# Numerical sign problem and the absence of the sign problem

It seems that the many-body fermionic system or many-body spin systems generically can have sign problems.

The sign problems occur in the fermionic matter: electrons, quarks, many-body/QFT include fermions.

The sign problems occur in the spin matter: Hubbard model.

(1) This problem seems to originate from the anti-commutative or non-commutative nature of fermionic operators $\{c_i,c_j\}=i \delta_{ij}$ and spin operators $[S_i,S_j]=2i\epsilon S_k$. Is this true?

In contrary, if we consider the bosonic model, in general, should that model have no sign problem by its nature? (Say bosonic degree of freedoms without spins.) Or is there any counter example?

(2) If we are able to absorb the anti-commutative or non-commutative nature of operators in a new formulation of the system with only a commutator or in a commutative nature, then we may be able to get rid of sign problem. Is this true? What are some solutions of sign-problem free models in the fermionic and spin matter? What are the ideas behind these sign-problem-free model?