Eigen vectors

Suppose we created a physical state $\phi$ (in the Laboratory, say) so that ``each time we observe the physical quality $A$ , we always obtain the same value $\lambda$ . In such a case, we have

$$E_\phi(p(A))=p(E_\phi(A))=p(\lambda)$$

for any polynomial $p$ . In other words, $E_\phi$ gives an representation of an algebra generated by $A$ .

Let us now assume that $O_A$ is a Hermitian operator. We put $m=E_\phi(A)$ . The variance of $A$ is given by

$$E_\phi(A^2)-E_\phi(A)^2=E_\phi((A-m)^2)=\vert\vert(O_A-m) \phi\vert\vert^2.$$

When $A$ always takes the same value, then as we have explained above, the variance should be zero. In that case, we have

$$O_A\phi=m \phi.$$

This means that $\phi$ is an eigen vector of $O_A$ which belongs to an eigen value $m$ .

Thus we come to a situation where term ``spectrum'' is used. Terms like ``spectrum of an operator", ``spectrum of a commutative ring'' are thus related. We may study in several directions. Namely, theory in $C^*$ -algebras, commutative algebras, operator theory, algebraic geometry, etc.

But we choose to continue a primitive approach where minimal knowledge is needed.