General Relativistic Quantum Mechanics Research Articles

True wavefunctions and antiparticles of Klein–Gordon equation: ℏ-conjugation, elimination of CPT-invariance, and anti-gravitation

While revisiting the Klein–Gordon relativistic quantum equation for spin-0 particles, we predicted that ℏ reverses its sign for negative energies, and formulated a universal symmetry rule, whereby all the parameters that couple particles to external fields reverse their sign along with ℏ at a particle ↔ antiparticle transformation; this in particular implies anti-gravitation between matter and antimatter. Our results suggest that the ℏ-conjugation principle and related invariance may replace CPT-invariance in general relativistic quantum mechanics.

Tales from the prehistory of Quantum Gravity

The main purpose of this paper is to analyse the earliest work of L\'eon Rosenfeld, one of the pioneers in the search of Quantum Gravity, the supposed theory unifying quantum theory and general relativity. We describe how and why Rosenfeld tried to face this problem in 1927, analysing the role of his mentors: Oskar Klein, Louis de Broglie and Th\'eophile De Donder. Rosenfeld asked himself how quantum mechanics should \textit{concretely} modify general relativity. In the context of a five-dimensional theory, Rosenfeld tried to construct a unifying framework for the gravitational and electromagnetic interaction and wave mechanics. Using a sort of "general relativistic quantum mechanics" Rosenfeld introduced a wave equation on a curved background. He investigated the metric created by what he called `quantum phenomena', represented by wave functions. Rosenfeld integrated Einstein equations in the weak field limit, with wave functions as source of the gravitational field. The author performed a sort of semi-classical approximation obtaining at the first order the Reissner-Nordstr\"om metric. We analyse how Rosenfeld's work is part of the history of Quantum Mechanics, because in his investigation Rosenfeld was guided by Bohr's correspondence principle. Finally we briefly discuss how his contribution is connected with the task of finding out which metric can be generated by a quantum field, a problem that quantum field theory on curved backgrounds will start to address 35 years later.

What is time in quantum mechanics?

Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by Kijowski and the event enhanced quantum theory (EEQT) method. It is suggested that for free particles the two methods may lead to the same result. On the other hand, the EEQT method can be easily geometrized within the framework of Galilei–Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.

Semiclassical scalar propagators in curved backgrounds: Formalism and ambiguities

The phenomenology of quantum systems in curved space-times is among the most fascinating fields of physics, allowing---often at the gedankenexperiment level---constraints on tentative theories of quantum gravity. Determining the dynamics of fields in curved backgrounds remains, however, a complicated task because of the highly intricate partial differential equations involved, especially when the space metric exhibits no symmetry. In this article, we provide---in a pedagogical way---a general formalism to determine this dynamics at the semiclassical order. To this purpose, a generic expression for the semiclassical propagator is computed and the equation of motion for the probability four-current is derived. Those results underline a direct analogy between the computation of the propagator in general relativistic quantum mechanics and the computation of the propagator for stationary systems in nonrelativistic quantum mechanics. A possible application of this formalism to curvature-induced quantum interferences is also discussed.

Multiple-event probability in general-relativistic quantum mechanics. II. A discrete model

We introduce a simple quantum mechanical model in which time and space are discrete and periodic. These features avoid the complications related to continuous-spectrum operators and infinite-norm states. The model provides a tool for discussing the probabilistic interpretation of generally covariant quantum systems, without the confusion generated by spurious infinities. We use the model to illustrate the formalism of general-relativistic quantum mechanics, and to test the definition of multiple-event probability introduced in a companion paper [Phys. Rev. D 75, 084033 (2007)]. We consider a version of the model with unitary time evolution and a version without unitary time evolution.

Multiple-event probability in general-relativistic quantum mechanics

We discuss the definition of quantum probability in the context of ``timeless'' general-relativistic quantum mechanics. In particular, we study the probability of sequences of events, or multievent probability. In conventional quantum mechanics this can be obtained by means of the ``wave function collapse'' algorithm. We first point out certain difficulties of some natural definitions of multievent probability, including the conditional probability widely considered in the literature. We then observe that multievent probability can be reduced to single-event probability, by taking into account the quantum nature of the measuring apparatus. In fact, by exploiting the von-Neumann freedom of moving the quantum/classical boundary, one can always trade a sequence of noncommuting quantum measurements at different times, with an ensemble of simultaneous commuting measurements on the joint system+apparatus system. This observation permits a formulation of quantum theory based only on single-event probability, where the results of the wave function collapse algorithm can nevertheless be recovered. The discussion also bears on the nature of the quantum collapse.