Violation Of Positivity Research Articles

Identification of in-sample positivity violations using regression trees: The PoRT algorithm

Abstract Background The positivity assumption is crucial when drawing causal inferences from observational studies, but it is often overlooked in practice. A violation of positivity occurs when the sample contains a subgroup of individuals with an extreme relative frequency of experiencing one of the levels of exposure. To correctly estimate the causal effect, we must identify such individuals. For this purpose, we suggest a regression tree-based algorithm. Development Based on a succession of regression trees, the algorithm searches for combinations of covariate levels that result in subgroups of individuals with a low (un)exposed relative frequency. Application We applied the algorithm by reanalyzing four recently published medical studies. We identified the two violations of the positivity reported by the authors. In addition, we identified ten subgroups with a suspicion of violation. Conclusions The PoRT algorithm helps to detect in-sample positivity violations in causal studies. We implemented the algorithm in the R package RISCA to facilitate its use.

Overcoming positivity violations for density matrices in surface hopping.

Fewest-switches surface hopping (FSSH) has emerged as one of the leading methods for modeling the quantum dynamics of molecular systems. While its original formulation was limited to adiabatic populations, the growing interest in the application of FSSH to coherent phenomena prompts the question of how one should construct a complete density matrix based on FSSH trajectories. A straightforward solution is to define adiabatic coherences based on wavefunction coefficients. In this paper, we demonstrate that inconsistencies introduced in the density matrix through such treatment may lead to a violation of positivity. We furthermore show that a recently proposed coherent generalization of FSSH results in density matrices that satisfy positivity while yielding improved accuracy throughout much (but not all) of parameter space.

Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics

Moments of the leptonic angular distribution in the Drell-Yan process have recently been shown to be sensitive probes of a specific class of dimension-8, four-fermion operators in the Standard Model Effective Field Theory, involving a pair of quarks and leptons. The same operators are also subject to positivity bounds, when requiring the associated (unknown) UV completion to obey basic principles of quantum field theory. We perform a phenomenological study to quantify the sensitivity of the high-luminosity LHC to this set of operators and, by extension, the positivity bounds. We further extend the angular basis of moments and consider double differential information to improve the ability to disentangle the different operators, leading to a sensitivity to new physics scales up to 3 TeV. We use this information to explore the violation of positivity at the LHC as a way to test the underlying principles of quantum field theory. Finally, we present a case study which combines our results with information from other (current and prospective) experiments, as well as the positivity cone to infer the properties of possible tree-level UV completions. The data lead to robust, model-independent lower bounds on the M/sqrt{g} combination of the particle mass and coupling, for states that couple to right-handed leptons and/or up quarks.

Reflection positivity and complex analysis of the Yang\u2013Mills theory from a viewpoint of gluon confinement

In order to understand the confining decoupling solution of the Yang–Mills theory in the Landau gauge, we consider the massive Yang–Mills model which is defined by just adding a gluon mass term to the Yang–Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev–Popov ghost term. First of all, we show that massive Yang–Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout–Englert–Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang–Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang–Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov–Stingl form. We try to understand these results in light of the Fradkin–Shenker continuity between confinement-like and Higgs-like regions in a single confinement phase in the complementary gauge-scalar model.

Microscopic Theory of Energy Dissipation and Decoherence in Solid-State Quantum Devices: Need for Nonlocal Scattering Models.

Energy dissipation and decoherence in state-of-the-art quantum nanomaterials and related nanodevices are routinely described and simulated via local scattering models, namely relaxation-time and Boltzmann-like schemes. The incorporation of such local scattering approaches within the Wigner-function formalism may lead to anomalous results, such as suppression of intersubband relaxation, incorrect thermalization dynamics, and violation of probability-density positivity. The primary goal of this article is to investigate a recently proposed quantum-mechanical (nonlocal) generalization (Phys. Rev. B 2017, 96, 115420) of semiclassical (local) scattering models, extending such treatment to carrier–carrier interaction, and focusing in particular on the nonlocal character of Pauli-blocking contributions. In order to concretely show the intrinsic limitations of local scattering models, a few simulated experiments of energy dissipation and decoherence in a prototypical quantum-well semiconductor nanostructure are also presented.

Reflection positivity in higher derivative scalar theories

Reflection positivity constitutes an integral prerequisite in the Osterwalder–Schrader reconstruction theorem which relates quantum field theories defined on Euclidean space to their Lorentzian signature counterparts. In this work, we rigorously prove the violation of reflection positivity in a large class of free scalar fields with a rational propagator. This covers, in particular, higher-derivative theories where the propagator admits a partial fraction decomposition as well as degenerate cases including, e.g., p4-type propagators.

Entanglement dynamics in a Kerr spacetime

We consider the entanglement dynamics between two-level atoms in a rotating black hole background. In our model the two-atom system is envisaged as an open system coupled with a massless scalar field prepared in one of the physical vacuum states of interest. We employ the quantum master equation in the Born-Markov approximation in order to describe the time evolution of the atomic subsystem. We investigate two different states of motion for the atoms, namely static atoms and also stationary atoms with zero angular momentum. The purpose of this work is to expound the impact on the creation of entanglement coming from the combined action of the different physical processes underlying the Hawking effect and the Unruh-Starobinskii effect. We demonstrate that, in the scenario of rotating black holes, the degree of quantum entanglement is significantly modified due to the phenomenon of superradiance in comparison with the analogous cases in a Schwarzschild spacetime. In the perspective of a zero angular momentum observer (ZAMO), one is allowed to probe entanglement dynamics inside the ergosphere, since static observers cannot exist within such a region. On the other hand, the presence of superradiant modes could be a source for violation of complete positivity. This is verified when the quantum field is prepared in the Frolov-Thorne vacuum state. In this exceptional situation, we raise the possibility that the loss of complete positivity is due to the breakdown of the Markovian approximation, which means that any arbitrary physically admissible initial state of the two atoms would not be capable to hold, with time evolution, its interpretation as a physical state inasmuch as negative probabilities are generated by the dynamical map.

Wigner-function formalism applied to semiconductor quantum devices: Need for nonlocal scattering models

In designing and optimizing new-generation nanomaterials and related quantum devices, dissipation versus decoherence phenomena are often accounted for via local scattering models, such as relaxation-time and Boltzmann-like schemes. Here we show that the use of such local scattering approaches within the Wigner-function formalism may lead to unphysical results, namely anomalous suppression of intersubband relaxation, incorrect thermalization dynamics, and violation of probability-density positivity. Furthermore, we propose a quantum-mechanical generalization of relaxation-time and Boltzmann-like models, resulting in nonlocal scattering superoperators that enable one to overcome such limitations.

Violation of positivity bounds in models of generalized parton distributions

As with parton distributions, flexible phenomenological parameterizations of generalized parton distributions (GPDs) are essential for their extraction from data. The large number of constraints imposed on GPDs make simple Lorentz covariant models viable; but, such models are often incomplete in that they employ the impulse approximation. Using the GPD of the pion as a test case, we show that the impulse approximation can lead to violation of the positivity bound required of GPDs. We focus on a particular model of the pion bound-state vertex that was recently proposed and demonstrate that satisfying the bound is not guaranteed by Lorentz covariance. Violation of the positivity bound is tied to a problematic mismatch between the behavior of the quark distribution at the endpoint and the crossover value of the GPD.

Gluons at Finite Temperature

The gluon propagator is investigated at finite temperature via lattice simulations. In particular, we discuss its interpretation as a massive-type bosonic propagator. Moreover, we compute the corresponding spectral density and study the violation of spectral positivity. Finally, we explore the dependence of the gluon propagator on the phase of the Polyakov loop.

Exploring complete positivity in hierarchy equations of motion

We derive a purely algebraic framework for the identification of hierarchy equations of motion that induce completely positive dynamics and demonstrate the applicability of our approach with several examples. We find bounds on the violation of complete positivity for microscopically derived hierarchy equations of motion and construct well-behaved phenomenological models with strongly non-Markovian revivals of quantum coherence.

