Flat Minkowski Spacetime Research Articles

Casimir effect on the radius stabilization of the noncommutative torus

We evaluate the one-loop correction to the spectrum of Kaluza–Klein system for the φ 3 model on R 1, d ×( T θ 2) L , where 1+ d dimensions are the ordinary flat Minkowski spacetimes and the extra dimensions are the L two-dimensional noncommutative tori with noncommutativity θ. The correction to the Kaluza–Klein mass spectrum is then used to compute the Casimir energy. The results show that when L>2 the Casimir energy due to the noncommutativity could give repulsive force to stabilize the extra noncommutative tori in the cases of d=4 n−2, with n a positive integral.

Finite-temperature Casimir effect on the radius stabilization of noncommutative torus

The one-loop correction to the spectrum of Kaluza-Klein system for the $\phi^3$ model on $R^{1,d}\times (T_\theta^2)^L$ is evaluated in the high temperature limit, where the $1+d$ dimensions are the ordinary flat Minkowski spacetimes and the $L$ extra two-dimensional tori are chosen to be the noncommutative torus with noncommutativity $\theta$. The corrections to the Kaluza-Klein mass formula are evaluated and used to compute the Casimir energy with the help of the Schwinger perturbative formula in the zeta-function regularization method. The results show that the one-loop Casimir energy is independent of the radius of torus if L=1. However, when $L>1$ the Casimir energy could give repulsive force to stabilize the extra noncommutative torus if $d-L$ is a non-negative even integral. This therefore suggests a possible stabilization mechanism of extra radius in high temperature, when the extra spaces are noncommutative.

A new family of gauges in linearized general relativity

For vacuum Maxwell theory in four dimensions, a supplementary condition exists (due to Eastwood and Singer) which is invariant under conformal rescalings of the metric, in agreement with the conformal symmetry of the Maxwell equations. Thus, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, one can consider, led by formal analogy, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. The admissibility of such gauges in the classical theory is first proven in the cases of linearized theory about flat Euclidean space or flat Minkowski spacetime. In the former, the general solution of the equation for the fulfillment of the gauge condition after infinitesimal diffeomorphisms involves a 3-harmonic 1-form and an inverse Fourier transform. In the latter, one needs instead the kernel of powers of the wave operator, and a contour integral. The analysis is also used to put restrictions on the dimensionless parameter occurring in the DeWitt supermetric, while the proof of admissibility is generalized to a suitable class of curved Riemannian backgrounds. Eventually, a non-local construction of the tensor field is obtained which makes it possible to achieve conformal invariance of the above gauges.

From Time Inversion to Nonlinear QED

In Minkowski flat space-time, it is perceived that time inversion is unitary rather than antiunitary, with energy being a time vector changing sign under time inversion. The Dirac equation, in the case of electromagnetic interaction, is not invariant under unitary time inversion, giving rise to a “Klein paradox.” To render unitary time inversion invariance, a nonlinear wave equation is constructed, in which the “Klein paradox” disappears. In the case of Coulomb interaction, the revised nonlinear equation can be linearized to give energy solutions for Hydrogen-like ions without singularity when nuclear number Z>137, showing a reversed energy order pending for experimental tests such as Zeeman effects. In non-relativistic limit, this nonlinear equation reduces to nonlinear Schrodinger equation with soliton-like solutions. Moreover, particle conjugation and electron-proton scattering with a nonsingular current-potential interaction are discussed. Finally the explicit form of gauge function is found, the uniqueness of Lorentz gauge is proven and the Lagrangian density of quantum electrodynamics (QED) is revised as well. The implementation of unitary time inversion leads to the ultimate derivation of nonlinear QED.

Background geometry of DLCQ M theory on ap-torus and holography

Via supergravity, we argue that the infinite Lorentz boost along the M theory circle in the manner of Seiberg toward the DLCQ M theory compactified on a $p$-torus $(p<5)$ implies the holographic description of the microscopic theory. This argument lets us identify the background geometries of DLCQ M theory on a $p$-torus; for $p=0(p=1),$ the background geometry turns out to be eleven-dimensional (ten-dimensional) flat Minkowski space-time, respectively. Holography for these cases results from the localization of the light-cone momentum. For $p=2,3,4,$ the background geometries are the tensor products of an anti--de Sitter space and a sphere, which, according to the AdS-CFT correspondence, have the holographic conformal field theory description. These holographic descriptions are compatible to the microscopic theory of Seiberg based on $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{M}$ theory on a spatial circle with the rescaled Planck length, giving an understanding of the validity of the AdS-CFT correspondence.

