This work addresses how classical-like behavior emerges in a relativistic quantum setting. Decoherence is described as the evolution of general quantum states belonging to a total state bundle toward a classical-like mixture of pointer states. The pointer states are geometrically supported by the relativistic quantum phase space (RQPS), defined as the subspace of the expectation-value space equipped with a ground variance field that saturates the uncertainty relations and satisfies an LCT-compatible de Sitter constraint.
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature
(1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as
Γ=L2/ℓ2 for states that saturate the uncertainty relations, where
L and
ℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit
ℓ→0, the equation reduces to the de Sitter spacetime equation; in the limit
L→∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both
ℓ→0 and
L→∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.
Quantum decoherence is both the fundamental mechanism underlying the quantum-to-classical transition and a major challenge for the development of scalable nanoscale quantum technologies. This work introduces a Quantum Phase Space (QPS) framework that provides a unified geometric description of decoherence. The framework is based on a dual structure consisting of an overcomplete continuous frame of minimum-uncertainty states that defines the QPS geometry, and an orthonormal basis that satisfies the strict mathematical requirements for decoherence within the Spectrum Broadcast Structure (SBS) objectivity criterion for pointer states. Within this framework, the variance-covariance matrix of the QPS ground states serves as a universal indicator of decoherence regimes:it is time-independent for Markovian (memoryless) dynamics, and time-dependent for non-Markovian dynamics with memory and information backflow. To illustrate the formalism, the Hu-Paz-Zhang (HPZ) model is generalized to include simultaneous position and momentum couplings to the environment, leading to a generalized non-Markovian master equation characterized by a spectral-density matrix. The corresponding evolution equations for the first moments
(⟨p(t)⟩,⟨x(t)⟩) and for the ground covariance matrix establish a direct connection between microscopic environmental properties, classical-like trajectories and the QPS geometry. The proposed framework provides a unified theoretical foundation for modeling decoherence in nanoscale systems involving simultaneous position and momentum interactions with the environment. The QPS framework may thus bridge fundamental theory and practical quantum engineering, offering a promising coherent pathway to understand, control, and exploit decoherence at the nanoscience frontier.
It is well-known that the pointer basis of a quantum system satisfies the condition to diagonalize the interaction Hamiltonian between the subsystems. We show that this condition can be translated into the form
δΛ=0, where
Λ, so-called the action, is the time integrated interaction energy: it is found out naturally in the phase of state vectors due to diagonal interaction. The careful treatment of a two states system demonstrates that the states of the total system branch into the states with different values of the action. Mathematically the pointer states are selected out by the saddle point condition on the phase
Λ. This study helps us to understand the precise mechanism and the general dynamics of decoherence.
The purpose of the present study is to derive the pointer states of a macroscopic system interacting with its environment, under the general assumptions, i.e., without assuming any form of the interaction Hamiltonian. The lowest order perturbation leads that the interaction energy shifts the phase factors of the state vectors. For a macroscopic system, these factors are the macroscopic quantities even for the very weak interaction. When we group the state vector of the total system by the view point of environmental side, the destructive interference occurs and the stationary phase approximation can be adopted. Only the pointer states then survive and the decoherence also occurs. The present approach is within the standard quantum mechanics as same as the standard decoherence theory, but the meaning of the classical state is much clear.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl,
SU(2),
SU(1,1), and
SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi
Q-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan
P-function to the
Q-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic
SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In
SU(2) and
SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale,
t∗, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.