Systems of covariance in quantum probability

John E. G. Farina

Stanley Gudder

S. Jay Olson

The formalism of covariant quantum theory, introduced by Reisenberger and Rovelli, casts the description of quantum states and evolution into a framework compatible with the principles of general relativity. The leap to this covariant formalism, however, outstripped the standard interpretation used to connect quantum theory to experimental predictions, leaving the predictions of the theory ambiguous. In particular, the absence of a pre-defined time variable or background causal structure resulted in an ``order of projections" ambiguity, in which the usual rule for multiple-measurement probabilities (obtained by time-ordered projections) is not defined. Equally troublesome, the probability postulate of Reisenberger and Rovelli fails to reproduce the Born interpretation for the case of simple quantum mechanical systems. Here, we develop an alternative quantum measurement formalism, based on basic principles of quantum information. After reviewing how this can be done in the context of the traditional formulation of quantum mechanics and noting its implications for the quantum measurement problem, we find that this approach can be generalized to the covariant setting, where it essentially solves the correspondence problems of covariant quantum theory. We show explicit agreement with the Born interpretation of standard quantum mechanics in the context of simple systems. We also demonstrate the origin of the quantum mechanical arrow of time within our framework, and use this to solve the order of projections ambiguity. In addition to compatibility with general covariance, we show that our framework has other attractive and satisfying features - it is fully unitary, realist, and self-contained. The full unitarity of the formalism in the presence of measurements allows us to invoke time-reversal symmetry to obtain new predictions closely related to the quantum Zeno effect.

Eduard Prugovečki

It is shown that to every Galilei-covariant nonrelativistic stochastic phase space representation of a system of massive particles, whose generator is rotationally invariant, corresponds a Poincaré-covariant relativistic representation sharing the same generator. The stochastic phase space probability densities of the two representations overlap in the limit of nonrelativistic velocities in the laboratory frame. The relativistic representations give rise to covariant and conserved probability currents at stochastic space–time points, in complete analogy with their nonrelativistic counterparts. This parallelism extends to the existence of a global representation of the proper Poincaré group in L2(Γ), which is reduced by each subspace of L2(Γ) spanned by the set of phase-space wavefunctions generated by some stochastic phase space representation of all pure states of the system.

Stanley Gudder

B. Roy Frieden

Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A One-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behavior of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe-particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle.