In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. We show that the appropriate class operators (the generalization of strings of projectors) in quantum cosmology are readily constructed using a complex potential. We derive the class operator for entering or not entering one or more regions in configuration space. They commute with the Hamiltonian, have a sensible classical limit and are closely related to intersection number operators. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the well-defined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology.
It is shown that $N^2$ is the upper limit for the number of histories in a decohering family of $N$-state quantum system. Simple criterion is found for a family of $N^2$ fine grained decohering histories of Gell-Mann and Hartle to be identical with a family of Griffiths' consistent quantum trajectories.
I give an informal overview of the decoherent histories approach to quantum mechanics, due to Griffiths, to Omnès, and to Gell-Mann and Hartle is given. Results on the connections between decoherence, records, correlation and entropy are described. The emphasis of the presentation is on understanding the broader meaning of the conditions of consistency and decoherence, and in particular, the extent to which they permit one to assign definite properties to the system. The quantum Brownian motion model is briefly discussed. (To appear in proceedings of the workshop, "Stochastic Evolution of Quan…
The predictions of the Bohmian and the decoherent (or consistent) histories formulations of the quantum mechanics of a closed system are compared for histories -- sequences of alternatives at a series of times. For certain kinds of histories, Bohmian mechanics and decoherent histories may both be formulated in the same mathematical framework within which they can be compared. In that framework, Bohmian mechanics and decoherent histories represent a given history by different operators. Their predictions for the probabilities of histories therefore generally differ. However, in an idealized mod…
Eternal inflation, the idea that there is always a part of the universe that is expanding exponentially, is a frequent feature of inflationary models. It has been argued that eternal inflation requires the violation of energy conditions, creating doubts for the validity of such models. We show that eternal inflation is possible without any energy condition violation, highlighting the important role of decoherence and the selection of states in the inflationary process.
The method of decoherent histories allows probabilities to be assigned to sequences of quantum events in systems, such as the universe as a whole, where there is no external observer to make measurements. This paper applies the method of decoherent histories to address cosmological questions. Using a series of simple examples, beginning with the harmonic oscillator, we show that systems in a stationary state such as an energy eigenstate or thermal state can exhibit decoherent histories with non-trivial dynamics. We then examine decoherent histories in a universe that undergoes eternal inflatio…
The consistent histories approach to quantum mechanics is traditionally based on linearly ordered sequences of events. We extend the histories formalism to sets of events whose causal ordering is described by directed acyclic graphs. The need for a global time is eliminated and our construction reflects the causal structure faithfully.
Inspired by quantum cosmology, in which the wave function of the universe is annihilated by the total Hamiltonian, we consider the internal dynamics of a simple particle system in an energy eigenstate. Such a system does not possess a uniquely defined time parameter and all physical questions about it must be posed without reference to time. We consider in particular the question, what is the probability that the system's trajectory passes through a set of regions of configuration space without reference to time? We first consider the classical case, where the answer has a variety of forms in …
The decoherent histories formalism, developed by Griffiths, Gell-Mann, and Hartle is a general framework in which to formulate a timeless, `generalised' quantum theory and extract predictions from it. Recent advances in spin foam models allow for loop gravity to be cast in this framework. In this paper, I propose a decoherence functional for loop gravity and interpret existing results as showing that coarse grained histories follow quasiclassical trajectories in the appropriate limit.
This paper discusses the relation between the decoherent histories approach to quantum mechanics that is based on coarse-grained decoherent histories of a closed system, and the approximate quantum mechanics of measured subsystems, as in the Copenhagen interpretation. We show how the a classical world used in such formulations is not to something to be postulated but rather explained by suitable sets of alternative histories of quasiclassical variables. We discuss the general definition of measurement, the collapse of the wave function, and irreversibility from the perspective of decoherent hi…
We discuss some features of the Decoherent Histories approach. We consider four assumptions, the first three being in our opinion necessary for a sound interpretation of the theory, while the fourth one is accepted by the supporters of the DH approach, and we prove that they lead to a logical contradiction. We discuss the consequences of relaxing any one of them.
We present a formulation of the decoherent (or consistent) histories quantum theory of closed systems starting with records of what histories happen. Alternative routes to a formulation of quantum theory like this one can be useful both for understanding quantum mechanics and for generalizing and extending it to new realms of application and experimental test.
