# Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined through the tensor product and reduction through partial trace. Operations on the system are represented by a set of operators $\{M_m\}$ verifying $\sum_m M_m^\dagger M_m=1$. This, in my understanding, includes all kinds of operations one can perform on a system, including measurements (see Kraus operators). If we dismiss measurement results or deal with noisy situations, one takes $\sum_m M_m^\dagger M_m\leq 1$.

Now, the only thing left for a complete description is dynamics, or more generally the action of a symmetry group $G$ on the system. In my understanding any element $g\in G$ should be represented by a unitary (or antiunitary) operator $U(g)$ acting on $\mathcal{H}$. The action on a state is represented as $\rho \overset{g}{\rightarrow} U(g)^{-1}\rho U(g)$, therefore $U(g)$'s only need to verify $$U(g_1\cdot g_2)=e^{i\theta(g_1,g_2)}U(g_1)U(g_2).$$ In other words, we have a projective unitary representation of $G$ on $\mathcal{H}$.

Question 1: Groups that are not simply connected cannot have a simple representation and superselection is necessary. However we can go to the universal covering group to have a simple representation and no superselection. Why do we consider that the Poincaré group $$\mathbb{R}(1,3) \ltimes O(1,3)$$ is THE symmetry group of any relativistic theory? This is, in my understanding, only the $(\frac{1}{2},\frac{1}{2})$ (or spin 1 = 4-vector) representation. We should rather say that the Poincaré algebra is the correct object encoding spacetime geometry constraints on physical systems.

Question 2: A confusion I have is the link between the $(\frac{1}{2},\frac{1}{2})$ representation and Minkowski space. Both are composed of 4-vectors, but the former components are amplitudes whereas for the latter we have coordinates.

Question 3: Back to the initial problem, I want to be more specific about $G=$connected Lie group because this is how we can get dynamics. Section 6 in http://en.wikipedia.org/wiki/Quantum_logic seems to point out that we need some conditions to have weak/strong continuity for the one-parameter group that will represent $G$, however I cannot find clear statements of the hypothesis and/or results.