Dirac Quantum Field Theory in Rindler Spacetime

The dynamical properties of Dirac particles inRindler spacetime are investigated. It is shown that thevacuum state of the Dirac field in Minkowski spacetimeappears to be a thermal state for a Rindler observer, and the usual thermal equilibriumstate of the Dirac field in Minkowski spacetime is aquasithermal equilibrium state, which is timeindependent and characterized by two quasi-temperatureparameters for a Rindler observer.     

Scalar Fields in Multidimensional Gravity. No-Hair and Other No-Go Theorems

Global properties of static, spherically symmetric configurations with scalar fields of sigma-model type with arbitrary potentials are studied in D dimensions, including models where the space-time contains multiple internal factor spaces. The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and the potential V includes a contribution from their curvatures. The following results generalize those known in four dimensions: (A) a no-hair theorem on the nonexistence, in case V 0, of asymptotically flat black holes with varying scalar fields or moduli fields outside the event horizon; (B) nonexistence of particlelike solutions in field models with V 0; (C) nonexistence of wormhole solutions under very general conditions; (D) a restriction on possible global causal structures (represented by Carter-Penrose diagrams). The list of structures in all models under consideration is the same as is known for vacuum with a cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild – de Sitter, and horizons which bound a static region are always simple. The results are applicable to various Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli fields.     

Four-Vertex Theorems, Sturm Theory and Lagrangian Singularities

We prove that the vertices of a curve γ⊂R ⁿ are critical points of the radius of the osculating hypersphere. Using Sturm theory, we give a new proof of the (2k+2)-vertex theorem for convex curves in the Euclidean space R ^2k . We obtain a very practical formula to calculate the vertices of a curve in R ⁿ . We apply our formula and Sturm theory to calculate the number of vertices of the generalized ellipses in R ^2k . Moreover, we explain the relations between vertices of curves in Euclidean n-space, singularities of caustics and Sturm theory (for the fundamental systems of solutions of disconjugate homogeneous linear differential operators L:C ^∞(S ¹)→C ^∞(S ¹)).     

Dirac Field on Moyal-Minkowski Spacetime and Non-commutative Potential Scattering

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov’s formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.     

Steps Towards the Axiomatic Foundations of the Relativistic Quantum Field Theory: Spin-Statistics, Commutation Relations, and CPT Theorems

A realistic physical axiomatic approach of the relativistic quantum field theory is presented. Following the action principle of Schwinger, a covariant and general formulation is obtained. The correspondence principle is not invoked and the commutation relations are not postulated but deduced. The most important theorems such as spin-statistics, and CPT are proved. The theory is constructed form the notion of basic field and system of basic fields. In comparison with others formulations, in our realistic approach fields are regarded as real things with symmetry properties. Finally, the general structure is contrasted with other formulations.     

Equivalence of Deviau's, Hestenes', and Parra's Formulations of Dirac Theory

Daviau showed the equivalence of matrix Dirac theory, formulated within a spinor bundle , to a Clifford algebraic formulation within space Clifford algebra Pauli algebra (matrices) biquaternions. We will show, that Daviau's map : is an isomorphism. It is shown that Hestenes' and Parra's formulations are equivalent to Daviau's Clifford algebra formulation, which uses outer automorphisms. The connection between different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra . The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau, and others to interprete the right action as the iso-spin freedom.     

Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

It is proved that there exists a vector representation of Dirac‘s spinor field andin one sense it is equivalent to biquaternion (i.e. complexified quaternion) representation. This can be considered as a generalization of Cartan‘s idea of triality to Dirac‘s spinors. In the vector representation the first-order Dirac Lagrangian is dual-equivalent to the two-order Lagrangian of topologically massive gauge field. The potential field which corresponds to the Dirac field is obtained by using master (or parent) action approach. The novel gauge field is self-dual and contains both anti-symmetric Lee and symmetric Jordan structure.     

Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

It is proved that there exists a vector representation of Dirac＇s spinor field and. in one sense it is equivalent to biquaternion （i.e. complexified quaternion） representation. This can be considered as a generalization of Cartan＇s idea of triality to Dirac＇s spinors. In the vector representation the first-order Dirac Lagrangian is dual-equivalent to the two-order Lagrangian of topologically massive gauge field. The potential field which corresponds to the Dirac field is obta/ned by using master （or parent） action approach. The novel gauge field is self-dual and contains both anti-symmetric Lee and symmetric Jordan structure.     

A Remark on Schwarz's Topological Field Theory

The standard evaluation of the partition function Z of Schwarz's topological field theory results in the Ray–Singer analytic torsion. Here we present an alternative evaluation which results in Z=1. Mathematically, this amounts to a novel perspective on analytic torsion: it can be formally written as a ratio of volumes of spaces of differential forms which is formally equal to 1 by Hodge duality. An analogous result for Reidemeister combinatorial torsion is also obtained.     

Connections on Naturally Reductive Spaces,Their Dirac Operator and Homogeneous Models in String Theory

 Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ ^t joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D ^t corresponding to t=1/3 is the so-called ``cubic' Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator on spinors and an eigenvalue estimate for the first eigenvalue of D <sup>1/3</sup>. This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T≠ 0 is a 3-form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and a detailed discussion of a 5-dimensional example. Received: 19 February 2002 / Accepted: 26 August 2002 Published online: 22 November 2002 RID="*" ID="*" This work was supported by the SFB 288 ``Differential geometry and quantum physics' of the Deutsche Forschungsgemeinschaft and the Max-Planck Society.     

A Search on Dirac Equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.     

Massive, Non-ghost Solutions for the Dirac Field Coupled Self-consistently to Gravity

We present new, massive, non-ghost solutions for the Dirac field coupled self-consistently to gravity. We employ a gauge-theoretic formulation of gravity which automatically identifies the spin of the Dirac field with the torsion of the gauge fields. Homogeneity of the field observables requires that the spatial sections be flat. Expanding and collapsing singular solutions are given, as well as a solution which expands from a singularity before recollapsing. Torsion effects are only important while the Compton wavelength of the Dirac field is larger than the Hubble radius. We study the motion of spinning point-particles in the background of the expanding solution. The anisotropy due to the torsion is manifest in the particle trajectories.     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.