Rigorous Version Research Articles

Almost Optimal Upper Bound for the Ground State Energy of a Dilute Fermi Gas via Cluster Expansion

AbstractWe prove an upper bound on the energy density of the dilute spin-$$\frac{1}{2}$$ 1 2 Fermi gas capturing the leading correction to the kinetic energy $$8\pi a \rho _\uparrow \rho _\downarrow $$ 8 π a ρ ↑ ρ ↓ with an error of size smaller than $$a\rho ^{2}(a^3\rho )^{1/3-\varepsilon }$$ a ρ 2 ( a 3 ρ ) 1 / 3 - ε for any $$\varepsilon > 0$$ ε > 0 , where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).

Adiabatic Evolution of Low-Temperature Many-Body Systems

We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

Likelihoodism and Guidance for Belief

Likelihoodism is the view that the degree of evidential support should be analysed and measured in terms of likelihoods alone. The paper considers and responds to a popular criticism that a likelihoodist framework is too restrictive to guide belief. First, I show that the most detailed and rigorous version of this criticism, as put forward by Gandenberger (2016), is unsuccessful. Second, I provide a positive argument that a broadly likelihoodist framework can accommodate guidance for comparative belief, even when objectively well-grounded prior probabilities are not available. As I show, the shift from non-relational to comparative probabilities opens up a new space for addressing the belief guidance problem for likelihoodism.

Functionalising the wavefunction

Functionalism is the view that being x is to play the role of x. This paper defends a functionalist account of three-dimensional entities in the context of Wave Function Realism (WFR), that can explain in details how we can recover three-dimensional entities out of the wavefunction. In particular, the essay advocates for a novel version of WFR in terms of a functional reductionist approach in the style of David Lewis. This account entails reduction of the upper entities to the bottom ones, when the latter behave appropriately. As applied to WFR, it shows how the wavefunction can turn out to be identical to three-dimensional objects, provided certain conditions. The first major goal of the paper is thus to put forward an improved and more rigorous version of WFR, which dissolves several extant issues about the theory, and can serve as a starting point for the future literature about the topic. Moreover, the second major goal of the article is to take WFR as a case study to demonstrate the pros of functional reductionism, especially in the form defended here, thereby helping to bring this view back in the philosophy of science debate. The positive upshots of this paper suggest a possible application of functional reductionism also to other contexts.

Identifying Health Literacy Responsiveness Improvement Ideas in Danish Health Centers: Initial Testing of the OS! Approach.

Background:Health services hold a critical role in mitigating the effect of health literacy challenges. Several tools and approaches have been developed to support health organizations in responding to their target population's health literacy needs. One of these is the OS! approach, which is an adapted and less rigorous version of the Australian Organizational health literacy responsiveness tool (Org-HLR). In this study, we aimed to report on our initial testing of the OS! approach in two Danish health care settings.Brief description:The OS! aim to prompt and support the development of local organizational health literacy improvement ideas. The process consists of three consecutive workshops focusing on reflection, self-evaluation, and prioritization. For each workshop tools are provided to support implementation.Implementation:The OS! approach was tested in two Danish municipal health centers (Tårnby Health Centre and Frederiksberg Health Centre), where both staff members and leaders took part in the processes (N = 62 and N = 84). In Tårnby, the process was closely facilitated by the research team, while local managers was in charge of the implementation in Frederiksberg.Results:In both settings, the process succeeded in identifying a list of relevant prioritized action ideas aimed at improving local health literacy responsiveness. In both instances, leaders were able to integrate some of these ideas in action plans for the organizations nearby future.Lessons learned:We found the OS! approach effective in its aim to identify health literacy responsiveness improvement ideas and based on our experience also less arduous to implement than the Org-HLR. We also learned that good implementation of the OS! approach depends strongly on careful facilitation including thorough introduction of the concepts of health literacy, openness, and creativity in relation to local adaptions within the overall framework of the approach, and the consolidation of consistent leader support throughout the organization. [HLRP: Health Literacy Research and Practice. 2022;6(3):e232–e238.]Plain Language Summary:This study describes the initial testing of the OS! approach. The OS! approach is based on the Australian methodology “Org-HLR” and use a participatory approach to prompt and support the identification of local health literacy responsiveness improvement ideas in health care organizations. The article provides valuable experiences regarding local development of organizational health literacy in practice.

