The problem with commutation relation

Question

Quantum mechanics states that operators of general coordinate $Q$ and conjugate momenta $P$ obey commutation relations of the type $QP-PQ= i a I$, where $a$ is a constant factor and $I$ is the eigenoperator. According to the theory, operators of physical quantities are Hermitian in the Hilbert space. They have real eigenvalues and their eigenvectors form an orthonormal basis.

From the commutation relation it follows the uncertainty principle $D_Q D_P$ >= positive constant, where $D$ is the standard deviation of measurement outcome in a given state. If $Q$ had an eigenvector, in that state $D_Q=0$, since $Q$ is well-defined. However, this statement contradicts the uncertainty principle. $0 \times D_P$ can not be positive, since zero times anything can not be positive. This means that $Q$ can not have an eigenvector. The same argument is true for $P$ as well. $Q$ and $P$ can not represent physical quantities. Hence, quantum theory is inconsistent. What is the resolution of this problem?

Answer 1

Classic error. The inequality holds only if the states are normalized, and the eigenstates of $Q$ are not - at least not normalized in the usual sense as they are outside the usual Hilbert space. The eigenstates of $P$ for instance are planes waves $\sim e^{i PQ/\hbar}$ and are clearly not normalizable in the usual sense.

So: no contradiction because the assumptions underlying the inequality are not satisfied. This is not the only situation where assumptions are violated; the most common assumptions is on self-adjointness of operators (see for instance this question ).