Exact quasinormal frequencies of the Dirac field in a Lifshitz black brane

In the zero momentum limit we exactly calculate the quasinormal frequencies of the massive Dirac field propagating in a Lifshitz black brane. Our results are an extension of those for the massive Klein–Gordon field in the zero momentum limit, but in contrast to the boson field for which only the Dirichlet boundary condition was imposed at the asymptotic region, here we impose two different boundary conditions far from the event horizon and compare the values obtained for the quasinormal frequencies. Furthermore, based on our results we determine some relevant limits, study the classical stability of the quasinormal modes of the Dirac field and determine its relaxation times.     

Boundary conditions as Dirac constraints

In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints, which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are non-commutative. Received: 16 October 2000 / Revised version: 8 January 2001 / Published online: 23 February 2001     

有限空间中Dirac场的正则量子化

研究当存在边界的情形下 Dirac场的正则量子化问题. 采用文献[1]的观点, 将边界条件当作Dirac初级约束.与已有研究不同的是, 本文从离散的角度研究此问题. 将Dirac场的拉氏量和内在约束进行离散化, 并且将离散的边界条件当作初级Dirac约束. 因此, 从约束的起源来看, 这个模型中存在两种不同的约束: 一种是由于模型的奇异性而带来的约束, 即内在约束; 另一种是边界条件. 在对此模型进行正则量子化过程中提出一种能够平等地处理内在约束和边界条件的方法. 为了证明该方法能够平等地对待这两种起源不同的约束, 在计算Dirac 括号时分别选取了两个不同的子集合来构造"中间Dirac括号", 最后得到了相同的结果. 关键词： 边界条件 Dirac约束 Dirac括号     

有限空间中经典场的正则量子化

从离散的角度研究带边界的1+1维经典标量场和Dirac场的正则量子化问题. 与以往不同的是, 这里将时间和空间两个变量同时进行变步长的离散, 应用变步长离散的变分原理, 得到离散形式的运动方程、边界条件和能量守恒的表达式. 然后, 根据Dirac理论, 将边界条件当作初级约束, 将边界条件和内在约束统一处理. 研究表明, 采用此方法, 不仅在每个离散的时空格点上能够建立起Dirac括号, 从而可以完成该模型的正则量子化;而且, 该方法还保持了离散情况下的能量守恒.     

Relative zeta-determinants of Dirac Laplacians on a half-infinite cylinder with boundary conditions in the smooth,self-adjoint Grassmannian

In this paper we compute the relative zeta-determinants of Dirac Laplacians on a half-infinite cylinder when the Dirichlet boundary condition and the boundary condition belonging to the smooth, self-adjoint Grassmannian are imposed on the bottom of the half-infinite cylinder. We next apply this result to compute the relative zeta-determinants of Dirac Laplacians on a manifold with cylindrical end when the boundary condition belonging to the smooth, self-adjoint Grassmannian is imposed on the bottom of the half-infinite cylinder.     

Quantize Classical Field Theories with Boundaries Canonically

We study the problem of quantizing the classical fields with intrinsic second class constraints in a finite volume in this paper. To illustrate our idea clearly, we study the classical Schrodinger field in a finite volume. We work in the discrete version and take the discrete boundary conditions (BCs) as primary Dirac constraints, both Dirichlet and Neumann BCs are considered. We find it is possible to treat the BCs and intrinsic constraints on the same footing.     

Dirac quantization of open p-brane

We give the scheme of Dirac quantization of open p-brane in the D-brane background. Treating the mixed boundary conditions as primary constraints, we get a set of secondary constraints, then the constraints conditions are shown to be equivalent to orbifold conditions imposed on normal p-brane modes.     

The Casimir effect for fermions in one dimension

We study the Casimir problem for a fermion coupled to a static background field in one space dimension. We examine the relationship between interactions and boundary conditions for the Dirac field. In the limit that the background becomes concentrated at a point (a “Dirac spike”) and couples strongly, it implements a confining boundary condition. We compute the Casimir energy for a masslike background and show that it is finite for a stepwise continuous background field. However the total Casimir energy diverges for the Dirac spike. The divergence cannot be removed by standard renormalization methods. We compute the Casimir energy density of configurations where the background field consists of one or two sharp spikes and show that the energy density is finite except at the spikes. Finally we define and compute an interaction energy density and the force between two Dirac spikes as a function of the strength and separation of the spikes.     