Reliability and validity of the 15-item Chronic Oral Mucosal Disease Questionnaire in China: A short-form instrument to measure quality of life.

Chronic oral mucosal diseases have been a common problem, which affected individuals' health and social functions seriously. Although there has been a rigorous Chinese version of the Chronic Mucosal Diseases Questionnaire (COMDQ-26) to measure the quality of life of patients with chronic oral mucosal diseases, 26 items were so redundant to be a burden for patients. It was necessary to make a brief, practical, and easy-to-use tool in China. This study aimed to test the validity and reliability of the Chinese version of the short-form Chronic Mucosal Diseases Questionnaire (COMDQ-15). A total of 165 respondents from a tertiary stomatological hospital were recruited in this study from July to December, 2020. Internal consistency, test-retest reliability, criterion validity, confirmatory factor analysis, and exploratory factor analysis were used to evaluate the reliability and validity of the Chinese version of Chronic Mucosal Diseases Questionnaire-15. The total Cronbach's α of 0.841 and subscale ranging from 0.736 to0.965 revealed good internal consistency of the reliability. The intraclass correlation coefficients exceeded 0.80 indicated excellent test-retest reliability. Compared with the Oral Health Impact Profile-14, the criterion validity was 0.549. The confirmatory factor analysis results supported acceptable validity of the Chinese version of Chronic Mucosal Diseases Questionnaire-15, and the exploratory factor analysis demonstrated that the 15 items were classified into four domains, with a cumulative proportion of 71.52%. The Chinese version of Chronic Mucosal Diseases Questionnaire-15 is a brief and low-burden tool, and has been proved reliable and valid to assess the quality of life of patients with chronic oral mucosal diseases in China (the Chinese Clinical Trial Registry (no. ChiCTR2000031607).

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

Maximum principle for discrete-time stochastic optimal control problem and stochastic game

<p style='text-indent:20px;'>This paper is first concerned with one kind of discrete-time stochastic optimal control problem with convex control domains, for which necessary condition in the form of Pontryagin's maximum principle and sufficient condition of optimality are derived. The results are then extended to two kinds of discrete-time stochastic games. Two illustrative examples are studied, for which the explicit optimal strategies are given. This paper establishes a rigorous version of discrete-time stochastic maximum principle in a clear and concise way and paves a road for further related topics.</p>

Bosonization of Fermionic Many-Body Dynamics

We consider the quantum many-body evolution of a homogeneous Fermi gas in three dimensions in the coupled semiclassical and mean-field scaling regime. We study a class of initial data describing collective particle-hole pair excitations on the Fermi ball. Using a rigorous version of approximate bosonization, we prove that the many-body evolution can be approximated in Fock space norm by a quasifree bosonic evolution of the collective particle-hole excitations.

Reappraisal of the normal ordered Jeziorski-Monkhorst ansatz in the UGA-OSCC theory for a study of IP, EA and EE

The Unitary Group Adapted Open Shell Coupled Cluster (UGA-OSCC) theory, which is the single reference CSF limit of the UGA-State Universal Multireference CC (UGA-SUMRCC) formulation by Mukherjee et al. [Sen, Shee, and Mukherjee, J. Chem. Phys. 137, 074104 (2012)], was proposed and successfully applied via a normal ordered Jeziorski-Monkhorst (J-M)-like CC ansatz for the wave operator. This generates a spin-adapted, size extensive energy. The computed energy differences of spectroscopic interest such as IP, EA and EE were consequently size intensive. Here we revisit the formulation by suggesting a rigorous version which completely bypasses the use of a sufficiency condition posited in our earlier formulation via an exact factorisation of the wave operator in the Bloch equation. This brings in certain connected composites involving contractions among the cluster operators in the resultant equations. This formulation is flexible enough to accommodate truncation of these composites while remaining rigorously size-extensive for each such approximant. We test the effect of these new terms in the Bloch equation for this formulation by evaluating IP, EA and EE for a test set of molecules, which shifts the respective results closer in the order of 0.1 to 0.001 eV towards the benchmark EOM-CCSDt or FCI or experimental values.

