Introduction of a classical level in quantum theory. Continuous monitoring
This paper is written in the framework of a peculiar form of \textit{Generalized Quantum Mechanics} (GQM) proposed years ago by the author's group with the aim of describing a quantum measurement process encompassing also the ``intrinsic classical level for some basic macroscopic quantities'' needed to interpret ``the setup and output of an experiment'' according to Bohr and von Neumann. As recalled in a final annex, within the Heisenberg picture, to a set \(A\) of \(p\) compatible observables, and to \(T\subseteq\mathbb R^p\), are associated a positive operator valued measure \(\hat F_A(T)\) (an operator on the Hilbert space \(\mathbb H\)), and an operation valued measure \(\mathcal F_{S_A}(T)\) (a mapping among trace class operators) where \(S_A\) represents the observation apparatus. If then \(\hat\rho\) is the statistical operator (the state of the system), while the probability of observing \(A \in T\) at the time \(t\) is \[ P(A \in T, t) = \mathrm {Tr} [\hat F_A(T,t)\hat\rho] \] the reduction of the state as a consequence of having observed \(A \in T\) at the time \(t_0\) produces the new statistical operator \(\mathcal F_{S_A}(T,t_0)\hat\rho\) after suitable normalization. It can be shown moreover that ``in GQM the observation of a sequence of results at certain successive times can be put on the same foot as the observation of \(A\) at a single time'' In the original papers ``The idea was that certain basic quantities should be chosen once for ever by an additional postulate and they should be thought as having at any time well defined values, considered as \textit{beables} in the sense of Bell, by treating them formally as continuously observed. Then any observation on a microscopic system should be expressed in terms of the modifications that its interaction with the remaining part of the world produces in the value of such basic macroscopic quantities. Obviously the theory remains statistic and to any possible evolution of the basic quantities a precise probability is assigned ...'' Now, an integration over all possible histories of the basic quantities in a given time interval gives rise in the Heisenberg picture to the evolution equation \[ \frac{\partial\hat\rho(t)}{\partial t}=-\sum_j\frac{\alpha_j}{4}\left[\hat A^j(t),\left[\hat A^j(t),\hat\rho(t)\right]\right] \] where \(\alpha_j\) are positive constants. This theory however meets two main difficulties: ``i) it is in conflict with basic conservation rules, specifically with energy conservation; ii) it does not seem it can be extended to relativistic field theories. Both these difficulties seem to be related to the requirement \(\alpha_1, \alpha_2, \ldots\) to be positive.'' The author's aim in the present paper is than to show that ``it is possible to release such requirement and significant models can be constructed in which ... a consistent positive probability distribution can be defined for the histories of the related basic quantities, the conservation rules respected and relativistic covariance achieved, when appropriate.'' To discuss the problem, first two elementary (\textit{pedagogical}) models are considered: the harmonic oscillator, and the relativistic scalar field; and then the more significant case of the Spinor Quantum electrodynamics is addressed. In this latter case the previous evolution equation must be replaced by a Tomonaga-Schwinger-like equation with a dimensionless constant \(\gamma\) playing the role of ``a new physical fundamental constant, that discriminates between classical and quantum scales.'' In short the author claims that ``[his] proposal corresponds to a specific choice of the `dissipative' term in the Liouville-von Newman equation, dropping the positivity requirement at the price of restricting the class of the observables.''
scientific article; zbMATH DE number 6829724
scientific article