Symmetries and conservation laws

The recognition of a certain symmetry in physics can provide a very

versatile tool in the description and understanding of various phenomena.

For instance, if a system is invariant under translations in space,

the Hamilton or Euler-Lagrange equations immediately imply that the

linear momentum is conserved.

Extending this to a 4-vector formalism of space-time shows that an

invariance under translations in time implies conservation of energy.

For an invariance under spatial rotations it turns out that the angular

momentum is conserved.

Similarly, the invariance of electrodynamics under gauge transformations

leads to conservation of charge.

In general, every symmetry of nature yields a conservation law and conversely,

every conservation law implies an underlying symmetry.

This is known as Noether's theorem.

In this lecture we will investigate symmetries on the Quantum level which

will reveal some unexpected phenomena that have a deep impact on our

understanding of nature.

Lecture Notes :

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