Non-unitary Evolution Research Articles

Robustness and diffusion of pointer states

Classical properties of an open quantum system emerge through its interaction with other degrees of freedom (decoherence). We treat the case where this interaction produces a Markovian master equation for the system. We derive the corresponding distinguished local basis (pointer basis) by three methods. The first demands that the pointer states mimic as closely as possible the local nonunitary evolution. The second demands that the local entropy production be minimal. The third imposes robustness on the inherent quantum and emerging classical uncertainties. All three methods lead to localized Gaussian pointer states, their formation and diffusion being governed by well-defined quantum Langevin equations.

Higher Order Quantum Onsager Coefficients from Dynamical Invariants

Functionals representing dynamical invariants under unitary quantum dynamics of open systems are used to derive Onsager coefficients for entropy production in irreversible processes if the nonunitary time evolution is determined by quantum dynamical semigroups. The procedure allows a derivation from first principles of the quantum analogue to the classical case.

Vacuum structure for expanding geometry

We consider gravitational wave modes in the Friedmann-Robertson-Walker metrics in a de Sitter phase and show that the state space splits into many unitarily inequivalent representations of the canonical commutation relations. Non-unitary time evolution is described as a trajectory in the space of the representations. The generator of time evolution is related to the entropy operator. The thermodynamic arrow of time is shown to point in the same direction as the cosmological arrow of time. The vacuum is a two-mode SU(1,1) squeezed state of thermo-field dynamics. The link between an expanding geometry, squeezing and thermal properties is exhibited.

Unitary and nonunitary evolution in quantum cosmology

We analyse when and why unitarity violations might occur in quantum cosmology restricted to minisuperspace. To this end we discuss in detail backscattering transitions between expanding and contracting solutions of the Wheeler-DeWitt equation. We first show that upon neglecting only backscattering, one obtains an intermediate regime in which matter evolves unitarily but which does not correspond to any Schr\"odinger equation in a given geometry since gravitational backreaction effects are taken into account at the quantum level. We then show that backscattering amplitudes are exponentially smaller than matter transition amplitudes. Both results follow from an adiabatic treatment valid for macroscopic universes. To understand how backscattering and the intermediate regime should be interpreted, we review the problem of electronic transitions induced by nuclear motion since it is mathematically very similar. In this problem, transition amplitudes are obtained from the conserved current. The same applies to quantum cosmology and shows that the unique consistent interpretation is based on the current when backscattering is neglected. We then review why, in a relativistic context, backscattering is interpreted as pair production whereas it is not in the non relativistic case. In each example the correct interpretation is obtained by coupling the system to an external quantum device. From the absence of such external systems in cosmology, we conclude that backscattering does not have a unique consistent interpretation in quantum cosmology.

Canonical quantization and expanding metrics

The operator formalism for expanding metrics scenarios is presented. Non-unitary time evolution implied by expanding geometry is described as a trajectory over unitarily inequivalent representations at different times of the canonical commutation relations.

QUANTUM CHRONOTOPOLOGY AND TIME ASYMMETRY IN FIELD THEORY

A new, physical definition of the time elements is mathematically given. Their union constitutes a disconnected time topological space, [Formula: see text]. Based on the properties of this time–space and using the theory of random quantum fields previously developed, a nonunitary evolution operator, [Formula: see text], is derived. [Formula: see text] breaks down into two disjunct, spontaneously renormalized evolution operators by quantizing the random field action integral: one operator is unitary, [Formula: see text], and the other non-measure-preserving, [Formula: see text]. [Formula: see text] coincides, apart from the renormalization, with the evolution operator of the standard QFT. U ump (τ) breaks time symmetry in QFT. It reconciles the time reversal invariance of the basic equations of quantum theory with the irreversibility of the macroscopic, and some microscopic, phenomena. It provides a basis for the CP violation in the neutral K0 meson decay. Functional integrals arising in the theory have in the limit countably additive measures.

