Measurement in quantum mechanics,pt.2.

Alexey N. Rubtsov

Stanley Gudder

Abstract We define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

Allen Parks, John Edward Gray, Garrett K. Josemans

Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested during the measurement process as pointer translations, which have vanishing imaginary or vanishing real parts. The consequence of this that one can characterize some of the aspects of these symmetries of the time dependent Hamiltonians that arise from time dependent weak values. This allows one to generalize some work on non-Hermitian variables in quantum mechanics (QM) due to Bender related to weak values and weak measurement. We also speculate how these symmetries might apply to distinguishing between the various two-time interpretations of QM in the Conclusions of this paper.

Alexey N. Rubtsov

Allen Parks

Quantum mechanical weak values and their measurement have been a focus of theoretical, experimental, and applied research for more than two decades. The concept of PT symmetry was also introduced into quantum mechanics during this time. This paper defines the notion of a weak value measurement pointer Hamiltonian and establishes equivalences between its Dirac symmetries, its PT symmetries, its eigenvalues, and the associated weak value. The affect of these symmetries upon measurement pointer observables is also identified.

Bartel Leenert van der Waerden