# Why do we not require physical laws to be invariant under the action of a bigger transformation group?

In Quantum Field Theory, Lagrangians are constructed such that the coordinates on which the quantum fields depend are invariant under action of the Poincare group $\mathbf{P}(1,3):=\mathbf{R}^{(1,3)} \rtimes \mathrm{SO}(1,3)$, where $\mathbf{R}^{(1,3)}$ is the abelian, additive group of translations and $\mathrm{SO}(1,3)$ are the Lorentz transformations which are rotations in Minkowski space if one defines a rotation to be a transformation that leaves the inner product of two vectors invariant and $\rtimes$ denotes the semidirect product.

Now this Poincare group is basically the group of all isometries that leaves the length of vectors invariant, i.e. it leaves expressions like $(t-t')^2-\sum_{i=1}^3 (x_i-x_i')^2$ invariant.

My question is: Why do we only demand our coordinates to be invariant under isometries? Would it not be possible that physical laws must be invariant also under e.g. rescaling or other operations?

Thanks.

EDIT: I just found a link with a proof that the laws of Electrodynamics are conformally invariant which is pretty interesting: https://www.academia.edu/1684509/Conformal_Invariance_of_Classical_Electrodynamics

Furthermore, it is true in 4 dimensions only! This seems to make 4 dimensions into a kind of priviliged setting for ED! EDIT2: Even though it is violated in the quantum theory due to symmetry breaking.

• We have observed that relativity is correct to within experimental error, so physical laws must be Poincare-invariant. Is there any observation that would indicate that nature follows some other symmetry? Nov 14 '17 at 23:27
• FWIW, there exist models invariant under bigger symmetry groups, such as, e.g., CFT & SUSY. Nov 14 '17 at 23:28
• Not sure if related, but I remember that my teacher said that numbers like the Reynolds number is important when making prototypes in fluid dynamics, which is a combination of not just geometry but dynamical variables too, if I remember correct.
– Emil
Nov 14 '17 at 23:28
• @probably_someone I also do not know of any direct experimental evidence that would force us to construct our Lagrangians according to another symmetry group but from another point of view, there also does not seem to be a reason why physical laws should change if everything in the universe would be, say, twice as large - and from that point of view it would be a symmetry that could be worth exploring, or not? Nov 15 '17 at 5:46
• It should be pointed out that this conformal invariance of electrodynamics is violated in the quantum version of the theory. Nov 18 '17 at 8:42

You can construct a theory which is also scale-invariant. Actually, mostly one assumes that a unitary and scale invariant theory is also Conformal invariant (I believe that there is a proof of this for $d=2$, but not for any other $d$), hence what you are asking for is conformal theories.