Jaynes-Cummings model under continuous measurement: Weak chaos in a quantum system induced by unitarity collapse

We show that a system managed by quantum theory bears chaotic behavior induced by unitarity collapse in its time development. Nonunitary time evolution of the Jaynes-Cummings (JC) model, the interacting system consisting of two subsystems (a two-level atom and a quantized photon field), under a continuous quantum-nondemolition (QND) measurement of the photon number is investigated. In the regime of weak coupling between the subsystems, the measured system shows the Rabi oscillation and the decrease of the photon number variance, which mimic the unmeasured JC system and the photon field under the continuous QND photodetection, respectively. In the strong-coupling regime, the quantum system shows a nonintegrable nature, that is, it yields weak chaos characterized by both the broad continuous power spectrum and the decaying correlation function due to the lack of quantum recurrence.

Geometry of the Hilbert space and the quantum Zeno effect

We show that the quadratic short time behaviour of transition probability is a natural consequence of the inner product of the Hilbert space of the quantum system. We prove that Schr\"odinger time evolution between two successive measurements is not a necessary but only a sufficient condition for predicting quantum Zeno effect. We provide a relation between the survival probability and the underlying geometric structure such as the Fubini-Study metric defined on the projective Hilbert space of the quantum system. This predicts the quantum Zeno effect even for systems described by non-linear and non-unitary evolution equations, within the collapse mechanism of the wavefunction during measurement process. Two examples are studied, one is non-linear Schr\"odinger equation and other is Gisin's equation and it is shown that one can observe quantum Zeno effect for systems described by these equations.

Testing causality violation on spacetimes with closed timelike curves

Generalized quantum mechanics is used to examine a simple two-particle scattering experiment in which there is a bounded region of closed timelike curves (CTCs) in the experiment's future. The transitional probability is shown to depend on the existence and distribution of the CTCs. The effect is therefore acausal, since the CTCs are in the experiment's causal future. The effect is due to the non-unitary evolution of the pre- and post-scattering particles as they pass through the region of CTCs. We use the time-machine spacetime developed by Politzer [1], in which CTCs are formed due to the identification of a single spatial region at one time with the same region at another time. For certain initial data, the total cross-section of a scattering experiment is shown to deviate from the standard value (the value predicted if no CTCs existed). It is shown that if the time machines are small, sparsely distributed, or far away, then the deviation in the total cross-section may be negligible as compared to the experimental error of even the most accurate measurements of cross-sections. For a spacetime with CTCs at all points, or one where microscopic time machines pervade the spacetime in the final moments before the big crunch, the total cross-section is shown to agree with the standard result (no CTCs) due to a cancellation effect.

Dynamical origin of the quantum Zeno effect

The quantum Zeno effect is often studied and understood in term of nonunitary evolutions, involving projections a la von Neumann (measurements). We propose a dynamical explanation of this effect, which involves only unitary operators. The limit of infinitely frequent measurements is critically discussed: it is unphysical, yet interesting and peculiar.

Initial conditions in quasi-classical field theory

We investigate the problem of divergences and renormalizations in the Hamiltonian formalism of quasiclassical field theory. This approach is known to involve divergences in the leading term of the expansion. Proposals have been made to eliminate the divergences by using nonequivalent representations of the canonical commutation relations at different moments of time. In this paper, we consider the Schrödinger equation with ultraviolet and infrared cutoffs. In order to remove the cutoffs, conditions are imposed on the initial state of the regularized theory in addition to the conditions imposed on the counterterms in the Hamiltonian. In the leading order of the quasi-classical expansion, we give the explicit form of these conditions, which is invariant under the evolution. This allows us to show that this approximation does not require the introduction of nonunitary evolution transformations.

