# Completeness relation for Hilbert space in quantum field theory

I'm studying chapter 7 section 1 of Peskin and Schroeder. On page 212, we have the one particle Hilbert space $$\tag{7.1} (1)_{\text{1-particle}}=\int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}|p\rangle\langle p|$$

From which Peskin and Schroeder derives the completeness relation for the entire Hilbert space (I guess this is the Fock space of the 1 particle space above?)

$$\tag{7.2} 1=|\Omega\rangle\langle\Omega|+\sum_{\lambda}\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p(\lambda)}|\lambda_p\rangle\langle \lambda_p|$$

here $$|\lambda_0\rangle$$ is an eigenvector of the energy $$H$$ with momentum zero, $$|\lambda_p\rangle$$ is the boost of $$|\lambda_0\rangle$$ with momentum $$p$$, and assume $$|\lambda_p\rangle$$ to be relativistically normalized. Let $$E_p(\lambda)=\sqrt{|\boldsymbol{p}|^2+m_\lambda^2}$$ where $$m_\lambda$$ is the mass of the state $$|\lambda_p\rangle$$, that is the energy of the state $$|\lambda_0\rangle$$

My questions are:

1. How do we derive 7.2 from 7.1?
2. By $$|\lambda_0\rangle$$, do we mean all eigenvectors of $$H$$? Does it include $$|\Omega\rangle$$ also? What does it mean for $$|\lambda_p\rangle$$ to be relativistically normalized?

1. By $$|\lambda_0\rangle$$, do we mean all eigenvectors of $$H$$? Does it include $$|\Omega\rangle$$ also? What does it mean for $$|\lambda_p\rangle$$ to be relativistically normalized?
$$|\lambda_0 \rangle$$ includes all eigenstates of $$H$$ and $$P$$, namely the total energy and momentum of all particles in the system. It does not include $$|\Omega \rangle \langle \Omega |$$ which is pulled out of the sum separately for convenience and to make it clear that it is in fact included.
For the $$|\lambda_0 \rangle$$ to be "relativistically normalized", the entire identity acting on some $$|\lambda'_{p'}\rangle$$ should, through a delta function and proper cancellation of the factor including the energy in the integrand, give back the same $$|\lambda'_{p'}\rangle$$. For this to happen, we need the normalization of $$|\lambda_{p}\rangle$$ to depend on the energy. The reason it would be called relativistically normalized is that it cancels out with the factor which makes the integrand relativistically covariant, namely
$$\frac{d^3 p}{(2 \pi)^3 2E_p(\lambda)}$$