To my understanding, I can (at least, formally) express the (unnormalized) PDF for a certain constituent of a composite state as $$ f(x)=f\left(\dfrac{k}{K}\right)=\sum_j m_j^{(k)}|\langle\psi_j^{(k)}|\Psi^{(K)}\rangle|^2 \quad, $$ where

  • $|\Psi^{(K)}\rangle$ is a bound state whose total momentum is $K$.
  • $|\psi^{(k)}_j\rangle$ are Fock space states containing $m^{(k)}_j$ constituent particles of momentum $k$.

Would you agree with such a definition?


Let me clarify what the definition above means using a simple example. Consider the Fock space of a purely bosonic $1+1$ dimensional system. Assume we want to study the bound state of total momentum $4$: \begin{equation} \begin{gathered} |\psi\rangle= c_1|1^4\rangle+ c_2|2^2\rangle+ c_3|3^1,1^1\rangle+ c_4|4^1\rangle+ c_5|1^2,2^1\rangle+\ldots \end{gathered} \end{equation} The sum contains infinitely more terms (with negative momenta) if one quantises in equal time (e.g. $|-1,2^2\rangle$); in the light front that would be pretty much it.

Then, my guess is that the PDF should be evaluated as follows: \begin{equation} \begin{alignedat}{8} \label{PDFex} f(1/4) &= 4\times|c_1|^2 + 1\times|c_3|^2 + 2\times |c_5|^2 +\ldots \quad&&,\\ f(2/4) &= 2\times|c_2|^2 + 1\times |c_5|^2+\ldots \quad&&,\\ f(3/4) &= 1\times|c_3|^2+\ldots \quad&&,\\ f(4/4) &= 1\times|c_4|^2+\ldots \quad&&. \end{alignedat} \end{equation} (again, in equal time one has infinitely more terms in each line)


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