The Kallen–Lehmann spectral representation for a quantum field $\phi(x)$ is normalized:
$$\int_0^\infty d\mu^2\, \rho(\mu^2) = 1 \ ,$$
up to factors of $2\pi$ that are a matter of convention. See, for example, Ref [1] equation (9.51) or Ref [2] equation (10.7.18).
One way to derive this is to use the equal time commutation relation for the interacting field $\phi(x)$:
\begin{align} \langle 0 | \left.[\phi(x),\pi(y)]\right|_{x^0= y^0} | 0 \rangle =\int_0^\infty d\mu^2\, \rho(\mu^2) \partial_{y^0} i \Delta(x-y ; \mu^2) = \int_0^\infty d\mu^2\, \rho(\mu^2) i \delta^{(3)}(\mathbf{x}-\mathbf{y}) \end{align} Since the left-hand side is $i \delta^{(3)}(\mathbf{x}-\mathbf{y})$ from the equal time commutation relation, we end up with the above normalization for $\rho$. We have used the spectral representation for the vev of the commutator, Ref. [1] eq. (9.56).
Question: does the normalization of $\rho(\mu^2)$ still hold when it is the spectrum between two composite operators $\mathcal O$? In this case it is not obvious to me that the equal time commutation relations are valid.
Related question: In Appendix A.1 of arXiv:2105.07035, the authors derive a 4D continuum spectrum by assuming a scalar field in an infinite, flat extra dimension and looking at the two-point function on a 4D plane $z=0$, $\langle \Phi(x^\mu, z=0)\Phi(y^\mu, z=0)\rangle$. The resulting spectrum is $$\rho(\mu) = \frac{1}{2\sqrt{\mu^2 +k^2}}$$ This spectrum is not obviously normalizable. Should it be? What goes wrong with the 4D equal time commutation relation argument above? Or is there some idea that the 5D theory is an infinite number of 4D fields which are separately normalized (e.g. in a Kaluza–Klein reduction of a compact extra dimension).
The second question relates to the first when modeling (as in the referenced paper) a 4D field with a continuum spectrum using a holographic extra dimension.
References:
[1] Greiner, Field Quantization, Chapter 9.4
[2] Weinberg, Quantum Theory of Fields, Vol. 1 Chapter 10.7
