# Is it necessary to prove the existence of an operator representing symmetry on Hilbert space?

Is there any need to prove the existence of an operator $U$ which represents the action of symmetry transformation on rays in Hilbert space? Or is it enough just to prove that it is unitary and linear or antiunitary and antilinear?

I do not see how we should go about finding this operator. I got the answer that this operator is given, but I don't see how. The only thing that is given is the symmetry transformation which takes the whole ray $R_1$ into, let's say, $R_2$. But what then is the action of the operator on this ray? Which vector in $R_1$ will be taken into which in $R_2$?

My guess is that we can show that coefficients of the old and a new transformed vector are the same, so we can just say that in this way the transformation is translated into an operator. Every orthonormal vector of the old basis has the same coefficient as the one in the new basis.

• I'm not sure if this is quite a duplicate, but it's very closely related to this question. It also follows up on this one but is not a duplicate of that. Mar 12, 2016 at 11:03
• The statement of Wigner's theorem is: "Given a ray transformation that is a symmetry, there is an (anti-)unitary operator that represents it on the Hilbert space". The proofs should construct said operator from the ray transformation, else they are not proofs. I do still not understand what your actual issue is. Mar 12, 2016 at 11:05
• Might be a duplacate, yes, I am sorry. And my apologies to ACuriousMind. I would love to be able to explain it better, but it seems that I am not. My problem is obviously that I dont see how is this operator constructed. In Steven Weinbergs book there is a proof but I do not see where construction of the operator happen. Could you please, in the form of an answer, point out where do we construct the operator? Of course, it might be that the entire proof IS the construction.... Mar 13, 2016 at 12:53