#011: Determine whether quantum predictions allow a joint distribution by solving the moment problem

Category: Probability theory - Tags: Quantum mechanics, Statistics, Measure theory

Determine whether the expectation of the polynomials of position and momentum give a sequence of moments that can solve the multivariate moment problem.

Mathematical problem. In probability theory (and statistics) the moments of a distribution over a stochastic variable \(X\) are the expectation values \(m_i = E[X^i]\) of all the powers of \(X\). The moment problem is determining whether given all the moments of the distribution one can reconstruct the probability distribution. That is, given a sequence \(\{m_i\}_{i=0}^{\infty}\), determine whether a positive Borel measure \(\mu\) exists such that: \(m_i = \int_{-\infty}^{+\infty} x^i d\mu\). Note that not all sequences \(\{m_i\}\) allow for a solution.

We are asking whether the corresponding multivariate problem is solvable when starting from \(m_{ij} = E[X^iP^j] = \langle \psi | (X^iP^j + P^jX^i)/2 | \psi \rangle\). We want to know whether we can find a suitable measure, such as the one determined by a joint cumulative distribution function \(F_{X,P} = P(X \leq x, P \leq p)\) such that \(m_{ij} = \int x^ip^j dF = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} x^ip^j \frac{\partial^2 F}{\partial x \partial p} dx dp\).

Physical significance. The conjecture is that the answer is in the negative: the quantum momentum problem cannot be solved. This would give a clear answer that hidden variable theories are ruled out. Note that we do not need to show that the problem never has a solution. It suffices to show that it does not have a solution in one physically relevant case. For example, if one could show that the problem is not solvable for a Gaussian wave packet, it would already be an interesting result.