اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMathematical Aspects of Vacuum Energy on Quantum Graphs
165
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We use quantum graphs as a model to study various mathematical aspects of the vacuum energy, such as convergence of periodic path expansions, consistency among different methods (trace formulae versus method of images) and the possible connection with the underlying classical dynamics. We derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph. For an important special case of graphs with equal bond lengths, we derive a simpler explicit formula. The main results are derived using the trace formula. We also discuss an alternative approach using the method of images and prove that the results are consistent. This may have important consequences for other systems, since the method of images, unlike the trace formula, includes a sum over special ``bounce paths. We succeed in showing that in our model bounce paths do not contribute to the vacuum energy. Finally, we discuss the proposed possible link between the magnitude of the vacuum energy and the type (chaotic vs. integrable) of the underlying classical dynamics. Within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy.
قيم البحث
اقرأ أيضاً
It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the n
odal surplus) for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graphs first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form --- it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.
In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreov
er, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the first one where a nontrivial example of QMC over non-regular graphs is given.
From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to say, the
square of the evolution matrix of a quantum walk. Then we give to such a function two types of determinant expressions and derive from it some geometric properties of a finite graph. As an application, we illustrate the distribution of poles of this function comparing with those of the usual Ihara zeta function.
We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain important features
of the prototypical quantum star graph model. In particular, we show the spectrum is encoded in a secular equation with similar features. The secular equation of a quantum circulant graph takes two forms depending on whether the edge lengths respect the cyclic symmetry of the graph. When all the edge lengths are incommensurate, the spectral statistics correspond to those of random matrices from the Gaussian Orthogonal Ensemble according to the conjecture of Bohigas, Giannoni and Schmit. When the edge lengths respect the cyclic symmetry the spectrum decomposes into subspectra whose corresponding eigenfunctions transform according to irreducible representations of the cyclic group. We show that the subspectra exhibit intermediate spectral statistics and analyze the small and large parameter asymptotics of the two-point correlation function, applying techniques developed from star graphs. The particular form of the intermediate statistics differs from that seen for star graphs or Dirac rose graphs. As a further application, we show how the secular equations can be used to obtain spectral zeta functions using a contour integral technique. Results for the spectral determinant and vacuum energy of circulant graphs are obtained from the zeta functions.
The purpose of this article is to exploit the geometric structure of Quantum Mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end poin
ts of geodesics in the classical setting coincide with the probability distributions that minimise Shannons Entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
