2 Symmetries and Conservation Laws

N. Panchapakesan

Learning Outcomes

1.1 Learn what are symmetries. Space-Time symmetries and others.

1.2 Learn how symmetries of the Lagrangian lead to Conservation Laws.

1.3 Symmetry or invariance of Action under space translation leads to Momentum conservation.

1.4 Noether’s Theorem

1.5 Invariance under time translation leads to Energy conservation.

1.6 Invariance under rotations lead to Angular momentum conservation.

1.7 Learn to formulate these laws in terms of second and third order tensors.

1.8 Learn what is a differential conservation law and how it leads to a conserved quantity like energy or angular momentum.

2. Introduction

Emmy Noether, a woman mathematician discovered a deep connection between symmetries obeyed by the action integral and conservation laws of a dynamic system in the years 1915-18. . Conservation laws for Energy – Momentum angular momentum are important examples of her idea, now called Noether’s theorem.

She used the Principle of stationary action which was used to derive the equation of motion . She calculated the change in action produced by the variation in a variable like space-time and equated it to zero. This led to a total derivative being zero. This is called a differential conservation law. This leads further to a conserved charge which is often called a Noether charge.

3. Diferential Conservation Law

3.1 The conservation laws can be obtained by dealing with the invariance of the action integral under some variartion of some variables like the field or the coordinates. They can be also obtained more simply by dealing with the variation in the Lagrangian only.

The first two terms combine to vanish using Euler Lagrange equation of motion.

If δℒ is zero or known eq. (3.1) leads to a conservation law.

3.2 Translation

We apply this method to the case of translation. Physical observations of an isolated system yields the same result when observed in Delhi, Chennai, Mumbai or anywhere. This is called space translation symmetry or invariance under space translation.

In the same way the time when you perform an experiment is not relevant for the observation or result. This is invariance under time translation. In four vector notation we can combine both to have symmetry under space time translation

Translation means shifting the co-ordinate system by a vector which may be denoted by ??, whose first component is time and the other three are space. Thus ?? → ?? + ?? and the change in Lagrangian is

The first integral is a volume integral over the whole volume, while the second is a surface integral over the boundary of the volume.

The vanishing of the variation over the boundary makes the second integral vanish, leading to Euler Lagrange equations . In our case the variation does not vanish on the boundary.

If variation keeps E-L equations unchanged then the first integral vanishes and we have

5. Internal Symmetry, Klein – Gordon equation with two fields Gauge invariance or Gauge transformation of the first kind

Summary

In this module we have learnt about an important theorem called Noether’s theorem. The theorem relates a symmetry (or an invariance under a variation) to a conservation law. Translation symmetry or invariance under shift of the origin of the coordinate system leads to conservation of energy and momentum of the whole system. Conservation of angular momentum is ensured by rotational Symmetry.

We proved it by equating to zero the change in the Lagrangian or the change in the action. The algebra involved integration by parts and Gauss’s theorem. We also found that conservation of angular momentum requires the energy momentum tensor to be symmetric in its indices. We learnt how to make it symmetric without changing its other properties.

When the continuous transformation is internal, that is, it does not involve space or time and action is invariant under the transformation we still get a conservation law, that of charge.

you can view video on Symmetries and Conservation Laws

Learn More