The Schrödinger operators on the Newtonian spacetime are defined in a way which make them independent of the class of inertial observers. In this picture the Schrödinger operators act not on functions on the spacetime but on sections of a certain one-dimensional complex vector bundle—the Schrödinger line bundle . This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations—the Schrödinger principal bundle . If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over spacetime, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wavefunctions' cannot be viewed as actual functions on the spacetime. In this approach, the change of an observer results not only in the change of actual coordinates in the spacetime but also in a change of the phase of wavefunctions. For the Schrödinger principal bundle, a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of the Laplace–Beltrami operator associated with a pseudo-Riemannian metric. The free Schrödinger operator turns out to be the Laplace–Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame—which is supported by producing right physics predictions—and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton–Jacobi equations that makes a bridge between the classical and quantum theory.
We provide a possible fully geometric formulation of the core idea of quantum reference frames (QRFs) as it has been applied in the context of gravity, freeing its definition from unnecessary (though convenient) ingredients, such as coordinate systems. Our formulation is based on two main ideas. First, a QRF encodes uncertainty about what is the observer's (and, hence, the measuring apparatus's) perception of time and space at each spacetime point (i.e., event). For this, an observer at an event
p is modeled, as usual, as a tetrad in the tangent space
Tp. So a QRF at an event
p is a complex function on the tetrads at
p. Second, we use the result that one can specify a metric on a given manifold by stipulating that a basis one assigns at each tangent space is to be a tetrad in the metric one wants to specify. Hence a spacetime, i.e. manifold plus metric, together with a choice of "point of view" on it, is represented by a section of the bundle of bases, understood as taking the basis assigned to each point to be a tetrad. Thus a superposition of spacetimes gets represented as, roughly speaking, an assignment of complex amplitudes to sections of this bundle. A QRF, defined here as the collection of complex amplitudes assigned to bases at events--i.e., a complex function defined on the bundle of bases of the manifold--can describe, in a local way (i.e., attributing the amplitudes to bases at events instead of to whole sections), these superpositions. We believe that this formulation sheds some light on some conceptual aspects and possible extensions of current ideas about QRFs. For instance, thinking in geometric terms makes it clear that the idea of QRFs applied to the gravitational scenarios treated in the literature (beyond linear approximation) lacks predictive power due to arbitrariness which, we argue, can only be resolved by some further input from physics.
We uncover the quantum geometric structure of graviton Wigner functions and stress-energy tensors via quantum field theory in curved spacetime, revealing the phase-space geometry of spacetime and relativistic quantum-state manifolds. We show that the polarization mode expansion of the underlying graviton field operator is fully captured by quantum geometry, as encoded in the spinor-helicity formalism, thereby establishing the geometric framework from the outset. Applying this to a weak gravitational background, we demonstrate the roles of the quantum metric and connection in describing graviton transport beyond the chiral vortical effect. We also clarify the role of the quantum metric in normalizing the perfect-fluid graviton stress tensor within this approach. This work paves the path for explorations of multistate Hilbert-space geometry in high-energy and gravitational physics. In addition, this framework naturally encompasses lower-spin excitations, allowing for a unified quantum geometric treatment of bosonic and fermionic many-body systems in condensed matter and particle physics.
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new type of nonminimal coupling of the fields and is shown to admit a canonical quantization procedure. For the resulting quantum theory we demonstrate the emergence of a particle interpretation, fully consistent with general relativistic geometry. The path dependency of parallel transport forces each observer to carry their own quantum state; we find that the communication of the corresponding quantum information may generate extra particles on curved spacetimes. A speculative link between quantum information and spacetime curvature is discussed which might lead to novel explanations for quantum decoherence and vanishing interference in double-slit or interaction-free measurement scenarios, in the mere presence of additional observers.
The theory of fiber bundles provides a natural setting for the description of macroscopic quantum systems, wherein their classical and quantum features are represented by actions on the base manifolds and the fibers, respectively, of the relevant bundles. We provide realizations of this picture in the description of (a) quasiparticle excitations of many-body systems, especially those in superfluid helium, (b) the interplay between the microscopic and macroscopic dynamics in certain irreversible processes, such as that of a laser, and (c) local thermodynamic equilibrium. In particular, (b) involves the treatment of a dynamical system which is defined on a vector bundle.
We extend the representation frame formalism, previously introduced to account for key cosmological observations in the Einstein static universe, to non-relativistic quantum mechanics. In this framework, each inertial observer is associated with a flat representation referential Robs, defined as the tangent space to the spatial manifold at the observer’s position, in which all measurements are represented. The Euclidean structure of Robs allows quantum systems to be described using the standard Schrödinger formalism, avoiding the technical ambiguities that arise when quantising directly on curved manifolds. We derive the relation between the Hamiltonian governing quantum dynamics in Robs and its counterpart defined on the physical manifold U, and show that curvature effects enter as observer-dependent modifications of effective potentials. Although the resulting quantum description depends on the observer’s representation frame, we show that this does not lead to contradictions between observers: consistency of measurement outcomes follows from the standard structure of quantum correlations established by physical interactions. We illustrate the formalism with explicit applications, including the hydrogen atom in an Einstein static universe and quantum systems in the vicinity of a black hole, highlighting how spacetime curvature manifests itself in the observer’s quantum description.