The Klein-Gordon equation is a useful test arena for quantum cosmological models described by the Wheeler-DeWitt equation. We use the decoherent histories approach to quantum theory to obtain the probability that a free relativistic particle crosses a section of spacelike surface. The decoherence functional is constructed using path integral methods with initial states attached using the (positive definite) ``induced'' inner product between solutions to the constraint equation. The notion of crossing a spacelike surface requires some attention, given that the paths in the path integral may cro…
We demonstrate a close connection between the decoherent histories (DH) approach to quantum mechanics and the quantum state diffusion (QSD) picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the QSD picture also define an approximately decoherent set of histories in the DH approach. The degree of localization is related to the degree of decoherence, and the probabilities for histories prescribed by each approach are essentially the same.
The decoherent histories approach to quantum theory is applied to a class of reparametrization invariant models, which includes systems described by the Klein-Gordon equation, and by a minisuperspace Wheeler-DeWitt equation. A key step in this approach is the construction of class operators characterizing the questions of physical interest, such as the probability of the system entering a given region of configuration space without regard to time. In non-relativistic quantum mechanics these class operators are given by time-ordered products of projection operators. But in reparametrization inv…
Decoherent histories quantum theory is reformulated with the assumption that there is one "real" fine-grained history, specified in a preferred complete set of sum-over-histories variables. This real history is described by embedding it in an ensemble of comparable imagined fine-grained histories, not unlike the familiar ensemble of statistical mechanics. These histories are assigned extended probabilities, which can sometimes be negative or greater than one. As we will show, this construction implies that the real history is not completely accessible to experimental or other observational dis…
We use the decoherent histories approach to quantum theory to compute the probability of a non-relativistic particle crossing $x=0$ during an interval of time. For a system consisting of a single non-relativistic particle, histories coarse-grained according to whether or not they pass through spacetime regions are generally not decoherent, except for very special initial states, and thus probabilities cannot be assigned. Decoherence may, however, be achieved by coupling the particle to an environment consisting of a set of harmonic oscillators in a thermal bath. Probabilities for spacetime coa…
I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed systems, up to and including the entire universe. It does not depend on an assumed separation of classical and quantum domains, on notions of measurement, or on collapse of the wave function. Its primary aim is to find sets of histories for closed systems exhibiting negligble interference, and therefore, to which probabilities may be assigned. Such sets of histo…
Quantum open systems are described in the Markovian limit by master equations in Lindblad form. I argue that common ``quantum jumps'' techniques, which solve the master equation by unraveling its evolution into stochastic trajectories in Hilbert space, correspond closely to a particular set of decoherent histories. This is illustrated by a simple model of a photon counting experiment.
The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of N items requires Ω(N) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in O{\sqrt{N}} queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alte…
We use the decoherent histories approach to quantum theory to derive the form of an effective theory describing the coupling of classical and quantum variables. The derivation is carried out for a system consisting of a large particle coupled to a small particle with the important additional feature that the large particle is also coupled to a thermal environment producing the decoherence necessary for classicality. The effective theory is obtained by tracing out both the environment and the small particle variables. It consists of a formula for the probabilities of a set of histories of the l…
We analyse the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival, in which one associates the usual Markovian master equation for the density operator with a class of stochastic non-linear Schrödinger equations. We find stationary solutions to the Ito equation which are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show that every initial state approaches these stationary solutions in the long time…
Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying oper…
The theory of decoherent histories is checked for the requirement of statistical independence of subsystems. Strikingly, this is satisfied only when the decoherence functional is diagonal in both its real a n d imaginary parts. In particular, the condition of consistency (or weak decoherence) required for the assignment of probabilities appears to be ruled out. The same conclusion is obtained independently, by claiming a plausible dynamical robustness of decoherent histories.
We investigate the possibility of assigning consistent probabilities to sets of histories characterized by whether they enter a particular subspace of the Hilbert space of a closed system during a given time interval. In particular we investigate the case that this subspace is a region of the configuration space. This corresponds to a particular class of coarse grainings of spacetime regions. We consider the arrival time problem and the problem of time in reparametrization invariant theories as for example in canonical quantum gravity. Decoherence conditions and probabilities for those applica…
We study and compare the decoherent histories approach, the environment-induced decoherence and the localization properties of thesolutions to the stochastic Schrödinger equation in quantum jump simulationand quantum state diffusion approaches, for a quantum two-level system model. We show, in particular, that there is a close connection between the decoherent histories and the quantum jump simulation, complementing a connection with the quantum state diffusion approach noted earlier by Diósi, Gisin, Halliwell and Percival. In the case of the decoherent histories analysis, the degree of approx…
The quantum mechanics of closed systems such as the universe is formulated using an extension of familiar probability theory that incorporates negative probabilities. Probabilities must be positive for sets of alternative histories that are the basis of fair settleable bets. However, in quantum mechanics there are sets of alternative histories that can be described but which cannot be the basis for fair settleable bets. Members of such sets can be assigned extended probabilities that are sometimes negative. A prescription for extended probabilities is introduced that assigns extended probabili…
We attempt a justification of a generalisation of the consistent histories programme using a notion of probability that is valid for all complete sets of history propositions. This consists of introducing Cox's axioms of probability theory and showing that our candidate notion of probability obeys them. We also give a generalisation of Bayes' theorem and comment upon how Bayesianism should be useful for the quantum gravity/cosmology programmes.
We investigate the idea that different interpretations of quantum mechanics can be seen as restrictions of the consistent (or decoherent) histories quantum mechanics of closed systems to particular classes of histories,together with the approximations and descriptions of these histories that the restrictions permit. (Talk at the the workshop Quo Vadis Quantum Mechanics, Center for Frontier Sciences, Temple University, Philadelphia, PA, September 24--27, 2002.)
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. We consider histories characterized by position samplings at $n$ moments of time. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, $F = m \ddot x + V'(x) $, at $n-2$ times. We find that it is peaked around $F=0$; thus classical correlations are exhibited. We show that the width of the peak $ ΔF$ is l…
We introduce a condition for the strong decoherence of a set of alternative histories of a closed quantum-mechanical system such as the universe. The condition applies, for a pure initial state, to sets of homogeneous histories that are chains of projections, generally branch-dependent. Strong decoherence implies the consistency of probability sum rules but not every set of consistent or even medium decoherent histories is strongly decoherent. Two conditions characterize a strongly decoherent set of histories: (1) At any time the operators that effectively commute with generalized records of h…
A model quantum cosmology is used to illustrate how arrows of time emerge in a universe governed by a time-neutral dynamical theory constrained by time asymmetric initial and final boundary conditions represented by initial and final density matrices. In a quantum universe universe arrows of time are described by the probabilities of appropriately coarse grained sets of histories of quantities like entropy that grow or decay. We show that the requirement of that these sets of histories decohere implies two things: (1) A time asymmetry between initial and final conditions that is a basis for ar…
Nonnegative probabilities that obey the sum rules may be assigned to a much wider family of sets of histories than decohering histories. The resulting {\it linearly positive histories} avoid the highly restrictive decoherence conditions and yet give the same probabilities when those conditions apply. Thus linearly positive histories are a broad extension of decohering histories. Moreover, the resulting theory is manifestly time-reversal invariant.
From data in the present we can predict the future and retrodict the past. These predictions and retrodictions are for histories -- most simply time sequences of events. Quantum mechanics gives probabilities for individual histories in a decoherent set of alternative histories. This paper discusses several issues connected with the distinction between prediction and retrodiction in quantum cosmology: the difference between classical and quantum retrodiction, the permanence of the past, why we predict the future but remember the past, the nature and utility of reconstructing the past(s), and in…
We continue our efforts to understand, within the framework of the quantum mechanics of the universe as a whole, the quasiclassical realm of familiar experience as a feature emergent from the Hamiltonian of the elementary particles and the initial condition of the universe. Quantum mechanics assigns probabilities to exhaustive sets of alternative decoherent histories of the universe. We introduce and define the notion of strong decoherence. We replace the notion of maximal sets of alternative decohering histories by defining the more useful concept of "full" sets of alternative strongly decohe…
We review the application of the consistent (or decoherent) histories formulation of quantum theory to canonical loop quantum cosmology. Conventional quantum theory relies crucially on "measurements" to convert unrealized quantum potentialities into physical outcomes that can be assigned probabilities. In the early universe and other physical contexts in which there are no observers or measuring apparatus (or indeed, in any closed quantum system), what criteria determine which alternative outcomes may be realized and what their probabilities are? In the consistent histories formulation it is t…
In a quantum theory of gravity spacetime behaves classically when quantum probabilities are high for histories of geometry and field that are correlated in time by the Einstein equation. Probabilities follow from the quantum state. This quantum perspective on classicality has important implications: (a) Classical histories are generally available only in limited patches of the configuration space on which the state lives. (b) In a given patch states generally predict relative probabilities for an ensemble of possible classical histories. (c) In between patches classical predictability breaks d…
