A non-relativistic theory of quantum mechanics and gravity with local modulus symmetry

Hans-Thomas Elze

Tao Zhou

We have constructed a non-relativistic theory of quantum mechanics based on local modulus symmetry. The resulting connection in the covariant derivative is identified as the escape velocity of the gravitational field. A new real and positive function called the quantum metric function is attached to the complex conjugate of the wave function to satisfy the local modulus symmetry requirement. In an expanding universe, these theoretical features produce new effects that deviate from predictions of conventional quantum mechanics and Newtonian gravity. The quantum metric function yields negligible change for microscopic objects but produces quantum pointer states for macroscopic objects, thus providing a solution to the quantum measurement problem. The time-reversal symmetry is broken in the new quantum kinematics, which has implications for the second law of thermodynamics. The modification of Newtonian gravity is negligible in smaller bound systems but can become significant at the galactic scales. Its potential association with the mass discrepancy problem in the galactic systems is discussed.

Claus Kiefer

Nicola Pranzini

Measurements play a crucial role in quantum mechanics because they serve two fundamental purposes. First, they enable the extraction of information about a quantum system; second, they give rise to new physical effects that do not appear in classical physics, one important example being the measurement-induced state update, where the act of measurement alters the system’s state. For these reasons, the mathematical description of measurements plays a primary role in the foundational postulates of the theory. However, the essence of what happens during a measurement remains poorly understood, even within the relatively easy realm of non-relativistic quantum theory. The situation becomes even less transparent in the relativistic setting, where the requirement to preserve causality and prevent faster-than-light signalling introduces both formal and interpretational challenges, complicating the very mathematical formulation of a measurement theory. As for quantum gravity, the idea of applying quantum measurements within or onto a quantum spacetime remains virtually unexplored, presenting a vast and uncharted frontier for theoretical inquiry. Building on recent advances across several areas at the intersection of quantum foundations, quantum information theory, relativistic quantum theory, and quantum gravity, the last few years saw a surge of interest in formulating a consistent framework for describing local measurements in a relativistic context. The main results of this endeavour are two distinct proposals for a relativistic measurement postulate for quantum field theory in generic spacetimes. While these two proposals are still being understood, and a growing research community is working to clarify their implications and potential interconnections, the field of relativistic quantum measurements has now reached a level of maturity that allows us to ask both formal and interpretive questions on a solid footing. All this only applies to the regime in which quantum field theory in curved spacetime remains valid: nothing can yet be said about measurements at the Planck energy, where quantum gravitational effects are expected to become important. This thesis presents an introduction to the theory of measurements in non-relativistic and relativistic quantum mechanics, along with four original research articles that explore aspects of measurements in both fields, as well as the potential extension of measurement theory to the domain of quantum gravity. The first article examines the limitations of realistic measurement apparatuses in non-relativistic quantum theory, establishing reasonable bounds for the precision that can be achieved with a given apparatus composed of many quantum components. The second article explores how to extract information from a newly introduced class of quantum systems, defined as those that cannot be replicated and thus fall outside the usual scope of quantum mechanics. Then, it applies the methodology to quantum field theory, demonstrating that classical results, such as the Unruh effect, can still be observed in this setting. The third article discusses detector-based measurements in quantum field theory. This is a recently proposed method that extends the non-relativistic measurement postulate to the relativistic regime by means of Unruh-de Witt detectors. The article highlights some conceptual and mathematical issues with the framework and proposes a solution using algebraic quantum field theory. The outcome is a self-consistent scheme for introducing local measurements in quantum field theory via measuring detectors. Finally, the fourth article uses the measurement scheme above to study how what we learned in the past years about local measurements can be extended to quantum gravity. This is achieved through the holographic principle realised in terms of the AdS-CFT correspondence. In conclusion, this thesis contributes to the development of both non-relativistic and relativistic measurement theories and introduces ideas and questions related to the role of measurement in quantum gravity. Though preliminary, this work opens a pathway toward studying measuring observers in quantum gravity.

J. S. Faulkner

Hans-Thomas Elze

Yes, there is. – A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schrödinger picture of a given field theory. While, for simplicity, we study the example of a (1) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, where it leads to the Schrödinger-Newton equations. Gravity is encoded here into a universal nonlinear extension of quantum theory. The probabilistic interpretation, i.e. Born's rule, holds provided the underlying model has only dimensionless parameters.