Saturday, 16 December 2017

electromagnetism - What is the symmetry which is responsible for preservation/conservation of electrical charges?


Another Noether's theorem question, this time about electrical charge.


According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.



What kind of symmetry creates the conservation of electrical charge?



Answer



Remember that voltage is always expressed as a "potential difference." You can't measure the absolute value of voltage because everything is invariant when you add a constant voltage everywhere. That expresses a symmetry just like time translation invariance.


When you bring in the magnetic field this invariance or symmetry can be generalised to a bigger gauge invariance transforming the electromagnetic potential as a vector field. Charge particles are also described by fields such as Dirac spinors, which are multiplied by a phase factor under the action of this symmetry, making it a U(1) invariance. Electric charge is the conserved quantity that Noether's theorem gives for this symmetry.


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