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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.

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  • $\begingroup$ 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? $\endgroup$
    – probably_someone
    Commented Nov 14, 2017 at 23:27
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    $\begingroup$ FWIW, there exist models invariant under bigger symmetry groups, such as, e.g., CFT & SUSY. $\endgroup$
    – Qmechanic ♦
    Commented Nov 14, 2017 at 23:28
  • $\begingroup$ 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. $\endgroup$
    – Emil
    Commented Nov 14, 2017 at 23:28
  • $\begingroup$ @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? $\endgroup$
    – exchange
    Commented Nov 15, 2017 at 5:46
  • $\begingroup$ It should be pointed out that this conformal invariance of electrodynamics is violated in the quantum version of the theory. $\endgroup$
    – Blazej
    Commented Nov 18, 2017 at 8:42

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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.

Conformal Field Theory (CFT) has a lot of applications. Most direct one is the explanation of critical phenomena: Physical systems in second order phase transitions (for example a magnet at its Curie temperature) are described by CFT's. Another usage of CFT's is via AdS/CFT correspondence, which roughly states that the boundaries of AdS curved spaces are described by CFT's which can be utilized to extract information about the bulk.

Lastly, String theory also uses CFT in the sense that so-called worldsheets on which strings live have conformal symmetry.

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  • $\begingroup$ That is pretty nice! I knew of CFT in String Theory but not as a method to describe 2nd order phase transitions. Although it does not directly answer the question with regard to QFT-Lagrangians. I guess, as long as one is not forced by experiment, one just does not want to restrict the construction possibilities? $\endgroup$
    – exchange
    Commented Nov 15, 2017 at 5:51
  • $\begingroup$ I just found a link with a proof that the laws of Electrodynamics are conformally invariant in 4 dimensions only which is pretty interesting: academia.edu/1684509/… $\endgroup$
    – exchange
    Commented Nov 15, 2017 at 18:47
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    $\begingroup$ Technically, if you do not have any scales in your Lagrangian; that is, if all of your couplings have zero-mass-dimension, you classically have scale invariance (and probably conformal invariance). This does not necessarily persists when quantum effects are included. $\endgroup$
    – SonerAlbayrak
    Commented Nov 16, 2017 at 6:30
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    $\begingroup$ Maxwell theory, and actually Yang-Mills theories in general, are classically scale invariant (hence conformal ?) as the gauge bosons are massless and gauge couplings have zero mass dimension. Even QCD is classically conformally invariant, though quantum effects (if I remember correctly those effects were due to some chiral symmetry breaking, but I may be completely wrong here) destroy conformal invariance. $\endgroup$
    – SonerAlbayrak
    Commented Nov 16, 2017 at 6:30
  • $\begingroup$ I found another interesting link that states that the hodge dual operator exchanges with conformal transformations: math.stackexchange.com/questions/2300970/… This means that all laws that can be formulated involving only differential forms and the hodge dual should be conformally invariant, as e.g. Yang Mills theory as done here: physics.stackexchange.com/questions/290091/… $\endgroup$
    – exchange
    Commented Nov 16, 2017 at 21:32

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