Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Exact phase space localized projectors from energy eigenstates

In investigations of the emergence of classicality from quantum theory, a useful step is the construction of quantum operators corresponding to the classical notion that the system resides in a region of phase space. The simplest such constructions are approximate projection operators. Here, we show how to construct exact projection operators which are localized on regions of phase. We elucidate the properties of such operators and explore their time evolution. For the harmonic oscillator we find sets of phase space localized histories which are exactly decoherent for any initial state and have probability 1 for classical evolution.     

Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity

Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I₂ direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give: Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(A). Received: 6 December 1996 / Accepted: 18 May 1997     

Time-ordering dependence of measurements in teleportation

We trace back the phenomenon of “delayed-choice entanglement swapping” as it was realized in a recent experiment to the commutativity of the projection operators that are involved in the corresponding measurement process. We also propose an experimental set-up which depends on the order of successive measurements corresponding to noncommutative projection operators. In this case entanglement swapping is used to teleport a quantum state from Alice to Bob, where Bob has now the possibility to examine the noncommutativity within the quantum history.     

Non-trivial saddle points and band structure of bound states of the two-dimensional O(N) vector model

We discuss O(N) invariant scalar field theories in 0 + 1 space-time dimensions (quantum mechanics) and in 1 + 1 space-time dimensions (field theory). Combining ordinary “Large N” saddle point techniques and simple properties of the diagonal resolvent of one-dimensional Schrödinger operators we find non-trivial (non-constant) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. In the “Large N” limit these saddle points are exact for the quantum mechanical case, but only approximate in the two-dimensional theory. In the latter case they are the leading contributions to the time evolution kernel at short times, or equivalently, the leading contribution to the high temperature expansion of partition function stemming from space dependent static configurations in case of the Euclidean theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra.     

Consistent Histories in Quantum Cosmology

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 universe 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 superpositions of expanding and contracting universes suggests that a “quantum bounce” is possible i.e. that the wave function of the universe may remain peaked on a non-singular classical solution throughout its history. However, a more careful consistent histories analysis shows that for arbitrary states in the physical Hilbert space the probability of this Wheeler-DeWitt quantum universe encountering the big bang/crunch singularity is equal to unity. A quantum Wheeler-DeWitt universe is inevitably singular, and a “quantum bounce” is thus not possible in these models.     

Timeless Configuration Space and the Emergence of Classical Behavior

The inherent difficulty in talking about quantum decoherence in the context of quantum cosmology is that decoherence requires subsystems, and cosmology is the study of the whole Universe. Consistent histories gave a possible answer to this conundrum, by phrasing decoherence as loss of interference between alternative histories of closed systems. When one can apply Boolean logic to a set of histories, it is deemed ‘consistent’. However, the vast majority of the sets of histories that are merely consistent are blatantly nonclassical in other respects, and further constraints than just consistency need to be invoked. In this paper, I attempt to give an alternative answer to the issues faced by consistent histories, by exploring a timeless interpretation of quantum mechanics of closed systems. This is done solely in terms of path integrals in non-relativistic, timeless, configuration space. What prompts a fresh look at such foundational problems in this context is the advent of multiple gravitational models in which Lorentz symmetry is not fundamental, but only emergent. And what allows this approach to overcome previous barriers to a timeless, conditional probabilities interpretation of quantum mechanics is the new notion of records—made possible by an inherent asymmetry of configuration space. I outline and explore consequences of this approach for foundational issues of quantum mechanics, such as the natural emergence of the Born rule, conservation of probabilities, and the Sleeping Beauty paradox.     

Probability sum rules and consistent quantum histories

An example shows that weak decoherence is more restrictive than the minimal logical decoherence structure that allows probabilities to be used consistently for quantum histories. The probabilities in the sum rules that define minimal decoherence are all calculated by using a projection operator to describe each possibility for the state at each time. Weak decoherence requires more sum rules. They bring in additional variables, that require different measurements and a different way to calculate probabilities, and raise questions of operational meaning. The example shows that extending the linearly positive probability formula from weak to minimal decoherence gives probabilities that are different from those calculated in the usual way using the Born and von Neumann rules and a projection operator at each time.     

Decoherence,correlation, and unstable quantum states in semiclassical cosmology

As almost any S-matrix of quantum theory possesses a set of complex poles (or branch cuts), it is shown using one example that this is the case in quantum field theory in curved space-time. These poles can be transformed into complex eigenvalues, the corresponding eigenvectors being Gamow vectors. This formalism, which is heuristic in ordinary Hilbert space, becomes a rigorous one within the framework of a properly chosen rigged Hilbert space. Then complex eigenvalues produce damping or growing factors and a typical two semigroups structure. It is known that the growth of entropy, decoherence, and the appearance of correlations, occur in the universe evolution, but this fact is demonstrated only under a restricted set of initial conditions. It is proved that the damping factors are mathematical tools that allow one to enlarge the set.     

Spacetime Coarse Grainings and the Problem of Time in the Decoherent Histories Approach to Quantum Theory

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, as a warm up, to deal with the problem of time in reparametrization invariant theories (as for example in canonical quantum gravity) which subsequently we examine. Decoherence conditions and probabilities for those application are derived. The resulting decoherence condition does not depend on the explicit form of the restricted propagator that was problematic for generalizations such as application in quantum cosmology. Closely related to our results, is the problem of tunnelling time as well as the quantum Zeno effect. Some interpretational comments conclude, and we discuss the applicability of this formalism to deal with the arrival time problem and the problem of time in general.     

Quantum Mechanical Histories and the Berry Phase

We elaborate on the distinction between geometric and dynamical phase in quantum theory and we show that the former is intrinsically linked to the quantum mechanical probabilistic structure. In particular, we examine the appearance of the Berry phase in the consistent histories scheme and establish that it is the basic building block of the decoherence functional. These results are consequences of the novel temporal structure of histories-based theories.     

Deformations of Quantum Field Theories and Integrable Models

Deformations of quantum field theories which preserve Poincaré covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an infinite class of explicit examples is constructed on the Borchers-Uhlmann algebra underlying Wightman quantum field theory. These deformations exist independently of the space-time dimension, and contain the recently studied warped convolution deformation as a special case. In the special case of two-dimensional Minkowski space, they can be used to deform free field theories to integrable models with non-trivial S-matrix.     

A self-consistent approach to quantum field theory for extended particles

A notion of quantum space-time is introduced, physically defined as the totality of all flows of quantum test particles in free fall. In quantum space-time the classical notion of deterministic inertial frames is replaced by that of stochastic frames marked by extended particles. The same particles are used both as markers of quantum space-time points as well as natural clocks, each species of quantum test particle thus providing a standard for space-time measurements. In the considered flat-space case, the fluctuations in coordinate values with respect to stochastic frames are described by coordinate probability amplitudes related to irreducible stochastic phase space representations of the Poincaré group. Lagrangian field theory on quantum space-time is formulated. The ensuing equations of motion for interacting fields contain no singularities in their nonlinear terms, and therefore can be handled by methods borrowed from classical nonlinear analysis.Supported in part by an NSERC grant.     

Einstein–Cartan gravity,Asymptotic Safety,and the running Immirzi parameter

In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein–Cartan theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (co-frame) which are invariant under both spacetime diffeomorphisms and local frame rotations. In the first part of the paper we develop a general methodology and corresponding calculational tools which can be used to analyze the flow equation for the pertinent effective average action for any truncation of this theory space. In the second part we apply it to a specific three-dimensional truncated theory space which is parametrized by Newton’s constant, the cosmological constant, and the Immirzi parameter. A comprehensive analysis of their scale dependences is performed, and the possibility of defining an asymptotically safe theory on this hitherto unexplored theory space is investigated. In principle Asymptotic Safety of metric gravity (at least at the level of the effective average action) is neither necessary nor sufficient for Asymptotic Safety on the Einstein–Cartan theory space which might accommodate different “universality classes” of microscopic quantum gravity theories. Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.     

Fundamental principles of quantum theory. II. From a convexity scheme to the DHB theory

Some classical and quantum theories are characterized within the convexity approach to probabilistic physical theories. In particular, the structure of the so-called DHB quantum theory will be analyzed. It turns out that the natural generalization of the standard Hubert space quantum mechanics, the operational one, is such a theory. The operational Hilbert space quantum theory will be reconstructed from the (weak) projection postulate and the complementarity principle. This is then used to argue that the DHB quantum theory is identical with the operational Hilbert space quantum theory.     

Relativistic quantum mechanics over stochastic phase space

Stochastic quantum mechanics is a quantum theory in which the basic limitations of real-world measuring instruments, due to their intrinsically quantum nature, are taken into account. Among other things this leads to a new operational definition of space-time, called quantum space-time. Fundamental to this approach is the formulation of quantum mechanics over phase space rather than just over position or momentum space. A concept of extended particle is a natural outgrowth of this development. Gauge and internal symmetry have a natural place within the theory, and preliminary computations combining some old ideas due to Born with more recent ideas on symmetry breaking suggest that the theory could lead to a mass formula compatible with known data on the low-lying baryons.Supported in part by NSERC Grant, No. A8403.     

Tensor formulation of spin-1 and spin-2 fields

Spin projection operators which constitute a resolution of the identity in the space of second rank tensor wave functions are constructed. These projectors are then used to establish Lagrangian quantum field theories for free massive particles with spin-1 (two equivalent formulations) and spin-2.