On the state space of the dipole ghost

A particular representation of SO(4, 2) is identified with the state space of the free dipole ghost. This representation is then given an explicit realization as the solution space of a 4th-order wave equation on a spacetime locally isomorphic to Minkowski space. A discrete basis for this solution space is given, as well as an explicit expression for its SO(4, 2) invariant inner product. The connection between the modes of dipole field and those of the massless scalar field is clarified, and a recent conjecture concerning the restriction of the dipole representation to the Poincaré subgroup is confirmed. A particular coordinate transformation then reveals the theory of the dipole ghost in Minkowski space. Finally, it is shown that the solution space of the dipole equation is not unitarizable in a Poincaré invariant manner.     

Spinor fields in the theory of relativity and a generalization of Heisenberg's nonlinear spinor equation

Some years ago it was shown that the nonlinear term of Heisenberg's spinor equation can be derived by torsion of the Minkowski space (Cartan space). This result is applied in the investigations of this paper. As the Heisenberg equation does not show any connection with recent phenomenological theories in high energy physics, like the parton or quark model, the problems of the metric of space-time are discussed from the aspect of fundamental axioms of topology (Hausdorff space). It will be shown that Feynman's relativistic parton theory can be derived by means of a quantised de Sitter space, where the constant curvature can assume only discrete values. It is also possible to derive the Dirac equation from the same mathematical considerations. A nonlinear spinor equation will be formulated which contains the parton theory and the nonlinear term of the Heisenberg equation as different approaches in the theory of elementary particles.     

Integrable systems from inelastic curve flows in 2– and 3– dimensional Minkowski space

Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2– and 3– dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrödinger (NLS) equation in 2– and 3– dimensional Euclidean space, respectively. In 2–dimensional Minkowski space, timelike/spacelike inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers’ equation and its symmetry integrability structure. In 3–dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.     

Langevin equation in field theory

A new approach to quantum field theory is developed based on the Langevin equation (stochastic quantization). Applications to conventional and gauge theories are discussed, as well as various extensions; the Langevin difference equation, the complex Langevin equation in Minkowski space, etc.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 66–76, March, 1986.     

Solutions of the Bethe–Salpeter Equation in Minkowski Space and Applications to Electromagnetic Form Factors

We present a new method for solving the two-body Bethe–Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe–Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe–Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.     

An Integral Transform in de Sitter Space

An integral transform on (4 + 1) de Sitter space, which is a generalization of the Fourier transform for a Dirac particle in Minkowski space, is considered, and proofs of its relevant properties are provided. With this integral transform we demonstrate explicitly that the momentum space eigenvalue equation for the second order Casimir operator of the de Sitter group is equivalent to a wave equation in Minkowski momentum space, which describes the mass-spin relation of a mechanical system in Minkowski space, consisting of two equal mass, point-like constituents rotating uniformly at a distance R from their geometric center, where R is the radius of the de Sitter space. Applications to the relativistic rotator are considered. Contrary to our previous results, we find that the relativistic rotator does not go into a structureless relativistic mass point in the elementary limit obtained by contracting the de Sitter group into the Poincaré group. Our analysis can be carried over, with relatively minor modifications, to anti-de Sitter space, and similar results hold there. Additional physical consequences are also discussed.     

Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass

The definition of quasi-local mass for a bounded space-like region Ω in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface \({\Sigma=\partial \Omega}\) and should be independent of whichever space-like region \({\Sigma}\) bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York [4,5] and Liu-Yau [16,17] (see also related works [3,6,9,12,14,15,28,32]) to define such notions using the isometric embedding theorem into the Euclidean three space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive [19]. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang’s [13] equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover a new expression of quasi-local mass for a large class of “admissible” surfaces (see Theorem A and Remark 1.1). The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial infinity and null infinity. Some of these results were announced in [29].     

Dirac Equation with Central Potential: Discrete Spectrum of the Hydrogen Atom in the Robertson–Walker Space–Time

The formulation of the Dirac equation withelectromagnetic field for a general space–time isspecialized to the Robertson–Walker metric. For aclass of physically meaningful electromagneticpotentials the angular part of the wave function separates asin the free-field case. The scheme is explicitly studiedfor a Coulomb potential. By using a realisticapproximation method one recovers the discrete energy levels of the hydrogen atom in Minkowski space.In case of static space–time, the result is exactfor zero curvature, while it is approximate for nonzerocurvature. The very good order of accuracy of the result is established by a comparison withsimilar qualitative and perturbative results.     

A canonical dilation of the Schrödinger equation

In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.     

相空间中离散完整系统的Noether对称性和Mei对称性

研究相空间中离散完整系统的Noether对称性、Mei对称性及其导致的守恒量.利用差分离散变分方法,给出相空间中离散完整系统的差分离散变分原理,建立系统的离散正则方程和能量演化方程;给出系统Noether对称性和Mei对称性的判定条件,得到系统离散形式的Noether守恒量和Mei守恒量及其存在的条件.举例说明结果的应用. 关键词： 相空间 离散完整系统 对称性 守恒量     

Intrinsic geometry of curves and the Minkowski force

The Frenet-Serret equations for a curve in a Riemann space are used to derive a theorem regarding the Minkowski Force. The consequence is that the well-known Lorentz-Dirac equation, involving radiation reaction is already implicit in the geometry.     

