The emergence of classical behavior in quantum theory is often ascribed to the interaction of a quantum system with its environment, which can be interpreted as environmental monitoring of the system. As a result, off-diagonal elements of the density matrix of the system are damped in the basis of a preferred observable, often taken to be the position, leading to the phenomenon of decoherence. This effect can be modeled dynamically in terms of a Lindblad equation driven by the position operator. Here the question of decoherence resulting from a monitoring of position and momentum, i.e., a phase-space measurement, by the environment is addressed. There is no standard quantum observable corresponding to the detection of phase-space points, which is forbidden by Heisenberg's uncertainty principle. This issue is addressed by use of a coherent-state-based positive operator-valued measure for modeling phase-space monitoring by the environment. In this scheme, decoherence in phase space implies the diagonalization of the density matrix in both position and momentum representations. This is shown to be linked to a Lindblad equation where position and momentum appear as two independent Lindblad operators.
The decoherence phenomenon arising from an environmental monitoring of the state of a quantum system, as opposed to monitoring of a preferred observable, is worked out in detail using two equivalent formulations, namely, repeated applications of universal tomographic measurements using positive operator-valued measures, and its continuous time unravelling from the Lindblad equation. The effect of decoherence is analysed by studying the evolution of Stratonovich-Weyl quasiprobability distributions on the state-space of the system. It is shown that decoherence makes an arbitrary-given quasiprobability distribution manifestly positive, thus modelling the emergence of classicality in some sense. The decoherence timescale, the minimum time that quasiprobability distributions of every initial state of the system become nonnegative, is shown to decrease in Hilbert-space dimension, and hence larger quantum systems decohere faster.
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.
The recent two papers have successively proposed the concepts and more precise expressions of strict conservation or statistical conservation of physical quantities, which are different from the expression of a conserved quantity in quantum mechanics. The Hamiltonian operator of the system discussed in this paper are independent of time, so it is commutative with the operators of an angular momentum itself, its component and its square, so that all these quantities are conserved quantities. In an isolated composite system, one of the angular momentum components of the total system are strictly conserved, while the angular momentum components of the subsystems can only be conserved statistically, both of which can be expressed in terms of eigenvalues and eigenfunctions. Generally, angular momentum itself does not have eigenvalues and eigenfunctions, so it cannot be expressed as strict conservation in terms of these concepts. Although there are eigenvalues and eigenfunctions for the square of angular momentum, they cannot express strict conservation either. And they can only be expressed as statistical conservation.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
The complementarity between time and energy, as well as between an angle and a component of angular momentum, is described at three different layers of understanding. The phenomena of super-resolution are readily apparent in the quantum phase representation which also reveals that entanglement is not required. We modify Schwinger's harmonic oscillator model of angular momentum to include the case of photons. Therein, the quantum angle measurement is shown to be equivalent to the measurement of the relative phase between the two oscillators. Two reasonable ways of dealing with degeneracy are shown to correspond to a conditional measurement which takes a snapshot in absolute time (corresponding to adding probability amplitudes) and a marginal measurement which takes an average in absolute time (corresponding to adding probabilities). The sense in which distinguishability is a ``matter of how long we look'' is discussed and the meaning of the general theory is illustrated by taking the two oscillators to be circularly polarized photons. It is shown that an odd number of
x-polarized photons will never have an angle in correspondence with the
y axis, but an even number of
x-polarized photons always can! The behavior of an
x-polarized coherent state is examined and the snapshot angular distributions are seen to evolve into two counter-rotating peaks resulting in considerable correspondence with the
y axis at the time for which a classical linear polarization vector would shrink to zero length. We also demonstrate how the probability distribution of absolute time (herein a measurable quantity, rather than just a parameter) has an influence on how these snapshot angular distributions evolve into a quantum version of the polarization ellipse.