Relativistic surface states of a semi-infinite chain acted on by an electric field

In this paper, we deal with the relativistic effect of surface states of a semi-infinite chain of the Kronig-Penney type potential acted on by an electric field. The crux of the problem is to solve Dirac equation with a linear potential. Combining the solution with other solutions in the various regions, known through the boundary conditions, we get a relativistic expression for energy of surface states. The form is similar to that obtained in the absence of an electric field, which the expression automatically approaches as the field vanishes.     

A non‐uniqueness problem of the Dirac theory in a curved spacetime

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four‐dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non‐uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.     

有界原子模型的原子结构计算

本文根据有界原子模型,通过对Dirac方程的求解,计算了H、Fe在不同物质密度、不同边界条件下的原子结构。结果发现:当物质密度较小时,近似于自由原子时的情况;当物质密度较大时,原子的外壳层电子能级有较大的变化。并且根据H在不同物质密度、不同边界条件下的原子结构的分析,展现了H的能带图象。     

Vacuum expectation values of the energy-momentum tensor of quantized fields on manifolds with different topologies and geometries. III

In a continuation of the earlier work [1, 2] into vacuum effects of quantized fields on bounded manifolds, the case of a spinor field is considered. The question of the boundary conditions for the Dirac equation is studied. Analytic results are obtained for a number of configurations. A comparison is made with the case of boson fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 9–13, July, 1980.     

带边界条件复标量场的正则量子化

研究了带边界条件有质量复标量场的量子化. 与把边界条件当作Dirac约束方法不同, 我们在经典解空间研究这个问题, 利用Fadeev-Jackiw(FJ)方法获得所有傅里叶模的对易关系, 避免用Dirac方法而产生的问题.     

The Casimir effect for fields with arbitrary spin

The Casimir force arises when a quantum field is confined between objects that apply boundary conditions to it. In a recent paper we used the two-spinor calculus to derive boundary conditions applicable to fields with arbitrary spin in the presence of perfectly reflecting surfaces. Here we use these general boundary conditions to investigate the Casimir force between two parallel perfectly reflecting plates for fields up to spin-2. We use the two-spinor calculus formalism to present a unified calculation of well-known results for spin-1/2 (Dirac) and spin-1 (Maxwell) fields. We then use our unified framework to derive new results for the spin-3/2 and spin-2 fields, which turn out to be the same as those for spin-1/2 and spin-1. This is part of a broader conclusion that there are only two different Casimir forces for perfectly reflecting plates—one associated with fermions and the other with bosons.     

Noncommutative geometric spaces with boundary: Spectral action

We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In particular we evaluate the spectral action corresponding to the noncommutative space of the standard model and show that the Einstein–Hilbert action gets modified by the addition of the extrinsic curvature terms with the right sign and coefficient necessary for consistency of the Hamiltonian. We also include effects due to the addition of a dilaton field.     

Setting the Quantum Integrand of M-Theory

In anomaly-free quantum field theories the integrand in the bosonic functional integral—the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand'. In the low-energy approximation to M-theory the E₈-model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction of heterotic M-theory on cylinders. The work of D.F. is supported in part by NSF grant DMS-0305505. The work of G.M. is supported in part by DOE grant DE-FG02-96ER40949     

Effect of the dependence of the specular reflectance on the angle of incidence of electrons on the impedance

We obtain the solution to the problem of the skin effect in a metal with specular-diffusion boundary conditions for arbitrary values of the anomaly parameter in the form of the Neumann series. For this purpose, we develop a method based on the idea of representation of not only the boundary condition imposed on the field (as is conventionally done), but also the boundary condition imposed on the distribution function, in the form of a source. The specular reflectance is an arbitrary function of the angle of incidence of electrons on the metal surface.     

Spin-1/2 relativistic particle in a magnetic field in NC phase space

This work provides an accurate study of the spin-l/2 relativistic particle in a magnetic field in NC phase space. By detailed calculation we find that the Dirac equation of the relativistic particle in a magnetic field in noncommutative space has similar behaviour to what happens in the Landau problem in commutative space even if an exact map does not exist. By solving the Dirac equation in NC phase space, we not only obtain the energy level of the spin-1/2 relativistic particle in a magnetic field in NC phase space but also explicitly offer some additional terms related to the momentum-momentum non-commutativity.