Spontaneous radiation of free electrons in a nonrelativistic collapse model

One way to solve the long-standing ``measurement''problem in quantum mechanics is to modify Schr\"odinger's equation so that the collapse of a state vector is the result of some intrinsic physical dynamics. The introduction of such collapse dynamics often results in energy nonconservation. In this paper, we first derive a general expression for the rate of change in the expectation value of energy in a general collapse model that supports a linear evolution of the density matrix. In particular, we show, under certain plausible assumptions, that energy nonconservation is an inevitable consequence of collapse. We then work with a specific model, namely, the nonrelativistic continuous spontaneous localization (CSL) model, to derive further consequences of collapse. In CSL, in a nonunitary evolution, a particle interacts with a fluctuating scalar field, leading to the collapse of the state vector. One consequence of this interaction is that a free electron can radiate spontaneously. We calculate the spectrum of this radiation. The result is then compared with the observed upper bound on spontaneous radiation in Ge and a constraint on the parameters of CSL is obtained.

Operator spectral states

In chemical physics, especially quantum mechanics, one encounters a myriad of phenomenological models based upon formal operator manipulations and calculations leading to spectral conclusions. Yet often the exact nature of such spectral results is unclear mathematically. On the other side, mathematical clarification can lead to greater physical insight. Two important illustrations of this symbiotic relationship are von Neumann's placing of the quantum mechanics of Schrödinger, Heisenberg, and others, into a Hilbert space framework, and Schwartz's rendering of the Dirac ket and delta notation into the theory of distributions, some thirty years later. With chemical physics now actively treating time-symmetry breakings into nonunitary evolutions, complex resonances, computations concerning the spectrum as caused by underlying chaotic dynamical systems, it seems useful to gather together here elements of operator spectral theory to provide a mathematical framework into which such spectral results can be placed or from which they may be viewed. A particularly useful, clarifying, but little-known device, is that of operator state diagram, which I will develop rather fully in this paper. All operator dualities and hence much of operator theory may be systematically portrayed by these diagrams. From these diagrams, I then discuss operator spectral states, scattering states available from the essential spectrum of an operator, spectral states of shift operators, and rigged Hilbert space spectral states. Only a few examples can be given here, for the rigorous working out of the state diagrams of a given new class of operators requires some care. Nonetheless, I hope that I will have provided some helpful clarifications of the instances given. Once the approach is seen, it is clear what needs to be done for other applications. Finally, I turn to what I shall call regular spectral states and probabilistic spectral states. These are new notions. Briefly, my view is that nature prefers regular spectral states, whereas we (humankind) prefer more specific spectral states. To reinforce this ansatz, I connect it to all other sections of this paper. A rule of probabilistically preferred spectral states is seen as an instance of the more general regularization principle which is an extensive irreversibility law. I include in each section a principle. This is meant to help the beginner come away with useful general notions, and it also permits one to express personal viewpoints.

Comment on “Spacetime information”

A recent paper by Hartle [Phys. Rev. D 51, 1800 (1995)] proposes a definition of "spacetime information" - the information available about a quantum system's boundary conditions in the various sets of decohering histories it may display - and investigates its properties. We note here that Hartle's analysis contains errors which invalidate several of the conclusions. In particular, the proof that the proposed definition agrees with the standard definition for ordinary quantum mechanics is invalid, the evaluations of the spacetime information for time-neutral generalized quantum theories and for generalized quantum theories with non-unitary evolution are incorrect, and the argument that spacetime information is conserved on spacelike surfaces in these last theories is erroneous. We show however that the proposed definition does, in fact, agree with the standard definition for ordinary quantum mechanics. Hartle's definition relies on choosing, case by case, a class of fine-grained consistent sets of histories. We supply a general definition of the relevant class of sets that agrees with Hartle's definition in the cases explicitly considered and that generalizes to other cases.