This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed a…
In this paper we are interested in the studying coarse-graining in field theories using the language of quantum open systems. Motivated by the ideas of Calzetta and Hu on correlation histories we employ the Zwanzig projection technique to obtain evolution equations for relevant observables in self-interacting scalar field theories. Our coarse-graining operation consists in concentrating solely on the evolution of the correlation functions of degree less than $n$, a treatment which corresponds to the familiar from statistical mechanics truncation of the BBKGY hierarchy at the n-th level. We der…
We use a $λΦ^4$ scalar quantum field theory to illustrate a new approach to the study of quantum to classical transition. In this approach, the decoherence functional is employed to assign probabilities to consistent histories defined in terms of correlations among the fields at separate points, rather than the field itself. We present expressions for the quantum amplitudes associated with such histories, as well as for the decoherence functional between two of them. The dynamics of an individual consistent history may be described by a Langevin-type equation, which we derive. \noindent {\it D…
Assignment of consistent quantum probabilities to events in a quantum universe is a fundamental challenge which every quantum cosmology/gravity framework must overcome. In loop quantum cosmology, this issue leads to a fundamental question: What is the probability that the universe undergoes a non-singular bounce? Using the consistent histories formulation, this question was successfully answered recently by the authors for a spatially flat FRW model in the canonical approach. In this manuscript, we obtain a covariant generalization of this result. Our analysis is based on expressing loop quant…
A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such universes is exhibited. We show how the probabilities of decoherent sets of alternative, coarse-grained histories of these model universes can be calculated. We consider in particular the probabilities for classical evolution defined by a suitable coarse-graining. For a restricted class of initial conditions and coarse grainings we exhibit the a…
Approximate Decoherence, Recoherence and Records in Isolated Quantum Systems
Instituto de Física de Cantabria (IFCA), Universidad de Cantabria–CSIC（坎塔布里亚物理研究所（IFCA），坎塔布里亚大学-西班牙国家研究委员会）；Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona（巴塞罗那自治大学物理系理论物理：信息与量子现象）；Department of Mathematics/Computer Science/Physics, University of Osnabrück（奥斯纳布吕克大学数学/计算机科学/物理系）；Department Mathematik/Informatik–Abteilung Informatik, Universität zu Köln（科隆大学数学/信息学系-信息学部）；ICREA – Institució Catalana de Recerca i Estudis Avançats（加泰罗尼亚高级研究与教育学院）
In this paper we elaborate on the structure of the continuous-time histories description of quantum theory, which stems from the consistent histories scheme. In particular, we examine the construction of history Hilbert space, the identification of history observables and the form of the decoherence functional (the object that contains the probability information). It is shown how the latter is equivalent to the closed-time-path (CTP) generating functional. We also study the phase space structure of the theory first through the construction of general representations of the history group (the …
We consider the problem of extracting physical predictions from the wave function of the universe in quantum cosmological models. We state the features of quantum cosmology an interpretational scheme should confront. We discuss the Everett interpretation, and extensions of it, and their application to quantum cosmology. We review the steps that are normally taken in the process of extracting predictions from solutions to the Wheeler-DeWitt equation for quantum cosmological models. Some difficulties and their possible resolution are discussed. We conclude that the usual wave function-based appr…
A fundamental issue for any quantum cosmological theory is to specify how probabilities can be assigned to various quantum events or sequences of events such as the occurrence of singularities or bounces. In previous work, we have demonstrated how this issue can be successfully addressed within the consistent histories approach to quantum theory for Wheeler-DeWitt-quantized cosmological models. In this work, we generalize that analysis to the exactly solvable loop quantization of a spatially flat, homogeneous and isotropic cosmology sourced with a massless, minimally coupled scalar field known…
We investigate the quantum theory of closed systems based on the linear positivity decoherence condition of Goldstein and Page. A quantum theory of closed systems requires two elements; 1) a condition specifying which sets of histories may be assigned probabilities that are consistent with the rules of probability theory, and 2) a rule for those probabilities. The linear positivity condition of Goldstein and Page is the weakest of the general conditions proposed so far. Its general properties relating to exact probability sum rules, time-neutrality, and conservation laws are explored. Its inco…
We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to stu…
The evolution of the universe is determined by its quantum state. The wave function of the universe obeys the constraints of general relativity and in particular the Wheeler-DeWitt equation (WDWE). For non-zero Λ, we show that solutions of the WDWE at large volume have two domains in which geometries and fields are asymptotically real. In one the histories are Euclidean asymptotically anti-de Sitter, in the other they are Lorentzian asymptotically classical de Sitter. Further, the universal complex semiclassical asymptotic structure of solutions of the WDWE implies that the leading order in \h…
The Consistent Histories (CH) formalism aims at a quantum mechanical framework which could be applied even to the universe as a whole. CH stresses the importance of histories for quantum mechanics, as opposed to measurements, and maintains that a satisfactory formulation of quantum mechanics allows one to assign probabilities to alternative histories of a quantum system. It further proposes that each realm, that is, each set of histories to which probabilities can be assigned, provides a valid quantum-mechanical account, but that different realms can be mutually incompatible. Finally, some of …
We introduce a framework for studying non-locality and contextuality inspired by the path integral formulation of quantum theory. We prove that the existence of a strongly positive joint quantum measure -- the quantum analogue of a joint probability measure -- on a set of experimental probabilities implies the Navascues-Pironio-Acin (NPA) condition $Q^1$ and is implied by the stronger NPA condition $Q^{1+AB}$. A related condition is shown to be equivalent to $Q^{1+AB}$.