Comments on the compatibility of thermodynamic equilibrium conditions with lattice propagators

In this paper the compatibility is analyzed of the non-perturbative equations of state of quarks and gluons arising from the lattice with some natural requirements for self-gravitating objects at equilibrium: the existence of an equation of state (namely, the possibility to define the pressure as a function of the energy density), the absence of superluminal propagation and Le Chatelier's principle. It is discussed under which conditions it is possible to extract an equation of state (in the above sense) from the non-perturbative propagators arising from the fits of the latest lattice data. In the quark case, there is a small but non-vanishing range of temperatures in which it is not possible to define a single-valued functional relation between density and pressure. Interestingly enough, a small change of the parameters appearing in the fit of the lattice quark propagator (of around 10~\%) could guarantee the fulfillment of all the three conditions (keeping alive, at the same time, the violation of positivity of the spectral representation, which is the expected signal of confinement). As far as gluons are concerned, the analysis shows very similar results. Whether or not the non-perturbative quark and gluon propagators satisfy these conditions can have a strong impact on the estimate of the maximal mass of quark stars.

Calcium and linoleic acid supplements in the prevention of pre-eclampsia.

Alzate et al.1, conducted a nested case-control study to (quote) “estimate the protective effect from calcium [supplement] alone [CC], compared to calcium plus conjugated linoleic acid [CC+CLA] in nulliparous women at risk of preeclampsia”. Based on a crude analysis of the data in Table 3,1 they concluded that neither CC nor CC+CLA reduced the risk of preeclampsia in the whole sample, but that CC+CLA significantly decreased risk among women 13-18 years old. A quick look analysis of the data in this table shows this conclusion is mostly based on the fact that none of the cases in 13-18 year old women was treated with CC+CLA. Contrary to the authors’ interpretation, this does not point to a protective effect of CC+CLA, it simply indicates that the assumption of positivity has being violated and, consequently, that an effect for this age group cannot be estimated2. In fact, the probability of getting no treated cases in this age-group was 28%, since only 15.5% of all women received CC+CLA. Also, accurate estimates of effect in women 34-45 years old were not possible, because there were only seven women who used CC+CLA in this age group. In spite of the limited sample size, the authors restricted their attention to the apparent protective effect of CC+CLA in 13-19 year old women, while ignoring apparent harmful effects in older women. I estimated age-specific rate ratios (RR) by fitting a saturated conditional complementary log-log3 to the data in Table 3 and found that CC+CLA was protective among women 13-19 (RR= 0.61, 95% CI: 0.41- 0.90), but harmful in women 19-34 (RR= 1.74, 95% CI: 1.21- 2.50) and 35-45 years old (RR= 4.98, 95% CI= 1.74-14.30). Of course, this approach is an improvement over a naive crude analysis, but does not solve the problem of violation and near violation of positivity described above. An overall age-adjusted RR was 1.02 (95% CI= 0.89-1.17; p= 0.756). Thus, this study provides no evidence of a beneficial effect of CC+CLA in preventing preeclampsia in any age group.

Exploring dynamical gluon mass generation in three dimensions

In the d=3 gluon mass problem in pure-glue non-Abelian $SU(N)$ gauge theory we pay particular attention to the observed (in Landau gauge) violation of positivity for the spectral function of the gluon propagator. This causes a large bulge in the propagator at small momentum. Mass is defined through $m^{-2}=\Delta (p=0)$, where $\Delta(p)$ is the scalar function for the gluon propagator in some chosen gauge, it is not a pole mass and is generally gauge-dependent, except in the gauge-invariant Pinch Technique (PT). We truncate the PT equations with a new method called the vertex paradigm that automatically satisfies the QED-like Ward identity relating the 3-gluon PT vertex function with the PT propagator. The mass is determined by a homogeneous Bethe-Salpeter equation involving this vertex and propagator. This gap equation also encapsulates the Bethe-Salpeter equation for the massless scalar excitations, essentially Nambu-Goldstone fields, that necessarily accompany gauge-invariant gluon mass. The problem is to find a good approximate value for $m$ and at the same time explain the bulge, which by itself leads, in the gap equation for the gluon mass, to excessively large values for the mass. Our point is not to give a high-accuracy determination of $m$ but to clarify the way in which the propagator bulge and a fairly accurate estimate of $m$ can co-exist, and we use various approximations that illustrate the underlying mechanisms. The most critical point is to satisfy the Ward identity. In the PT we estimate a gauge-invariant dynamical gluon mass of $m \approx Ng^2/(2.48 \pi)$. We translate these results to the Landau gauge using a background-quantum identity involving a dynamical quantity $\kappa$ such that $m=\kappa m_L$, where $m_L^{-2}\equiv \Delta_L(p=0)$. Given our estimates for $m,\kappa$ the relation is fortuitously well-satisfied for $SU(2)$ lattice data.

Non-Markovian quantum measurements in the non-invasive limit

In many real quantum systems the measurement is much less invasive than described by the projection postulate. For example, it is performed by weakly coupling a detector for a finite time to the system. The finite coupling time might lead to different measurement outcomes depending on the details of the interaction. We first investigate a generalized measurements scheme by means of positive operator-valued measures in the weak coupling limit and discuss the consequences appearing due to the non-Markovian interaction. Second we discuss the resulting quasiprobabilities, which in the case of a Markovian measurement scheme obey the so-called weak positivity, i.e. second-order correlation functions are positive semi-definite. In particular, we show a possible violation of weak positivity by second-order correlation functions for a frequency dependent detector measuring a spin-1/2 system. When both are themselves in an equilibrium state but at different respective temperatures, we find that the detection of the quantum behavior vanishes if the system temperature is larger than that of the detector.

POSITIVITY VIOLATIONS IN QCD

Both lattice simulations and theoretical studies show that the spectral function of the gluon propagator of QCD (in various gauges, as well as for the gauge-invariant Pinch Technique, or PT, propagator) is not nonnegative everywhere, although it should be if it has a physical interpretation as in QED. Theory says moreover that the non-positive spectral function of the Landau-gauge or of the PT gluon propagator is further constrained to obey a superconvergence relation (the integral of the spectral function vanishes). We review the theoretical and lattice evidence for violation of positivity as well as various interpretations of this violation, and consider methods for checking superconvergence on the lattice (so far undone). The most common interpretation is that positivity violation implies confinement of gluons, so the gluon propagator does not describe processes with physical gluons. Another more direct and gauge-invariant interpretation arises from the PT: Asymptotic freedom alone demands non-positivity and superconvergence.

Solution of the Quark Gap Equation by using Gluon Propagator Models inspired by Lattice QCD

In this work we present the solution of the Schwinger-Dyson equation (SDE) for the quark propagator with gluon propagator models inspired by Lattice Quantum Chromodynamics (QCD) and a model of running coupling. Mass function M(p2) of the quarks and wave function renormalization F(p2) are obtained for the different models respectively. We also calculated the chiral quark condensate for each model and test for confinement through violation of positivity of the dynamical fermion propagator.

Extrapolated high-order propagators for path integral Monte Carlo simulations

We present a new class of high-order imaginary time propagators for path integral Monte Carlo simulations that require no higher order derivatives of the potential nor explicit quadratures of Gaussian trajectories. Higher orders are achieved by an extrapolation of the primitive second-order propagator involving subtractions. By requiring all terms of the extrapolated propagator to have the same Gaussian trajectory, the subtraction only affects the potential part of the path integral. The resulting violation of positivity has surprisingly little effects on the accuracy of the algorithms at practical time steps. Thus in principle, arbitrarily high order algorithms can be devised for path integral Monte Carlo simulations. We verified the fourth, sixth, and eighth order convergences of these algorithms by solving for the ground state energy and pair distribution function of liquid (4)He, which is representative of a dense, and strongly interacting, quantum many-body system.

Landau gauge gluon and ghost propagators in the refined Gribov-Zwanziger framework in 3 dimensions

In previous works, we have constructed a refined version of the Gribov-Zwanziger action in 4 dimensions, by taking into account a novel dynamical effect. In this paper, we explore the 3-dimensional case. Analogously to 4 dimensions, we obtain a ghost propagator behaving like $1/{p}^{2}$ in the infrared, while the gluon propagator reaches a finite nonvanishing value at zero momentum. Simultaneously, a clear violation of positivity by the gluon propagator is also found. This behavior of the propagators turns out to be in agreement with the recent numerical simulations.

Dynamics beyond completely positive maps: Some properties and applications

Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of the principal system with the environment, or if the applied transformation is correlated with the state of the system. We provide a physically motivated definition of accessible non-CP maps and derive two necessary conditions for a map to be accessible. We also show that entanglement between the system and the environment is not necessary to generate a non-CP dynamics. We describe two simple approximations that may be sufficient for some problems in process tomography, and then outline what these methods may be able to tell us in other situations where non-CP dynamics naturally arise.