The is-ought gap and the substitution criterion

Ever since its formulation by Hume, the idea of an inferential barrier between non-ethical (“is”) propositions and ethical (“ought”) propositions (also known as Hume’s is-ought thesis) has received much philosophical attention. Prior’s Paradox appears to demonstrate that the ban on “is”-“ought” inferences is violated in every possible instance, from which it follows that Hume’s is-ought thesis must be false. In this article, I will formulate a logically rigorous version of Hume’s is-ought thesis, introduce Prior-style counterexamples, and suggest how they might demonstrate the falsity of Hume’s is-ought thesis, identify the substitution criterion, and develop at length the unrestricted and restricted versions of this criterion as it is applied by Pigden and Schurz. Thereafter, I will outline Schurz’s defence of the is-ought gap in accordance with his interpretation of the substitution criterion, before introducing a more sophisticated version of Prior-style objections, proposing a possible response from Pigden, identifying the costs of Pigden’s response and counter proposing a response on behalf of Schurz through an extension of his approach, and offering concluding remarks.

Exploration of interlacing and avoided crossings in a manifold of potential energy curves by a unitary group adapted state specific multi-reference perturbation theory (UGA-SSMRPT).

The Unitary Group Adapted State-Specific Multi-Reference Perturbation Theory (UGA-SSMRPT2) developed by Mukherjee et al. [J. Comput. Chem. 36, 670 (2015)] has successfully realized the goal of studying bond dissociation in a numerically stable, spin-preserving, and size-consistent manner. We explore and analyze here the efficacy of the UGA-SSMRPT2 theory in the description of the avoided crossings and interlacings between a manifold of potential energy curves for states belonging to the same space-spin symmetry. Three different aspects of UGA-SSMRPT2 have been studied: (a) We introduce and develop the most rigorous version of UGA-SSMRPT2 that emerges from the rigorous version of UGA-SSMRCC utilizing a linearly independent virtual manifold; we call this the "projection" version of UGA-SSMRPT2 (UGA-SSMRPT2 scheme P). We compare and contrast this approach with our earlier formulation that used extra sufficiency conditions via amplitude equations (UGA-SSMRPT2 scheme A). (b) We present the results for a variety of electronic states of a set of molecules, which display the striking accuracy of both the two versions of UGA-SSMRPT2 with respect to three different situations involving weakly avoided crossings, moderate/strongly avoided crossings, and interlacing in a manifold of potential energy curves (PECs) of the same symmetry. Accuracy of our results has been benchmarked against IC-MRCISD + Q.

Stability and Instability of Self-Gravitating Relativistic Matter Distributions

We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein–Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein–Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift kappa >0. We prove their linear instability when kappa is sufficiently large, i.e., when they are strongly relativistic, and prove that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel’dovich & Podurets (for the Einstein–Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein–Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein–Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

Consistency Proof for Multi-time Schr\xf6dinger Equations with Particle Creation and Ultraviolet Cut-Off

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

The algebra of anomaly interplay

We give a general description of the interplay that can occur between local and global anomalies, in terms of (co)bordism. Mathematically, such an interplay is encoded in the non-canonical splitting of short exact sequences known to classify invertible field theories. We study various examples of the phenomenon in 2, 4, and 6 dimensions. We also describe how this understanding of anomaly interplay provides a rigorous bordism-based version of an old method for calculating global anomalies (starting from local anomalies in a related theory) due to Elitzur and Nair.