We consider notions of physical equivalence of sets of histories in the quantum mechanics of a closed system. We show first how the same set of histories can be relabeled in various ways, including the use of the Heisenberg equations of motion and of passive transformations of field variables. In the the usual approximate quantum mechanics of a measured subsystem, two observables re- presented by different Hermitian operators are physically distinguished by the different apparatus used to measure them. In the quantum mechanics of a closed system, however, any apparatus is part of the system an…
Recently it has been claimed that the Wheeler-DeWitt quantization of gravity is unable to avoid cosmological singularities. However, in order to make this assertion, one must specify the underlying interpretation of quantum mechanics which has been adopted. For instance, several nonsingular models were obtained in Wheeler-DeWitt quantum cosmology in the framework of the de Broglie-Bohm quantum theory. Conversely, there are specific situations where the singularity cannot be avoided in the framework of the Consistent Histories approach to quantum mechanics. In these specific situations, the mat…
A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for whic…
Classical mechanics and standard Copenhagen quantum mechanics respect subspace implications. For example, if a particle is confined in a particular region $R$ of space, then in these theories we can deduce that it is confined in regions containing $R$. However, subspace implications are generally violated by versions of quantum theory that assign probabilities to histories, such as the consistent histories approach. I define here a new criterion, ordered consistency, which refines the criterion of consistency and has the property that inferences made by ordered consistent sets do not violate s…
We illustrate the crucial role played by decoherence (consistency of quantum histories) in extracting consistent quantum probabilities for alternative histories in quantum cosmology. Specifically, within a Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmological model sourced with a free massless scalar field, we calculate the probability that the univese is singular in the sense that it assumes zero volume. Classical solutions of this model are a disjoint set of expanding and contracting singular branches. A naive assessment of the behavior of quantum states which are sup…
We sketch a quantum mechanical framework for the universe as a whole. Within that framework we propose a program for describing the ultimate origin in quantum cosmology of the quasiclassical domain of familiar experience and for characterizing the process of measurement. Predictions in quantum mechanics are made from probabilities for sets of alternative histories. Probabilities can be assigned only to sets of histories that approximately decohere. Decoherence is defined and the mechanism of decoherence is reviewed. Decoherence requires a sufficiently coarse-grained description of alternative …
Probabilities for observations in cosmology are conditioned both on the universe's quantum state and on local data specifying the observational situation. We show the quantum state defines a measure for prediction through such conditional probabilities that is well behaved for spatially large or infinite universes when the probabilities that our data is replicated are taken into account. In histories where our data are rare volume weighting connects top-down probabilities conditioned on both the data and the quantum state to the bottom-up probabilities conditioned on the quantum state alone. W…
Measurement is a fundamental notion in the usual approximate quantum mechanics of measured subsystems. Probabilities are predicted for the outcomes of measurements. State vectors evolve unitarily in between measurements and by reduction of the state vector at measurements. Probabilities are computed by summing the squares of amplitudes over alternatives which could have been measured but weren't. Measurements are limited by uncertainty principles and by other restrictions arising from the principles of quantum mechanics. This essay examines the extent to which those features of the quantum mec…
