The forum discussion centers on the interpretation of the uncertainty principle in quantum mechanics, specifically regarding simultaneous measurements of observables A and B. Participants reference the paper "Simultaneous Measurability in Quantum Theory" by James L. Park and Henry Margenau, which critiques von Neumann's proof that if A and B are simultaneously measurable, then their commutator [A,B] equals zero. The discussion highlights the need to re-examine the uncertainty relation σ(A)σ(B) ≥ 1/2 |μ([A,B])| and its implications for simultaneous measurability, emphasizing that the principle does not prevent accurate measurements of observables but rather limits simultaneous precision. Counter-examples to von Neumann's theorem are also presented, questioning the validity of the assumption that every observable corresponds to a Hermitian operator.
PREREQUISITESQuantum physicists, graduate students in physics, and researchers interested in the foundations of quantum mechanics and the implications of the uncertainty principle.
bhobba said:Basically yes.
It's an operator O = ∑ yi |bi><bi|, yi real. It means the resolution of the identity |bi><bi| describes some observation such that the probability of outcome i depends only on the |bi><bi|. Given any resolution of the identity we can in principle find an observation that describes it. The yi are the arbitrary real numbers associated with each outcome and the operator from that association is O = ∑ yi |bi><bi|. Via the spectral theorem the two are in 1-1 correspondence meaning each implies the other.
In fact, via Gleason the above implies the Born rule.
For foundational issues its easier to consider finite spaces and extend it via the RHS formalism. That way subtle issues of what exactly is a Hermitian operator is avoided.
Thanks
Bill