Letters in Mathematical Physics最新文献

We present an abstract approach to Lorentzian Gromov–Hausdorff distance and convergence, and an alternative approach to Lorentzian length spaces that does not use auxiliary “positive signature” metrics or other unobserved fields. We begin by defining a notion of (abstract) bounded Lorentzian metric space which is sufficiently general to comprise compact causally convex subsets of globally hyperbolic spacetimes and causets. We define the Gromov–Hausdorff distance and show that two bounded Lorentzian metric spaces at zero GH distance are indeed both isometric and homeomorphic. Then we show how to define from the Lorentzian distance, beside topology, the causal relation and the causal curves for these spaces, obtaining useful limit curve theorems. Next, we define Lorentzian (length) prelength spaces via suitable (maximal) chronal connectedness properties. These definitions are proved to be stable under GH limits. Furthermore, we define bounds on sectional curvature for our Lorentzian length spaces and prove that they are also stable under GH limits. We conclude with a (pre)compactness theorem.

We solve for quantum geometrically realised pre-spectral triples or ‘Dirac operators’ on the noncommutative torus ({mathbb {C}}_theta [T^2]) and on the algebra (M_2({mathbb {C}})) of (2times 2) matrices with their standard quantum metrics and associated quantum Riemannian geometry. For ({mathbb {C}}_theta [T^2]), we obtain a standard even spectral triple but now uniquely determined by full geometric realisability. For (M_2({mathbb {C}})), we are forced to a particular flat quantum Levi-Civita connection and again obtain a natural fully geometrically realised even spectral triple. In both cases there is an odd spectral triple for a different choice of a sign parameter. We also consider an alternate quantum metric on (M_2({mathbb {C}})) with curved quantum Levi-Civita connection and find a natural 2-parameter family of Dirac operators which are almost spectral triples, where fails to be antihermitian. In all cases, we split the construction into a local tensorial level related to the quantum Riemannian geometry, where we classify the results more broadly, and the further requirements relating to the pre-Hilbert space structure. We also illustrate the Lichnerowicz formula for which applies in the case of a full geometric realisation.

We propose a new notion of singularity in general relativity which complements the usual notions of geodesic incompleteness and curvature singularities. Concretely, we say that a spacetime has a volume singularity if there exist points whose future or past has arbitrarily small spacetime volume: in particular, smaller than a Planck volume. From a cosmological perspective, we show that the (geodesic) singularities predicted by Hawking’s theorem are also volume singularities. In the black hole setting, we show that volume singularities are always hidden by an event horizon, prompting a discussion of Penrose’s cosmic censorship conjecture.

We introduce and study a general notion of spatial localization on spacelike smooth Cauchy surfaces of quantum systems in Minkowski spacetime. The notion is constructed in terms of a coherent family of normalized POVMs, one for each said Cauchy surface. We prove that a family of POVMs of this type automatically satisfies a causality condition which generalizes Castrigiano’s one and implies it when restricting to flat spacelike Cauchy surfaces. As a consequence, no conflict with Hegerfeldt’s theorem arises. We furthermore prove that such families of POVMs do exist for massive Klein–Gordon particles, since some of them are extensions of already known spatial localization observables. These are constructed out of positive definite kernels or are defined in terms of the stress–energy tensor operator. Some further features of these structures are investigated, in particular the relation with the triple of Newton–Wigner selfadjoint operators and a modified form of Heisenberg inequality in the rest 3-spaces of Minkowski reference frames.

In this paper, we study the genus two G-function which was introduced by Dubrovin, Liu and Zhang for the cubic elliptic orbifold. As results, we first prove the quasi-modularity for the descendant correlation functions of certain type in all genus. Then we prove any derivatives of the genus two G-function of certain type are quasi-modular forms after a mirror transformation. In particular, we compute the explicit closed formula for its certain first derivative. Our proof mainly relies on two techniques: Givental quantization formalism for semisimple Frobenius manifold and the tautological relations on the moduli space of stable curves.

We consider the 3D Mikhalev system,

which has first appeared in the context of KdV-type hierarchies. Under the reduction (w=f(u)), one obtains a pair of commuting first-order equations,

which govern simple wave solutions of the Mikhalev system. In this paper we study higher-order reductions of the form

which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at (epsilon ^n) are assumed to be differential polynomials of degree n in the x-derivatives of u. We will view w as an (infinite) formal series in the deformation parameter (epsilon ). It turns out that for such a reduction to be non-trivial, the function f(u) must be quadratic, (f(u)=lambda u^2), furthermore, the value of the parameter (lambda ) (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, (lambda =1) and (lambda =3/2), as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of linear degeneracy of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.

This short letter considers the case of acoustic scattering by several obstacles in (mathbb {R}^{d+r}) for (r,d ge 1) of the form (Omega times mathbb {R}^r), where (Omega ) is a smooth bounded domain in (mathbb {R}^d). As a main result, a von Neumann trace formula for the relative trace is obtained in this setting. As a special case, we obtain a dimensional reduction formula for the Casimir energy for the massive and massless scalar fields in this configuration (Omega times mathbb {R}^r) per unit volume in (mathbb {R}^r).

This paper continues with Part I. We define the module for a (frac{{mathbb Z}}{2})-graded meromorphic open-string vertex algebra that is twisted by an involution and show that the axioms are sufficient to guarantee the convergence of products and iterates of any number of vertex operators. A module twisted by the parity involution is called a canonically ({mathbb Z}_2)-twisted module. As an example, we give a fermionic construction of the canonically ({mathbb Z}_2)-twisted module for the (frac{{mathbb Z}}{2})-graded meromorphic open-string vertex algebra constructed in Part I. Similar to the situation in Part I, the example is also built on a universal ({mathbb Z})-graded non-anti-commutative Fock space where a creation operator and an annihilation operator satisfy the fermionic anti-commutativity relation, while no relations exist among the creation operators or among the zero modes. The Wick’s theorem still holds, though the actual vertex operator needs to be corrected from the naïve definition by normal ordering using the (exp (Delta (x)))-operator in Part I.

We derive an expression for the spectral determinant of a second-order elliptic differential operator ( mathcal {T} ) defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation ( mathcal {T} u=0). Examples of application of the resulting formula include the explicit calculation of the determinant of harmonic and anharmonic oscillators with an added bounded potential with compact support.

In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator L generated by the differential expression of odd order n with the (mtimes m) periodic matrix coefficients, where (n>1.) We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set (left[ (2pi N)^{n},infty right) cup (-infty ,(-2pi N)^{n}]) belongs to at least m bands, where N is the smallest integer satisfying (Nge pi ^{-2}M+1) and M is the sum of the norms of the coefficients. Moreover, we prove that if (Mle pi ^{2}2^{-n+1/2}), then each point of the real line belong to at least m bands.