A fully -dimensional Regge calculus model of the Kasner cosmology

We describe the first discrete-time four-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of four-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built from tetrahedra. We implement a novel 2-surface initial-data prescription for Regge calculus, and provide the first fully four-dimensional application of an implicit decoupled evolution scheme (the `Sorkin evolution scheme'). We benchmark this code on the Kasner cosmology - a cosmology which embodies generic features of the collapse of many cosmological models. We (i) reproduce the continuum solution with a fractional error in the 3-volume of after evolution steps; (ii) demonstrate stable evolution; (iii) preserve the standard deviation of spatial homogeneity to less than and (iv) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.

Quantum Fields in the Spacetime Tangent Bundle

Maximal-acceleration invariant quantum fields are formulated in terms of the differential geometric structure of the spacetime tangent bundle. The simple special case is considered of a flat Minkowski space-time for which the bundle is also flat. The field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.

On the construction of a Dirac spinor that generates a specified tetrad

It is well known that a complex four-component Dirac spinor defines an orthonormal tetrad on flat Minkowski spacetime. For example, a spinor solution to the Dirac equation determines the four-velocity and Pauli-Lubanski spin vector, comprising the timelike and first spacelike members of the particle’s tetrad, as well as two other spacelike members that are rarely discussed. Also, of particular note is the complex null tetrad formalism that provides a map from a two-component SL(2, C) spinor and its complex conjugate to a tetrad. The inverse problem is studied here. Given a tetrad, a complex Dirac spinor valued function of the tetrad is explicitly defined in such a way that certain sums of bilinear products of the components of this spinor exactly reproduce the tetrad. This spinor may be used in the Feynman propagator representation of the solution to the Dirac equation to generate a Dirac wave spinor with desired initial properties. The mappings studied here represent a new class of nonlinear mappings SO(3,1)+↑→SO(3,1)¯.

Internal symmetry of hadrons: Finsler geometrical origin

The microlocal space of hadronic matter extension has recently been characterized as a Finsler space. This consideration of hadrons extended as composites of constituents can give rise to a dynamical theory of hadrons. The macrospaces, the space-time of common experience (the Minkowski flat space-time) and the Robertson-Walker background space-time of the universe, are found to appear as the “averaged” space-times of the Finsler space that describes the anisotropic nature of the microdomain of hadrons. From the assumed property of the fields of the constituents in the microspace it is possible to find the field (or wave) equations of the particles (or constituents) through the quantization of space-time at small distances (to an order of or less than a fundamental length). If the field (or wave) function is separable in the functions of the coordinates of the underlying manifold and the directional arguments of the Finsler space, then the former part of the field function is found to satisfy the Dirac equation in the Minkowski space-time or in the Robertson-Walker space-time according to the nature of the underlying manifold. In the course of finding a solution for the other part of the field function a relation between the mass of the particle and a parameter in the metric of the space-time has been obtained as a byproduct. This mass relation has cosmological implications and is relevant in the very early stage of the evolution of the universe. In fact, it has been shown elsewhere that the universe might have originated from a nonsingular origin with entropy and matter creations that can account for the observed photon-to-baryon ratio and total particle number of the present universe. The equations in the directional arguments for the constituents in the hadron configuration are found here and give rise to an additional quantum number in the form of an “internal” helicity that can generate the internal symmetry of hadron if one incorporates the arguments of Budini in generating the internal isospin algebra from the conformal reflection group. This consideration can also account for the meson-baryon mass differences.

Mass spectrum of strings in anti-de Sitter spacetime.

We perform string quantization in anti de Sitter (AdS) spacetime. The string motion is stable, oscillatory in time with real frequencies $\omega_n= \sqrt{n^2+m^2\alpha'^2H^2}$ and the string size and energy are bounded. The string fluctuations around the center of mass are well behaved. We find the mass formula which is also well behaved in all regimes. There is an {\it infinite} number of states with arbitrarily high mass in AdS (in de Sitter (dS) there is a {\it finite} number of states only). The critical dimension at which the graviton appears is $D=25,$ as in de Sitter space. A cosmological constant $\Lambda\neq 0$ (whatever its sign) introduces a {\it fine structure} effect (splitting of levels) in the mass spectrum at all states beyond the graviton. The high mass spectrum changes drastically with respect to flat Minkowski spacetime. For $\Lambda<0,$ we find $<m^2>\sim \mid\Lambda\mid N^2,$ {\it independent} of $\alpha',$ and the level spacing {\it grows} with the eigenvalue of the number operator, $N.$ The density of states $\rho(m)$ grows like $\mbox{Exp}[(m/\sqrt{\mid\Lambda\mid}\;)^{1/2}]$ (instead of $\rho(m)\sim\mbox{Exp}[m\sqrt{\alpha'}]$ as in Minkowski space), thus {\it discarding} the existence of a critical string temperature. For the sake of completeness, we also study the quantum strings in the black string background, where strings behave, in many respects, as in the ordinary black hole backgrounds. The mass spectrum is equal to the mass spectrum in flat Minkowski space.

