# Can one prove the full spin-statistics theorem from the spin 0, 1/2 and 1 cases?

Using second quantization for scalar field, spinor field and vector fields, we can get commutation and anticommutation relations for the birth and destruction operators of the fields, which leads us to the Bose or to Fermi statistics. Is it possible to expand these results on a field of arbitrary spin (integer or half-integer), using the basic idea that each field can be built by combination of spinor $\frac{\hbar }{2}$ fields?

Given that assumption, the full theorem follows directly from the spin-1/2 case. Any spin can be realized by coupling spin 1/2's. Given that spin 1/2 has an eigenvalue of $-1$ under particle exchange, coupling $n$ of them produces a composite system that has an eigenvalue of $(-1)^n$.
• "...There's nothing fancy going on...", - so, one of deuteron's state, of course, can't be interpreted as representation $\left( \frac{1}{2}, 0\right) \times \left( \frac{1}{2}, 0\right)$? "...It doesn't change the commutation relations...", - thank you, I understand this now. – user8817 Aug 20 '13 at 9:44