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.     

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     

Canonical Quantization of Field Theories with Boundaries

The problem of canonical quantization of singular systems in a finite volume is studied by analysing a non-relativistic field theory. Firstly, we take the boundary conditions (BCs) as primary Dirac constraints. The quantization is performed canonically using Dirac’s procedure. Then, we quantize this model canonically in the classical solution space. We show that these two different quantization schemes are equivalent although they start from different settings.     

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.     

Eigenvalue Boundary Problems for the Dirac Operator

 On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors. Received: 12 November 2001 / Accepted: 25 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967 and by European Union FEDER funds     

On the boundary conditions as Dirac constraints

In this paper, the canonical quantization of singular Lagrangian defined in a finite volume is discussed by studying a 1 + 1 dimensional free Schrödinger field. We take the boundary conditions (BCs) as Dirac constraints, and show that those BCs as well as the intrinsic constraints (which are introduced by the singularities of Lagrangian) form the second class constraints. The quantization is performed canonically.Received: 30 August 2004, Revised: 2 October 2004, Published online: 17 December 2004PACS: 11.25-W, 04.60.D, 11.10.E     

An action principle for the Neveu-Schwarz-Ramond string and other systems using supernumerary variables

An action principle which gives rise to the equations of motion and boundary conditions for the free relativistic string with fermionic degrees of freedom is presented. With the aid of extra variables, some of which are Grassmann functions, all the gauge generators are obtained as secondary constraints. The consistency of the system is demonstrated using a generalised Poisson bracket operation. The theory is quantised with Dirac brackets and the fermionic fields become elements of a Clifford algebra. The methods are also used to formulate the theory of the Klein-Gordon and Dirac point particles and the relativistic string and membrane without intrinsic spin. Under certain circumstances we show that the supernumerary variables may be removed entirely from the original Lagrangian.     

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary. Received: 22 August 2000 / Accepted: 15 March 2001     

BRST symmetries in SU(3) linear sigma model

We study the BRST symmetries in the SU(3) linear sigma model which is constructed through the introduction of a novel matrix for the Goldstone boson fields satisfying geometrical constraints embedded in a SU(2) subgroup. To treat these constraints we exploit the improved Dirac quantization scheme. We also discuss phenomenological aspects in the mean field approach to this model. Received: 16 November 2001 / Revised version: 23 February 2002 / Published online: 26 July 2002     

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.     

Generally covariant quantization and the Dirac field

Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.     

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.     

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.     

Entanglement entropy for a Dirac fermion in three dimensions: Vertex contribution

In three dimensions there is a logarithmically divergent contribution to the entanglement entropy which is due to the vertices located at the boundary of the region considered. In this work we find the corresponding universal coefficient for a free Dirac field, and extend a previous work in which the scalar case was treated. The problem is equivalent to find the conformal anomaly in three-dimensional space where multiplicative boundary conditions for the field are imposed on a plane angular sector. As an intermediate step of the calculation we compute the trace of the Green function of a massive Dirac field in a two-dimensional sphere with boundary conditions imposed on a segment of a great circle.     

Geometrization of the physics with teleparallelism. II. Towards a fully geometric Dirac equation

In an accompanying paper (I), it is shown that the basic equations of the theory of Lorentzian connections with teleparallelism (TP) acquire standard forms of physical field equations upon removal of the constraints represented by the Bianchi identities. A classical physical theory results that supersedes general relativity and Maxwell-Lorentz electrodynamics if the connection is viewed as Finslerian. The theory also encompasses a short-range, strong, classical interaction. It has, however, an open end, since the source side of the torsion field equation is not geometric.In this paper, Kaehler's partial geometrization of the Dirac equation is taken as a starting point for the development of fully geometric Dirac equations via the correspondence principle given in I. For this purpose, Kaehler's calculus (where the spinors are differential forms) is generalized so that it also applies when the torsion is not zero. The point is then made that the forms can take values in tangent Clifford algebras rather than in tensor algebras. The basic Eigenschaft of the Kaehler calculus also is examined from the physical perspective of dimensional analysis.Geometric Dirac equations of great structural simplicity are finally inferred from the standard Dirac equation by using the aforementioned correspondence principle. The realm of application of the Dirac theory is thus enriched in principle, though only at an abstract level at this point: the standard spinors, which are scalar-valued forms in the Kaehler version of that theory, become Clifford-valued. In addition, the geometrization of the Dirac equation implies a geometrization of the Dirac current. When this current is replaced in the field equations for the torsion, the theory of Paper I becomes fully geometric.     

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.     

On ghost fermions

The path integral for ghost fermions, which is heuristically made use of in the Batalin-Fradkin-Vilkovisky approach to quantization of constrained systems, is derived from first principles. The derivation turns out to be rather different from that of physical fermions since the definition of Dirac states for ghost fermions is subtle. With these results at hand, it is then shown that the nonminimal extension of the Becchi-Rouet-Stora-Tyutin operator must be chosen differently from the notorious choice made in the literature in order to avoid the boundary terms that have always plagued earlier treatments. Furthermore it is pointed out that the elimination of states with nonzero ghost number requires the introduction of a thermodynamic potential for ghosts; the reason is that Schwarz's Lefschetz formula for the partition function of the time-evolution operator is not capable, despite claims to the contrary, to get rid of nonzero ghost number states on its own. Finally, we comment on the problems of global topological nature that one faces in the attempt to obtain the solutions of the Dirac condition for physical states in a configuration space of nontrivial geometry; such complications give rise to anomalies that do not obey the Wess-Zumino consistency conditions. Received: 4 May 2001 / Revised version: 10 October 2001 / Published online: 8 February 2002     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Poisson brackets,strings and membranes

We construct Poisson brackets at the boundaries of open strings and membranes with constant background fields which are compatible with their boundary conditions. The boundary conditions are treated as primary constraints which give infinitely many secondary constraints. We show explicitly that we need only two (the primary one and one of the secondary ones) constraints to determine the Poisson brackets of strings. We apply this to membranes by using canonical transformations. Received: 2 May 2002 / Revised version: 29 May 2002 / Published online: 16 August 2002 RID="a" ID="a" e-mail: tezuka@physics.s.chiba-u.ac.jp