四维Minkowski空间中具有立方非线性复标量场方程的平面与球面波解

本文在四维Minkowski空间中对具有立方非线性复标量场方程进行了研究,并给出了场方程的两种精确解,这两种解相当于场的平面与球面波解。 关键词：     

Solving Bethe–Salpeter Equation for Scattering States

We present the Minkowski space solutions of the inhomogeneous Bethe–Salpeter equation for spinless particles with a ladder kernel. The off-mass shell scattering amplitude is first obtained.     

The Δ- Method for the Boltzmann Equation

A new numerical method is proposed to solve the Boltzmann equation. A frame is set up by using a discrete velocity approximation in the infinite velocity space, but by considering only those distribution function points which are not too small. The distribution function points may occur anywhere in the infinite discrete velocity space and are not constrained to a pre-specified region. A fourth-order finite difference is used for the convection terms. A Monte Carlo-like method is applied to the discrete velocity model of the collision integral. The effort of the method is proportional to the number of discrete points. Numerical examples are given for the full Boltzmann equation and results for some benchmark problems are compared with analytical or prior solutions.     

On multimeron solutions of the classical Yang-Mills equation

The double-meron solution and the solution indicated by Protogenov [1] of the SU(2) Yang-Mills equation in four-dimensional euclidean space are generalised. It is noted that the double-meron solution has an analogous solution in Minkowski space which has a finite energy integral.     

Reconstruction of the standard model in a generalized differential geometry based on the real structure

The standard model is reconstructed in a generalized differential geometry (GDG) based on the idea of a real structure as proposed by Coquereaux et al. and Connes. The GDG considered in this article is a kind of non-commutative geometry (NCG) on the discrete space that successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory. Here, a GDG is a direct generalization of the differential geometry on an ordinary continuous manifold to the product space of this manifold with a discrete manifold. In a GDG, a one-form basis on the discrete space is incorporated in addition to the one-form basis on Minkowski space, rather than as in Connes's original work. Although the Lagrangians obtained in this way are the same as those obtained in our previous formulation of GDG, the basic formalism becomes very simply and clear. Received: 31 January 2000 / Revised version: 28 July 2000 / Published online: 25 April 2001     

Solving the Bethe-Salpeter equation for two fermions in Minkowski space

The method of solving the Bethe-Salpeter equation in Minkowski space, developed previously for spinless particles (V.A. Karmanov, J. Carbonell, Eur. Phys. J. A 27, 1 (2006)), is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original BS amplitudes. The numerical solutions for the J = 0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. The stability of the scalar and positronium states without vertex form factor is discussed. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained.     

基于相空间的复杂物理场建模与分析方法

评述两个基于相空间的建模与分析方法及其应用。第一个是基于闵可夫斯基泛函的形态分析方法，第二个是基于离散玻尔兹曼方程的建模与分析方法。两者均是统计物理学相空间描述方法的进一步发展：以相对独立的行为特征量为基，构建相空间，使用该相空间和其子空间来描述系统的行为特征；该相空间中的一个点对应系统的一组行为特征；两点间的距离d可用来描述两组行为特征的差异，其倒数可用来描述两组行为特征的相似度（S=1/d）；一段时间内两点间距离的平均值${\bar d}$可用来描述两个动理学过程的差异，其倒数可用来描述这两个动理学过程的相似度（S_p=1/${\bar d}$）。从历史角度，基于闵可夫斯基泛函的形态相空间分析方法在先，接受其启发是离散玻尔兹曼方法朝着相空间描述方法发展过程中的关键环节。形态分析方法独立于数据来源，因而离散玻尔兹曼模拟得到的结果，除了可以使用其自带的分析功能之外，还可进一步使用形态分析方法获得另一个层面或视角的认识。在复杂介质动理学研究中，这两个方法从不同的视角，使得许多以前无法提取的信息得以分层次、定量化研究。     

Renormalization and scaling behavior of non-Abelian gauge fields in curved spacetime

In this article we discuss the one loop renormalization and scaling behavior of non-Abelian gauge field theories in a general curved spacetime. A generating functional is constructed which forms the basis for both the perturbation expansion and the Ward identities. Local momentum space representations for the vector and ghost particles are developed and used to extract the divergent parts of Feynman integrals. The one loop diagram for the ghost propagator and the vector-ghost vertex are shown to have no divergences not present in Minkowski space. The Ward identities insure that this is true for the vector propagator as well. It is shown that the above renormalizations render the three- and four-vector vertices finite. Finally, a renormalization group equation valid in curved spacetimes is derived. Its solution is given and the theory is shown to be asymptotically free as in Minkowski space.     

Quantum Minkowski Space and the q-Analogue of Dirac Equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices. The quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects a q-analogue of Dirac equation follows directly.