Letters in Mathematical Physics最新文献

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves, we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case.

The quantization of chiral fermions on a 3-manifold in an external gauge potential is known to lead to an abelian extension of the gauge group. In this article, we concentrate on the case of (Omega ^3 G) of based smooth maps on a 3-sphere taking values in a compact Lie group G. There is a crossed module constructed from an abelian extension (widehat{Omega ^3 G}) of this group and a group of automorphisms acting on it as explained in a recent article by Mickelsson and Niemimäki. We shall construct a representation of this crossed module in terms of a representation of (widehat{Omega ^3 G}) on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of (widehat{Omega ^3 G}) as outer automorphisms of the canonical anticommutation relations algebra in the Fock space.

This note is concerned with the linear BBM equation on the half-line. Its nonlinear counterpart originally arose as a model for surface water waves in a channel. This model was later shown to have considerable predictive power in the context of waves generated by a periodically moving wavemaker at one end of a long channel. Theoretical studies followed that dealt with qualitative properties of solutions in the idealized situation of periodic Dirichlet boundary conditions imposed at one end of an infinitely long channel. One notable outcome of these works is the property that solutions become asymptotically periodic as a function of time at any fixed point x in the channel, a property that was suggested by the experimental outcomes. The earlier theory is here generalized using complex-variable methods. The approach is based on the rigorous implementation of the Fokas unified transform method. Exact solutions of the forced linear problem are written in terms of contour integrals and analyzed for more general boundary conditions. For (mathcal C^infty )-data satifisying a single compatibility condition, global solutions obtain. For Dirichlet and Neumann boundary conditions, asymptotic periodicity still holds. However, for Robin boundary conditions, we find not only that solutions lack asymptotic periodicity, but they in fact display instability, growing in amplitude exponentially in time.

We present an algorithm for determining the minimal order differential equations associated with a given Feynman integral in dimensional or analytic regularisation. The algorithm is an extension of the Griffiths–Dwork pole reduction adapted to the case of twisted differential forms. In dimensional regularisation, we demonstrate the applicability of this algorithm by explicitly providing the inhomogeneous differential equations for the multi-loop two-point sunset integrals: up to 20 loops for the equal-mass case, the generic mass case at two- and three-loop orders. Additionally, we derive the differential operators for various infrared-divergent two-loop graphs. In the analytic regularisation case, we apply our algorithm for deriving a system of partial differential equations for regulated Witten diagrams, which arise in the evaluation of cosmological correlators of conformally coupled (phi ^4) theory in four-dimensional de Sitter space.

We discuss dense coding with n copies of a specific preshared state between the sender and the receiver when the encoding operation is limited to the application of group representation. Typically, to act on multiple local copies of these preshared states, the receiver needs quantum memory, because in general the multiple copies will be generated sequentially. Depending on available encoding unitary operations, we investigate what preshared state offers an advantage of using quantum memory on the receiver’s side.

Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare D-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic D-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. https://doi.org/10.1103/PhysRevA.95.032308), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.

Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of non-abelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of non-abelian theta functions and the bundle of WZW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya–Ueno–Yamada. As an application, we give a geometric construction of the Knizhnik–Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations.