Is there a conservation law associated with C4 symmetry?

[csmallw]

I know--because of Noether's theorem--that continuous rotational symmetry implies conservation of angular momentum, and that continuous translational symmetry implies conservation of linear momentum. It also turns out that the discrete translational symmetry exhibited by a Bravais lattice implies conservation of crystal momentum.

Are there conservation laws associated with discrete rotational symmetries (like C4 symmetry)?

I was originally thinking about Cooper pairs in a superconductor when this question came to mind. If there is no conservation law associated with discrete rotational symmetry, then it seems to me that the concept of the orbital angular momentum of a Cooper pair ("s-wave," "d-wave," etc.) might be kind of meaningless.

 

[DrDu]

Note that the conservation of crystal momentum is only up to a reciprocal lattice vector.
Even with full rotational symmetry a scattering like d+d->s+g would be in accordance with angular momentum conservation.
But you are right that in principle it would be more precise to use group labels from C4 (or whatever applies to the crystals under consideration) instead of labels from continuous rotation groups.

 

[M Quack]

Discrete rotation symmetries can be used to predict certain macroscopic properties of crystals.

For example, ferroelectric polarization can occur only parallel to a C4 axis (or C2 or C3 for that matter).

Consequently, crystals with two non-collinear rotation axes cannot be ferroelectric.

If you want to state this in the form of a conservation law, then the ferroelectic moment perpendicular to the axis is conserved -at zero.

This has no direct connection to superconductivity that I am aware of.