#012: Product of observables
Category: Quantum mechanics - Tags: Linear operators, Hilbert spaces, \*-algebra
Make a decisive argument as to whether the product of observables is well-defined and what it is.
Mathematical problem. Given that the product of two self-adjoint operators is not necessarily self-adjoint, the questions is whether there exists a canonical way to create the product of two quantum observables. The issue is related to positing a *-algebra of observables, which is the starting point of the algebraic approach to quantum theory. The idea is to understand both at a mathematical and physical level what the product of observable is (and why it is not unique).
The problem can be seen from two direction: how should we construct the product and how the product can be identified. For example, consider the sum. The sum of two self-adjoint operators is itself self-adjoint, so we have a straight forward way to construct it. Moreover, given the observable A, B and C, we have \(C=A+B\) if and only if \(\langle C \rangle = \langle A \rangle + \langle B \rangle\) for all states, so we have a straight forward way to identify the sum.
For constructing the product, note that the function \(f\) applied to an operator corresponds to the self-adjoint operator where the eigenvectors are the same while the eigenvalues are given by applying the function to the related eigenvalues. Therefore \(A+B\) and \(f(A)\) are allowed. Note that \(\frac{(a+b)^2 - a^2 -b^2}{2}=a*b\). The left side is in terms of operations allowed, therefore a product consistent with those rules must be in terms of the anti-commutator \(A*B = \frac{\{A, B\}}{2}\).
The issue is that the operation so defined is not associative: \(\{\{A, B\}, C\} - \{A, \{B, C\}\} = A [ B, C ] - [B, A] C\). However, for position and momentum we have \([Q, P] = \imath \hbar I\) and therefore \(Q [Q, P] - [Q, P] Q = 0\). By recursively applying the product, it would seem we are able to construct all polynomials of \(Q\) and \(P\) and therefore (through Taylor expansion) generic functions. Are there any problems?
For identification in terms of properties of expectation values, it is not clear. It is not clear for a generic function of one observable either. Ideally, we would like to say that an observable is a function of other observables if the expectation of one has some property with respect with the expctation of the others.
Physical significance. This would seem like a basic problem that should have a settled question in the literature, yet we were not able to find an in depth treatment. If the operation is well-defined, it would also be interesting to understand what symmetry it would generate and what the conjugate variable would be. If not, it would be interesting to understand exactly the root of the problem.