time. So far, we have elaborated on the relation between space-time translational symmetry and the resultant energy-momentum conservation. However, it is a common feature of symmetry. Similar relations can be derived for a variety of symmetry transformations. We list some of them here in Table (in this chapter) before proceeding further. Symmetries come in two categories, first, those related to space-time structure, which are often referred to as external, and second, those related to internal symmetries, such as electric charge, isospin and color charge. Mathematically speaking, the latter leave the Lagrangian density invariant after the symmetry operation apart from a total derivative. It gives a surface integral that vanishes under normal circumstances. Space-time symmetries change the coordinate values, so only the action, which is an integral of a Lagrangian over all space-time, remains invariant. In other classifications, they are either continuous or discrete. We discuss the continuous space-time symmetries first.

Lesson Plan:

Lecture 01: Continuous Symmetries - I

Lecture 02: Continuous Symmetries - II

Lecture 03: Discrete Symmetries - I

Lecture 04: Discrete Symmetries - II

Lecture 05: Discrete Symmetries - II

Lecture 06: Internal Symmetries