Character Randomized Benchmarking for Non-Multiplicity-Free Groups With Applications to Subspace, Leakage, and Matchgate Randomized Benchmarking

Randomized benchmarking (RB) is a powerful method for determining the error rate of experimental quantum gates. Traditional RB, however, is restricted to gatesets, such as the Clifford group, that form a unitary 2-design. The recently introduced character RB can benchmark more general gates using techniques from representation theory; up to now, however, this method has only been applied to "multiplicity-free" groups, a mathematical restriction on these groups. In this paper, we extend the original character RB derivation to explicitly treat non-multiplicity-free groups, and derive several applications. First, we derive a rigorous version of the recently introduced subspace RB, which seeks to characterize a set of one- and two-qubit gates that are symmetric under SWAP. Second, we develop a new leakage RB protocol that applies to more general groups of gates. Finally, we derive a scalable RB protocol for the matchgate group, a group that like the Clifford group is non-universal but becomes universal with the addition of one additional gate. This example provides one of the few examples of a scalable non-Clifford RB protocol. In all three cases, compared to existing theories, our method requires similar resources, but either provides a more accurate estimate of gate fidelity, or applies to a more general group of gates. In conclusion, we discuss the potential, and challenges, of using non-multiplicity-free character RB to develop new classes of scalable RB protocols and methods of characterizing specific gates.

Rigorous implementation of the Galerkin method for stepped structures needs generalized functions

In this paper we study free vibrations of stepped structures, specifically for longitudinal vibrations of bars and flexural vibrations of rectangular plates, providing two versions of the Galerkin method. Specifically, we first apply the straightforward version of the Galerkin method which stipulates the employment of the Galerkin procedure to be conducted in each subdomain, or step, of the structure. Second, the rigorous realization of the Galerkin method is presented where the structural parameters, like rigidity and mass, are treated as generalized functions over the entire domain. This latter implementation utilizes unit step functions, as well as the Dirac's delta function, and its derivative to treat the changes of the structural parameters across the steps.It turns out that this rigorous implementation leads to additional terms that do not appear in the straightforward (or “naïve”) realization of the Galerkin method. Both versions of Galerkin methods are compared with the exact solutions of the considered problems. It turns out that with increase of number of terms in the expansion, the rigorous, generalized-functions based Galerkin method tends to exact solution. In contrast the naïve realization of Galerkin's method, which is usually utilized in literature, does not tend to exact solution. This study demonstrates that extreme care must be taken when implementing the Galerkin's method for stepped structures, and only the rigorous version should be employed.

Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations

Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any quantization scheme, this algebra is inherently non-commutative and comprises a large set of dynamics. In contrast to other approaches, the generating elements of the algebra are not interpreted as observables, but as operations on the underlying system; they describe the impact of temporary perturbations caused by the surroundings. In accordance with the doctrine of Niels Bohr, the operations carry individual names of classical significance. Without stipulating from the outset their “quantization”, their concrete implementation in the quantum world emerges from the inherent structure of the algebra. In particular, the Heisenberg commutation relations for position and velocity measurements are derived from it. Interacting systems can be described within the algebraic setting by a rigorous version of the interaction picture. It is shown that Hilbert space representations of the algebra lead to the conventional formalism of quantum mechanics, where operations on states are described by time-ordered exponentials of interaction potentials. It is also discussed how the familiar statistical interpretation of quantum mechanics can be recovered from operations.

A C*-algebraic Approach to Interacting Quantum Field Theories

A novel C*-algebraic framework is presented for relativistic quantum field theories, fixed by a Lagrangean. It combines the postulates of local quantum physics, encoded in the Haag–Kastler axioms, with insights gained in the perturbative approach to quantum field theory. Key ingredients are an appropriate version of Bogolubov’s relative S-operators and a reformulation of the Schwinger–Dyson equations. These are used to define for any classical relativistic Lagrangean of a scalar field a non-trivial local net of C*-algebras, encoding the resulting interactions at the quantum level. The construction works in any number of space-time dimensions. It reduces the longstanding existence problem of interacting quantum field theories in physical spacetime to the question of whether the C*-algebras so constructed admit suitable states, such as stable ground and equilibrium states. The method is illustrated on the example of a non-interacting field and it is shown how to pass from it within the algebra to interacting theories by relying on a rigorous local version of the interaction picture.

On the Correlation Energy of Interacting Fermionic Systems in the Mean-Field Regime

We consider a system of $$N\gg 1$$ interacting fermionic particles in three dimensions, confined in a periodic box of volume 1, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their N-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED.