# Does quantum manybody theory with 2nd quantization completely equivalent to 1st quantization formulation?

by equivalent I mean:

1. the Hilbert spaces are isomorphic;
2. the operators can therefore be mapped 1:1
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If the Hamiltonian commutes with the particle number operator then no particles are created or destroyed. In that case you can restrict the Fock space to the sector with a specified number of particles (e.g., all states with $N$ particles). The second quantized theory on this sector of Fock space is isomorphic to the first quantized theory described by the many-body wavefunction $\Psi(r_1, r_2,\ldots,r_N)$.