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Physics

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    $\begingroup$ Eq. $(5)$ does not make sense at all, mathematically. (at least in the usual understanding of the notation). $\endgroup$ Commented Oct 27 at 12:25
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    $\begingroup$ Seems you're just lost in notation; there is no paradox. If you have 3 particles, you can't conflate their Hilbert spaces. $\endgroup$ Commented Oct 27 at 12:27
  • $\begingroup$ Up to (4) seems ok, numerically. Step (3) is just renaming variable. So Eqn. (4) expresses 3 particles into two-particle operations. Eqn. (5) is odd. That is the problem. It may be understood as $\int dr_1 \int dr_2 \phi_p(1)^* \phi_q(2)^* r_{12} \phi_q (2)\phi_r(2)^* r_{12}^{-1} \phi_p(1) \phi_r(2) $. $\endgroup$ Commented Oct 27 at 12:48
  • $\begingroup$ I think it is just Eqn. (4), two "2" are independent. Like tensor contraction between $A = a_i e^i$, $B = b^i e_i$. If we compute $AB$, better rename as $i$ and $j$. Also, in Eqn. (4), if insert completeness relation, name two "2" as "2" and "3". $\endgroup$ Commented Oct 27 at 13:16