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## Main Question or Discussion Point

My question concerns both quantum theory and relativity. But since I came up with this while studying QFT from Weinberg, I post my question in this sub-forum.

As I gather, we first work out the representation of Poincare group (say ##\mathscr{P}##) in ##\mathbb{R}^4## by demanding the Minkowski norm of 4 vector be invariant under homogeneous Lorentz transformations. Since ##\mathscr{P}## is a continuous symmetry group, we expand the elements of the group in terms of continuous parameters and discover that the Minkowski metric imposes constraints on the elements and we end up with 10 independent parameters. So far so good.

Now we work out the unitary representation of ##\mathscr{P}## in the Hilbert space. Here we

My question: Is there some way to work out the unitary representation of the Poincare group without working out the ##\mathbb{R}^4## representation first? If the answer is no, then it seems to me that we are giving a "preference" to the ##\mathbb{R}^4## representation.

As I gather, we first work out the representation of Poincare group (say ##\mathscr{P}##) in ##\mathbb{R}^4## by demanding the Minkowski norm of 4 vector be invariant under homogeneous Lorentz transformations. Since ##\mathscr{P}## is a continuous symmetry group, we expand the elements of the group in terms of continuous parameters and discover that the Minkowski metric imposes constraints on the elements and we end up with 10 independent parameters. So far so good.

Now we work out the unitary representation of ##\mathscr{P}## in the Hilbert space. Here we

__use the same parameters__we obtained in ##\mathbb{R}^4## representation (and the corresponding algebra) to obtain the generators of ##\mathscr{P}## in Hilbert space. And finally by using the continuity constraint, we end up with the commutation relations of the ##\hat{J}^{\mu\nu}## and ##\hat{P}^{\mu}## (and obtain the Poincare algebra).My question: Is there some way to work out the unitary representation of the Poincare group without working out the ##\mathbb{R}^4## representation first? If the answer is no, then it seems to me that we are giving a "preference" to the ##\mathbb{R}^4## representation.