Why do the eigenvalues of the 4-momentum operator organize themselves into hyperboloids?

Question

Specifically I'm asking for the motivation behind figure 7.1 in page 213 of the QFT textbook by Peskin and Schroeder. In that section they just consider eigenstates of the 4-momentum operator $P^\mu=(H,\textbf{P})$,

$$H|\lambda _\textbf{p}\rangle = E_{\textbf{p}}(\lambda)|\lambda _\textbf{p}\rangle$$ $$\textbf{P}|\lambda _\textbf{p}\rangle = \textbf{p}|\lambda _\textbf{p}\rangle$$

and these eigenstates can all be reached by boosting (via $U(\Lambda)$) from some eigenstate $|\lambda_0\rangle $ with eigenvalues $\textbf{p}=\textbf{0}$ and $E_{\textbf{p}}(\lambda)=m_{\lambda}$ for some given $m_\lambda$. Since boosting leaves $p^2$ invariant we then have (assuming $p^0>0$)

$$E_{\textbf{p}}(\lambda)=+\sqrt{|\textbf{p}|^2+m_{\lambda}^2}$$

so mathematically this seems to suggest the shape of a hyperboloid but where does the distinction between "single-particle" states and "multi-particle" states come from? and why for a specific multi-particle state like a state with two particles for example do we have a continuum of hyperboloids? (not to mention, what does a multi-particle state or single particle state even mean in this context of an interacting theory) Also why is there a gap between the first isolated hyperboloid and the other ones? if the only restriction on $m_\lambda$ is that $m_\lambda>0$, then we should just have a continuum of hyperboloids starting from the origin in that figure. I would appreciate any clarifications.

Answer 1

Why do the eigenvalues organise themselves into hyperboloids? It’s a consequence of Poincare invariance, nothing more complicated.

Acting on the Hamiltonian (the generator of time translations, with the minus sign) by the boosts. Use the structure of the Poincaré group to derive the transformation of the Hamiltonian after the transformation. Then demonstrate that any eigenvalue of the Hamiltonian gives rise to a transformed eigenvalue of the boosted Hamiltonian.

Next step is to realise that since momentum and the Hamiltonian commute (again due to the structure of the Poincaré group), your momentum eigenvalues can exist simultaneously with the energy eigenvalues. Imagine a (generalised) state that’s the eigenstate of all 4 translation generators with a 4-tuple of corresponding eigenvalues. Derive how the boosted translation generators have transformed eigenvalues that must lie on hyperboloid.

A more hand wavy but also more intuitive argument is: since all inertial reference frames are equally good a description, any point on the hyperboloid must be an (generalised) eigenvalue tuple if any of them is.