Does any linear unitary operator stand for a symmetry transformation?

[wangyi]

Hi, i am confused on this question: Wigner proved that a symmetry is represented by either a linear unitary operator or an anti-linear anti-unitary operator. But does it's inverse right? i.e.
Does any linear unitary operator stand for a symmetry transformation?

It seems to be right, as a unitary operator does not change the inner product of two states, and it holds as time flowing.

But this result is too strong to believe, because there are infinity number of linear unitary operators, are there also so many symmetry?

Best wishes.
Thank you!

 

[Careful]

** Hi, i am confused on this question: Wigner proved that a symmetry is represented by either a linear unitary operator or an anti-linear anti-unitary operator. **

Hi some authors DEFINE a symmetry simply as a unitary or anti unitary operator, see e.g. arxiv.org/abs/gr-qc/9607051. However the more common point of view is that a symmetry is a unitary or anti-unitary operator that commutes with the quantum Hamiltonian. I do not remember precisely how Wigner has shown this, but I guess that the logical way to do this would be to start from a classical Hamiltonian system with a continuous symmetry and notice that an infinitesimal symmetry transformation corresponds to the Poisson derivation with respect to the Noether current (which is conserved on shell). Next, one should apply the Dirac quantization rule and exponentiate the corresponding expression which gives the unitary operation. However, this does not explain why discrete symmetry groups should have a unitary or anti-unitary representation: the motivation here probably is that the modulus of the scalar product should be a preserved quantity, something which is an *ad hoc* restriction on the representation of the Weyl algebra AFAIK (however this introduces a nonlinear phase factor and does not relate in any obvious way to the classical theory). However, computation of some examples shows that the latter requirement makes sense. For example it is easy to show that finite spatial symmetry groups (like finite rotation groups) are unitarily represented in the *standard* Schroedinger picture.

Hope to have answered your question.

Cheers,

Careful

 

[wangyi]

Thank you! I think now I am clear. The article you suggested is of great help. In that article, the author does not simply define all symmetry as linear unitary operator, but treat mathematical and physical symmetry differently. A symmetry of any physical sense need to preserve Hamiltonian.