Work on the mathematical foundations of quantum mechanics by Arthur Wightman lead to a theorem that stated that the expectation value of the product of two fields, ϕ ϕ ( x ) ϕ ϕ ( y ) , could be analytically continued to all separations ( x − − y ) . (The first two postulates of the Pauli era proofs involve the vacuum state and fields at separate locations.) The new result allowed more rigorous proofs of the spin-statistics theorems by Gerhart Luders and Bruno Zumino and by Burgoyne. In 1957 Res Jost derived the CPT theorem using the spin-statistics theorem and Burgoyne's proof the spin-statistics theorem in 1958 required no constraints on the interactions nor on the form of the field theories. These results are among the most rigorous practical theorems.