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Authors:Christian Brennecke, Marco Caporaletti, Benjamin Schlein Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider Bose gases of [math] particles in a box of volume one, interacting through a repulsive potential with scattering length of order [math], for [math]. Such regimes interpolate between the Gross–Pitaevskii and thermodynamic limits. Assuming that [math] is sufficiently small, we determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing in the limit [math]. Citation: Reviews in Mathematical Physics PubDate: 2022-08-03T07:00:00Z DOI: 10.1142/S0129055X22500271

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Authors:Thomas Norman Dam, Benjamin Hinrichs Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider a model for a massive uncharged non-relativistic particle interacting with a massless bosonic field, widely referred to as the Nelson model. It is well known that an ultraviolet renormalized Hamilton operator exists in this case. Further, due to translation-invariance, it decomposes into fiber operators. In this paper, we treat the renormalized fiber operators. We give a description of the operator and form domains and prove that the fiber operators do not have a ground state. Our results hold for any non-zero coupling constant and arbitrary total momentum. Our proof for the absence of ground states is a new generalization of methods recently applied to related models. A major enhancement we provide is that the method can be applied to models with degenerate ground state eigenspaces. Citation: Reviews in Mathematical Physics PubDate: 2022-08-03T07:00:00Z DOI: 10.1142/S0129055X22500337

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Authors:J. Dimock Abstract: Reviews in Mathematical Physics, Ahead of Print. We study the radiation of photons from a classical charged particle. We particularly consider a situation where the particle has a constant velocity in the distant past, then is accelerated, and then has a constant velocity in the distant future. Starting with no photons in the distant past we seek to characterize the quantum state of the photon field in the distant future. Working in the Coulomb gauge and in a [math]-algebra formulation, we give sharp conditions on whether this state is or is not in Fock space. Citation: Reviews in Mathematical Physics PubDate: 2022-07-29T07:00:00Z DOI: 10.1142/S0129055X22500325

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Authors:Nima Moshayedi Abstract: Reviews in Mathematical Physics, Ahead of Print. We study the behavior of Donaldson’s invariants of 4-manifolds based on the moduli space of anti-self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable [math]-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when [math]. Additionally, we relate these constructions to Nekrasov’s partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena. Citation: Reviews in Mathematical Physics PubDate: 2022-07-22T07:00:00Z DOI: 10.1142/S0129055X22500295

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Authors:Francesco Fidaleo Abstract: Reviews in Mathematical Physics, Ahead of Print. After introducing the infinite Fermi [math]-tensor product of a single [math]-graded [math]-algebra as an inductive limit, we systematically study the structure of the so-called symmetric states, that is those which are invariant under the group consisting of all finite permutations of a countable set. Among the obtained results, we mention the extension of De Finetti theorem which asserts that a symmetric state is a “mixture” of product states, each of which is a product of a single even state. This result induces a canonical morphism of the simplexes made of the symmetric even states on the usual infinite [math]-tensor product and the symmetric states on the infinite Fermi [math]-tensor product. We then extend the so-called Klein transformation to the infinite Fermi [math]-tensor product, available when the parity automorphism is inner. In such a situation, we investigate further properties of product states, the last being the extremal symmetric states on such an infinite Fermi [math]-tensor product [math]-algebra. This paper is complemented with a finite dimensional illustrative example for which the Klein transformation is not implementable, and then the Fermi tensor product might not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be easily reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties. Citation: Reviews in Mathematical Physics PubDate: 2022-07-18T07:00:00Z DOI: 10.1142/S0129055X22500301

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Authors:Jean-François Bougron, Alain Joye, Claude-Alain Pillet Abstract: Reviews in Mathematical Physics, Ahead of Print. We study a class of dynamical semigroups [math] that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system [math] driven by a Markov chain [math]. We show that the almost sure large time behavior of the system can be extracted from the large [math] asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator [math]. As a physical application, we consider the case where the [math]’s are the reduced dynamical maps describing the repeated interactions of a system [math] with thermal probes [math]. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas. Citation: Reviews in Mathematical Physics PubDate: 2022-07-08T07:00:00Z DOI: 10.1142/S0129055X22500283

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Authors:Joren Brunekreef, Luca Lionni, Johannes Thürigen Abstract: Reviews in Mathematical Physics, Ahead of Print. Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods. Citation: Reviews in Mathematical Physics PubDate: 2022-07-04T07:00:00Z DOI: 10.1142/S0129055X2250026X

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Authors:Young Jin Suh Abstract: Reviews in Mathematical Physics, Ahead of Print. First, we want to give a complete classification of Yamabe solitons and gradient Yamabe solitons for real hypersurfaces in the complex hyperbolic two-plane Grassmannians [math]. Next, as an application we also give a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on real hypersurfaces in the complex hyperbolic two-plane Grassmannians [math]. Citation: Reviews in Mathematical Physics PubDate: 2022-06-23T07:00:00Z DOI: 10.1142/S0129055X22500246

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Authors:Luc Vinet, Alexei Zhedanov Abstract: Reviews in Mathematical Physics, Ahead of Print. The operator that intertwines between the [math]-Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl intertwining operator maps the Hermite polynomials on the generalized Hermite polynomials. Citation: Reviews in Mathematical Physics PubDate: 2022-06-23T07:00:00Z DOI: 10.1142/S0129055X22500258

