Permanent spatial decomposition (PSD) is the (hypothesized) property of the wave function of a macroscopic system of decomposing into localized permanently non-overlapping parts when it spreads over a macroscopic region. The typical example of this phenomenon is the measurement process, in which the wave function of the laboratory (quantum system + apparatus + environment) decomposes into n parts, corresponding to the n outcomes of the measurement: the parts are non-overlapping, because they represent a macroscopic pointer in different positions, and they are permanently non-overlapping due to the irreversible interaction with the environment. PSD is often mentioned in the literature, but until now no formal definition or systematic study of this phenomenon has been undertaken. The aim of this paper is to partially fill this gap by giving a formal definition of PSD and studying its possible connection with scattering theory. The predictive and explanatory powers of this phenomenon are also discussed and compared with those of Bohmian mechanics.
Permanent spatial decomposition (PSD) is the (hypothesized) property of the wave function of a macroscopic system of decomposing into localized permanently non-overlapping parts when it spreads over a macroscopic region. The typical example of this phenomenon is the measurement process, in which the wave function of the laboratory (quantum system + apparatus + environment) decomposes into n parts, corresponding to the n outcomes of the measurement: the parts are non-overlapping, because they represent a macroscopic pointer in different positions, and they are permanently non-overlapping due to the irreversible interaction with the environment. PSD is often mentioned in the literature, but until now no formal definition or systematic study of this phenomenon has been undertaken. The aim of this paper is to partially fill this gap by giving a formal definition of PSD and studying its possible connection with scattering theory. The predictive and explanatory powers of this phenomenon are also discussed and compared with those of Bohmian mechanics.
The preferred basis problem is mentioned in the literature in connection with the measurement problem and with the Many World Interpretation. It is argued that this problem actually corresponds to two inequivalent problems: (i) the preferred-decomposition problem, i.e., what singles out a preferred decomposition of a suitable state vector of a system as the sum of a finite or countable set of vectors?, and (ii) the preferred-representation problem, i.e., what singles out a preferred representation for the Hilbert space of a system? In this paper the preferred-decomposition problem is addressed and two processes, namely decoherence and permanent spatial decomposition (PSD), are examined and compared as possible solutions to this problem. It is shown that, perhaps contrary to common belief, in realistic situations decoherence is not sufficient to solve the preferred-decomposition problem. PSD is the (hypothesized) tendency of the wave function of the universe to decompose into permanently non-overlapping wave packets. Three phases can be roughly identified as composing PSD: Microscopic decomposition, amplification, and interaction with the environment. Decoherence theory considers only the interaction with the environment and ignores the first two phases. For this reason PSD is fundamentally different from decoherence and, unlike decoherence, provides a simple and non-elusive solution to the preferred-decomposition problem.
One of the main goals of no-collapse interpretations of quantum mechanics is to show how results of measurements are objectified, or, in a more poignant terminology, how in a measurement situation facts arise. One needs to show how in their dynamical evolution the measured system and, perhaps more importantly, the measuring apparatus acquire, or at least appear to acquire, the definite properties that are otherwise explained by the collapse of the state. This can be seen as a special case of the problem whether quantum mechanics alone can account for the classical behaviour of macroscopic systems; in particular, the permanent definiteness of at least those properties of macroscopic systems that are directly observable, such as the position of the pointer of a measuring apparatus. In other words, whether quantum mechanics can account for macrofacts of the familiar kind.
In the measurement process, certain degrees of freedom of the observables are selected and correlations are produced with corresponding degrees of freedom of the measuring macroscopic device. These correlations are described, in the present paper, by using the Schmidt decomposition, in which the eigenvalues of the reduced density matrices of these two systems are equal. The measurement process is related to the trace operation over the observable degrees of freedom, since we do not directly measure the microscopic system but its corresponding macroscopic part in the Schmidt decomposition. This trace operation leads only to partial information on the microscopic system referred to as the “collapse” of the wave function. This interpretation is demonstrated by analyzing Stern-Gerlach experiments.
When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops "branches": a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.
We systematically analyze Holland's local expectation values within Bohmian mechanics, referring to them as weak actual values to emphasize their connection with weak measurement theory. We derive the exact time evolution equation for these quantities along a Bohmian trajectory and formally establish their correspondence with the real part of the weak value under position postselection. The explanatory power of this framework is demonstrated by revisiting a recent waveguide experiment: we show that the measured quantity corresponds to the imaginary part of the momentum weak value, which reflects the spatial decay of the wave field, not the particle velocity. This cleanly clarifies the distinct physical roles of the real and imaginary parts of weak values.
We show how the correlations of a quantum system with other quantum systems may cause one of its observables to behave in a classical manner. In particular, "reduction of the wave packet," postulated by von Neumann to explain definiteness of an outcome of an individual observation, can be explained when a realistic model of an apparatus is adopted. Instead of an isolated quantum apparatus with a number of states equal to the number of possible distinct outcomes of the measurement, discussed by von Neumann, we consider an apparatus interacting with other physical systems, described here summarily as "environment." The interaction of the quantum apparatus with the environment results in correlations. Correlations impose effective superselection rules which prevent apparatus from appearing in a superposition of states corresponding to different eigenvalues of the privileged pointer observable. It is the propagation of the correlations with the pointer basis states which is ultimately responsible for the choice of the pointer observable. It can be thought of as a process of amplification in which the state of many distinct physical systems becomes correlated with the pointer basis state. Whether these environment systems are regarded as a part of the apparatus setup, or as a part of its environment is irrelevant. What is crucial is the redundancy of the record concerning the pointer observable which is imprinted into the correlations. Eigenspaces of the pointer observable provide a natural basis for the pointer of the quantum apparatus and determine the to-be-measured observable of the quantum system. Decay of those elements of the apparatus-system density matrix, which are off-diagonal in the pointer observable, is caused by the natural evolution of the combined system-apparatus-environment object. For a hypothetical finite environment with N distinct eigenvalues of the apparatus-environment interaction Hamiltonian, off-diagonal terms will decay to become of the order of ${N}^{\ensuremath{-}\frac{1}{2}}$, and will recur only on a Poincar\'e time scale. No recurrences will be observed in realistic circumstances. As the correlations spread through the environment on a time scale typically much shorter than the recurrence time scale calculated for the environment already correlated with the pointer observable, the measurement becomes effectively irreversible. Relevance of this model of the measurement process for the understanding of the second law of thermodynamics and its relation to Bohr's "irreversible act of amplification" is briefly discussed. The emergence of the pointer observable can be interpreted as a clue about the resolution of the measurement problem in case of no environment. It points towards the possibility that properties of quantum systems have no absolute meaning. Rather, they must be always characterized with respect to other physical systems.