quantum field theory - Physical Interpretation of Lorentz-transformed Single Particle states being linear

As in this question, let $\psi_{p,\sigma}$ be a single-particle 4-momentum eigenstate, with $\sigma$ being a discrete label of other degrees of freedom.

Weinberg discusses the effect of a homogenous Lorentz transformation $U(\Lambda, 0)$ or $U(\Lambda)$ on these states, and concludes that $U(\Lambda)\psi_{p,\sigma}$ is a linear combination of $\psi_{\Lambda p,\sigma'}$.

$$U(\Lambda)\psi_{p,\sigma} = \sum_{\sigma'} C_{\sigma'\sigma}(\Lambda, p)\psi_{\Lambda p,\sigma'}$$

Again, is there any physical information that we can extract from this? (I realize that $\psi_{\Lambda p,\sigma'}$ represent physical states after Lorentz transformation).

Answer

Yes, indeed, first of all it tells you that a particle of momentum $p$ in one frame looks like a particle of momentum $\Lambda p$ in another frame which is related to the first by Lorentz transformation $\Lambda$. That means $p$ is a Lorentz vector and hence gives you more certainty that it indeed captures momentum of the particle.

If you read a few more pages in the book, you will see that $\sigma$ would correspond to spin for a massive particle and helicity for a massless particle. So, the above equation is telling us that not only do you find a different momentum for the particle but also that you might find the particle carrying a different spin (or more concretely the probabilities to find the particle in different spin states would be changed) when you do a Lorentz transformation.

Compare this with a usual case one encounters in QM. We have a massive motionless particle as our physical system. And we now want to study how the physical state transforms under a rotation $$U(R)\psi_{p=0,\sigma} = C_{\sigma',\sigma}(R,0)\psi_{p=0,\sigma'}$$ Here $\sigma$ labels spin of the particle and $U(R)$ forms a representation of rotation group which is generated by the angular momenta $J_1, J_2, J_3$.

So, this equation is telling you that transformation of single particle states in relativistic quantum mechanics is a straightforward union of the transformations that we expect from our earlier study of relativity and quantum mechanics.