Annales Henri Poincaré最新文献

We consider the transport properties of non-interacting, gapless one-dimensional quantum systems and of the edge modes of two-dimensional topological insulators, in the presence of time-dependent perturbations. We prove the validity of Kubo formula, in the zero temperature and infinite volume limit, for a class of perturbations that are weak and slowly varying in space and in time, in an Euler-like scaling. The proof relies on the representation of the real-time Duhamel series in imaginary time, which allows to prove its convergence uniformly in the scaling parameter and in the size of system, at low temperatures. Furthermore, it allows to exploit a suitable cancellation for the scaling limit of the model, related to the emergent anomalous chiral gauge symmetry of relativistic one-dimensional fermions. The cancellation implies that as the temperature and the scaling parameter are sent to zero, the linear response is the only contribution to the full response of the system. The explicit form of the leading contribution to the response function is determined by lattice conservation laws. In particular, the method allows to prove the quantization of the edge conductance of 2d quantum Hall systems from quantum dynamics.

In this paper we study strong coupling asymptotic expansions of (mathcal {N}=2) four-dimensional SU(2) gauge theory partition functions in general (Omega )-background. This is done by refining the Painlevé/gauge theory correspondence in terms of quantum Painlevé equations, obtained from (mathbb {C}^2/mathbb {Z}_2) blowup relations. We present a general ansatz and a systematic analysis of the expansions of the gauge theory partition functions by solving the above equations around the strong coupling singularities, including Argyres–Douglas points. We check our results via refined holomorphic anomaly equations and compare with irregular Virasoro conformal blocks.

We show that the momentum, the density, and the electromagnetic field associated with the massive Klein–Gordon–Maxwell equations converge in the semiclassical limit toward their respective equivalents associated with the relativistic pressureless Euler–Maxwell equations. The proof relies on a modulated stress–energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic pressureless Euler–Maxwell equations and show how this system, and so the semiclassical limit of Klein–Gordon–Maxwell, is related to the relativistic massive Vlasov–Maxwell equations.

We develop a general framework to study quantum trajectories resulting from repeated random measurements subject to stationary noise and generalize results of Kümmerer and Maassen [33] to this setting. The resulting trajectory of quantum states is a time-inhomogeneous Markov chain in a random environment. Kümmerer and Maassen introduced the concept of dark subspaces for noise-free processes, establishing that their absence is equivalent to asymptotic purification of the system state. We clarify the notion of dark subspaces in the disordered setting by defining a measurable correspondence consisting of a collection of random subspaces satisfying a darkness condition. We further prove that asymptotic purification occurs if and only if this collection of random dark subspaces is empty. Several examples of these phenomena are provided.

We consider Dirac-type operators on manifolds with boundary and set out to determine all local smooth boundary conditions that give rise to (strongly) regular self-adjoint operators. By combining the general theory of boundary value problems for Dirac operators as in [4] and pointwise considerations, for local smooth boundary conditions the question of being self-adjoint resp. regular is fully translated into linear-algebraic language at each boundary point. We analyze these conditions and classify them in low dimensions and ranks. In particular, we classify all local self-adjoint regular boundary conditions for Dirac spinors (four spinor components) in dimensions 3 and 4. With the same techniques we can also treat transmission boundary conditions.

We show that a symmetric informationally complete positive operator-valued measure exists in a given dimension d if and only if there exists a (d^2)-dimensional operator system satisfying certain order-theoretic conditions. We also describe a method of constructing such an operator system and demonstrate that the first step of this construction can be carried out successfully. We obtain analogous results for the existence of (d+1) mutually unbiased bases in a given dimension.

We argue that the spectrally cut-off Gaussian free field (Phi _Lambda ) on a compact Riemannian manifold or on ({textbf {R}}^d) cannot satisfy the spatial Markov property. Moreover, when the manifold is reflection positive, we show that (Phi _Lambda ) fails to be reflection positive. We explain the difficulties one encounters when trying to deduce the reflection positivity property of the measure exp((-Vert rho Phi _Lambda Vert _{L^4}^4) mu _{text {GFF}}(textrm{d}Phi )) from the reflection positivity property of the Gaussian free field measure (mu _{text {GFF}}) in a naive way. These issues are probably well-known to experts of constructive quantum field theory but to our knowledge, no detailed account can be found in the literature. Our pedagogical note aims to fill this small gap.

Within the class of Dereziński–Enss pair-potentials which includes Coulomb potentials, a stationary scattering theory for N-body systems was recently developed (Skibsted in Commun Math Phys 401:2193–2267, 2023). In particular, the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy, and this holds without imposing any a priori decay condition on channel eigenstates. In this paper, we improve for the case of 3-body systems on the known weak continuity properties in that we show that all non-threshold energies are stationary complete in this case, resolving a conjecture from Skibsted (Commun Math Phys 401:2193–2267, 2023) in the special case (N=3). A consequence is that the above scattering quantities depend strongly continuously on the energy parameter at all non-threshold energies, hence not only almost everywhere as previously demonstrated (for an arbitrary N). Another consequence is the unitary of the scattering matrix at any such energy. As a side result, we give an independent stationary proof of asymptotic completeness for 3-body systems with long-range pair-potentials. This is an alternative to the known time-dependent proofs (Dereziński in Ann Math 38:427–476, 1993; Enss in Long-range scattering of two- and three-body quantum systems, Publications ecole polytechnique, Palaiseau, 1989).

We consider a tracer particle coupled to a Bose scalar field and study the regime where the field’s propagation speed approaches infinity. For initial states devoid of field excitations, we introduce an effective approximation of the time-evolved wave function and prove its validity in Hilbert space norm. In this approximation, the field remains in the vacuum state, while the tracer particle propagates with a modified dispersion relation. Physically, the new dispersion relation can be understood as the effect of radiative corrections due to interactions with virtual bosons. Mathematically, it is defined as the solution of a self-consistent nonlinear equation, whose form depends on the relevant time scale.

To a smooth variety X with simple normal crossings divisor D, we associate a sheaf of vertex algebras on X, denoted (Omega ^{ch}_{X}(operatorname {log}D)), whose conformal weight 0 subspace is the algebra (Omega _{X}(operatorname {log}D)) of forms with log poles along D. We prove various basic structural results about (Omega ^{ch}_{X}(operatorname {log}D)). In particular, if (X^{*}=Xsetminus D) has a volume form then we show that (Omega ^{ch}_{X}(operatorname {log}D)) admits a topological structure of rank (d=operatorname {dim}(X)), which is enhanced to an extended topological structure if (Dsim -K_{X}) is in fact anticanonical. In this latter case, we also show that the resulting (q, y) character (operatorname {Ell}(X,D)(q,y)) is a section of the line bundle (Theta ^{otimes d}) on the elliptic curve (E=textbf{C}^{*}/q^{textbf{Z}}). We further show how (Omega ^{ch}_{X}(operatorname {log}D)) can be understood in terms of a simple birational modification of the space of jets into X.