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Authors:Haruya Mizutani, Nico M. Schiavone Abstract: Reviews in Mathematical Physics, Ahead of Print. In this paper we are interested in generalizing Keller-type eigenvalue estimates for the non-selfadjoint Schrödinger operator to the Dirac operator, imposing some suitable rigidity conditions on the matricial structure of the potential, without necessarily requiring the smallness of its norm. Citation: Reviews in Mathematical Physics PubDate: 2022-06-09T07:00:00Z DOI: 10.1142/S0129055X22500234

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Authors:Euijung Chang, Jaeyoung Kim, Hyesun Kwak, Hun Hee Lee, Sang-Gyun Youn Abstract: Reviews in Mathematical Physics, Ahead of Print. We investigate information theoretic properties of low rank (less than or equal to 3) quantum channels with [math]-symmetry, where we have a complete description. We prove that PPT property coincides with entanglement-breaking property and that degradability seldomly holds in this class. In connection with these results, we will demonstrate how we can compute Holevo and coherent information of those channels. In particular, we exhibit a strong form of additivity violation of coherent information, which resembles the superactivation of coherent information of depolarizing channels. Citation: Reviews in Mathematical Physics PubDate: 2022-06-01T07:00:00Z DOI: 10.1142/S0129055X22500210

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Authors:W. A. Zúñiga-Galindo Abstract: Reviews in Mathematical Physics, Ahead of Print. We construct in a rigorous mathematical way interacting quantum field theories on a [math]-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The advantage of the approach presented here is that all the perturbation calculations can be carried out in the standard way using functional derivatives, but in a mathematically rigorous way. Citation: Reviews in Mathematical Physics PubDate: 2022-05-30T07:00:00Z DOI: 10.1142/S0129055X22500222

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Authors:Ivan Novikov Abstract: Reviews in Mathematical Physics, Ahead of Print. We study Feynman checkers, the most elementary model of electron motion introduced by Feynman. For the model, we prove that the probability to find an electron vanishes nowhere inside the light cone. We also prove several results on the average electron velocity. In addition, we present a lot of identities related to the model. Citation: Reviews in Mathematical Physics PubDate: 2022-05-14T07:00:00Z DOI: 10.1142/S0129055X22500209

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Authors:Matteo Gallone, Alessandro Michelangeli, Eugenio Pozzoli Abstract: Reviews in Mathematical Physics, Ahead of Print. We classify the self-adjoint realizations of the Laplace–Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realizations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterize all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one. Citation: Reviews in Mathematical Physics PubDate: 2022-04-30T07:00:00Z DOI: 10.1142/S0129055X22500180

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Authors:Fei Han, Varghese Mathai Abstract: Reviews in Mathematical Physics, Ahead of Print. This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here, we prove T-duality and Hori formulae for winding [math]-loop spaces, which are infinite dimensional subspaces of loop spaces. Citation: Reviews in Mathematical Physics PubDate: 2022-04-18T07:00:00Z DOI: 10.1142/S0129055X22500192

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Authors:Wojciech Dybalski, Alessandro Pizzo Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider the massless Nelson model with two types of massive particles which we call atoms and electrons. The atoms interact with photons via an infrared regular form-factor and thus they are Wigner-type particles with sharp mass-shells. The electrons have an infrared singular form-factor and thus they are infraparticles accompanied by soft-photon clouds correlated with their velocities. In the weak coupling regime, we construct scattering states of one atom and one electron, and demonstrate their asymptotic clustering into individual particles. The proof relies on the Cook’s argument, clustering estimates, and the non-stationary phase method. The latter technique requires sharp estimates on derivatives of the ground state wave functions of the fiber Hamiltonians of the model, which were proven in the earlier papers of this series. Although we rely on earlier studies of the atom–atom and electron–photon scattering in the Nelson model, the paper is written in a self-contained manner. A perspective on the open problem of the electron–electron scattering in this model is also given. Citation: Reviews in Mathematical Physics PubDate: 2022-04-08T07:00:00Z DOI: 10.1142/S0129055X22500143

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Authors:Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider the quantum evolution [math] of a Gaussian coherent state [math] localized close to the classical state [math], where [math] denotes a self-adjoint realization of the formal Hamiltonian [math], with [math] the derivative of Dirac’s delta distribution at [math] and [math] a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (with respect to the [math]-norm, uniformly for any [math] away from the collision time) by [math], where [math], [math] and [math] is a suitable self-adjoint extension of the restriction to [math], [math], of ([math] times) the generator of the free classical dynamics. While the operator [math] here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489], in the present case the approximation gives a smaller error: it is of order [math], [math], whereas it turns out to be of order [math], [math], for the delta potential. We also provide similar approximation results for both the wave and scattering operators. Citation: Reviews in Mathematical Physics PubDate: 2022-03-31T07:00:00Z DOI: 10.1142/S0129055X22500155

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Authors:L. Amour, J. Nourrigat Abstract: Reviews in Mathematical Physics, Ahead of Print. The purpose of this paper is to give different interpretations of the first non-vanishing term (quadratic) of the ground state asymptotic expansion for a spin system in quantum electrodynamics, as the spin magnetic moments go to [math]. One of the interpretations makes a direct link with some classical physics laws. A central role is played by an operator [math] acting only in the finite-dimensional spin state space and making the connections with the different interpretations, and also being in close relation with the multiplicity of the ground state. Citation: Reviews in Mathematical Physics PubDate: 2022-03-29T07:00:00Z DOI: 10.1142/S0129055X22500179