Completely positive dynamical maps and the neutral kaon system

The evolution of quantum irreversible processes can be effectively described using dynamical semigroups. They involve non-unitary time evolutions of density matrices that are generated by completely positive maps. In the case of the neutral kaon system such an approach leads to a modification of the standard Weisskopf-Wigner evolution, characterized by six phenomenological parameters. Bounds on the magnitude of these parameters could be given using available experimental data. In this respect, it is remarkable that the condition of complete positivity might be put to a precise test in future more refined experiments.

Complete positivity and the [formula omitted] system

Models that provide experimentally testable violations of ordinary quantum mechanics have been recently proposed. These models are based on non-unitary time evolutions of density matrices that are generated by linear positive maps. We discuss the consequences of imposing a stronger condition on those maps, known as complete positivity. It turns out that experimental data on the neutral kaon system giving upper bounds to the parameters characterizing positive maps, also give bounds to those determining completely positive ones.

Berry's phase and Aharonov-Anandan's phase

The Aharonov-Anandan geometric phase is generalized to non-unitary evolution, and is shown to be always real. By using a counter-example, which is exactly solvable, it is shown that Berry's geometric phase is not always the adiabatic limit of Aharonov-Anandan's geometric phase for a non-Hermitian driving Hamiltonion. The reason is then given.

Classical and quantum initial value problems for models of chronology violation.

We study the classical and quantum theory of a class of nonlinear differential equations on chronology violating spacetime models in which space consists of finitely many discrete points. Classically, in the linear and weakly nonlinear regimes (for generic choices of parameters) we prove existence and uniqueness of solutions corresponding to initial data specified before the dischronal region; however, uniqueness (but not existence) fails in the strongly coupled regime. The evolution preserves the symplectic structure. The quantum theory is approached via the quantum initial value problem (QIVP); that is, by seeking operator-valued solutions to the equation of motion with initial data representing the canonical (anti)commutation relations. Using normal operator ordering, we construct solutions to the QIVP for both Bose and Fermi statistics (again for generic choice of parameters) and prove that these solutions are unique. For models with two spatial points, the resulting evolution is unitary; however, for a more general model the evolution fails to preserve the (anti)commutation relations and is therefore nonunitary. We show that this nonunitary evolution cannot be described using a superscattering operator with the usual properties. We present numerical evidence to show that the bosonic quantum theory can pick out a unique classical limit for certain ranges of the coupling strength, even when there are many classical solutions. We show that the quantum theory depends strongly on the choice of operator ordering. In addition, we show that our results differ from those obtained using the ``self-consistent path integral''. It follows that the path integral evolution does not correspond to a solution of the equation of motion.

Decoherence and classical predictability of phase-space histories.

We consider the decoherence of phase-space histories in quantum Brownian motion models, consisting of a particle moving under a potential $V(x)$ in contact with a heat bath of temperature $T$ and dissipation constant $\ensuremath{\gamma}$ in the Markovian regime. The phase-space histories are described by quasiprojectors consisting of Gaussian density matrices smeared over large phase-space cells and are characterized by the size [$\ensuremath{\Gamma}$] of the phase-space cell together with the size [$M$] of the margin (the region at the boundary of $\ensuremath{\Gamma}$ in which the Weyl symbol of the projector goes from 1 to 0). We generalize earlier results of Hagedorn to show that an initial Gaussian density matrix remains approximately Gaussian under nonunitary time evolution deriving a bound giving the validity of this approximation. Following the earlier work of Omnes [J. Stat. Phys. 51, 351 (1989)] we use this result to compute the time evolution of the phase-space projectors under the master equation and show that histories of phase-space samplings approximately decohere, and that the probabilities for these histories are peaked about classical dissipative evolution, but with an element of unpredictability which is reflected in the increase of the size of the margin.

Unitarity of quantum theory and closed timelike curves.

Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTC's) are subject to a nonunitary evolution X. Recently, a prescription has been proposed that restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more nonunitary evolutions. We propose an alternative methd by which unitarity of the evolution may be regained by extending X to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localized in regions spacelike separated form the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal.