If one reads eg page 32 of Srednicki where he says:
In quantum theory, symmetries are represented by unitary (or antiunitary) operators. This means that we associate a unitary operator U(Λ) to each proper, orthochronous Lorentz transformation Λ. These operators must obey the composition rule...
Where does all this come from? Is it Wigners theorem? Probability conservation?
For instance equation 2.15 in the same link:
$$U(\Lambda)^{-1}P^{\mu}U(\Lambda) = \Lambda^{\mu}_{\nu}P^{\nu} $$
I understand the right hand side no problem: The Lorentz matrix acts on the four vector and "rotates it" end of story. But why do we need the left hand side? What does the left hand side even mean, what is $U$, except being a unitary operator?
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