Finite Energy Solutions of the Yang-Mills Equations in ℝ 3+1

Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the size of the energy norm of the data. Since the energy norm is left invariant by the Yang-Mills flow the local solution is automatically extended to the entire space-time. Thus our results, which settle a problem stated in [Str], imply the well-known, fundamental, regularity result of Eardley and Moncrief [E-M]. That result, proved in the temporal gauge, requires a higher degree of smoothness for the data. Our main result, proved also in the temporal gauge, allows us to extend the concept of solutions to arbitrary finite energy initial data. The solutions are automatically unique in the class of solutions obtained by our procedure. Moreover, the global regularity proof given by Eardley and Moncrief depends in an essential way on the specific properties of the fundamental solution of the wave equation in the flat Minkowski space-time R3+1, namely the strong Huygens Principle. Indeed due mainly to this fact their proof does not seem to extend to general curved space-times. We have reasons to hope that the very different approach we take here will resolve this difficulty. The basic ingredients of our method are: 1. The introduction of appropriate local Coulomb gauges adapted to the causal structure of the equations. 2. An appropriate method of localizing the new space-time estimates for

Cosmic string nucleation near the inflationary phase boundary

We investigate the nucleation of circular cosmic strings in models of generalized inflationary universes with an accelerating scale factor. We consider toy cosmological models of a smooth inflationary exit and transition into a flat Minkowski spacetime. Our results establish that an inflationary expanding phase is necessary but not sufficient for quantum nucleation of circular cosmic strings to occur.

Radiating sphere in a Kaluza–Klein space–time

The well-known solution of Vaidya for a radiating sphere in a Kaluza–Klein-type of space–time is extended. The field equations yield a two-parameter solution—either an exactly four-dimensional Vaidya metric with a flat extra space tucked to it or a flat Minkowski space–time. The geodesic trajectories of test particles in this field are studied and some astrophysical consequences due to the presence of the extra space are also briefly discussed.

Tree scattering amplitudes of the spin-4/3 fractional superstring. I. The untwisted sectors.

Scattering amplitudes of the spin-4/3 fractional superstring are shown to satisfy spurious state decoupling and cyclic symmetry (duality) at tree-level in the string perturbation expansion. This fractional superstring is characterized by the spin-4/3 fractional superconformal algebra---a parafermionic algebra studied by Zamolodchikov and Fateev involving chiral spin-4/3 currents on the world-sheet in addition to the stress-energy tensor. Examples of tree scattering amplitudes are calculated in an explicit c=5 representation of this fractional superconformal algebra realized in terms of free bosons on the string world-sheet. The target space of this model is three-dimensional flat Minkowski space-time with a level-2 Kac-Moody so(2,1) internal symmetry, and has bosons and fermions in its spectrum. Its closed string version contains a graviton in its spectrum. Tree-level unitarity (i.e., the no-ghost theorem for space-time bosonic physical states) can be shown for this model. Since the critical central charge of the spin-4/3 fractional superstring theory is 10, this c=5 representation cannot be consistent at the string loop level. The existence of a critical fractional superstring containing a four-dimensional space-time remains an open question.

Tree scattering amplitudes of the spin-4/3 fractional superstring. II. The twisted sectors.

The spin-4/3 fractional superstring is characterized by a world-sheet chiral algebra involving spin-4/3 currents. The discussion of the tree-level scattering amplitudes of this theory presented in hepth/9310131 is expanded to include amplitudes containing two twisted-sector states. These amplitudes are shown to satisfy spurious state decoupling. The restriction to only two external twisted-sector states is due to the absence of an appropriate dimension-one vertex describing the emission of a single twisted-sector state. This is analogous to the ``old covariant'' formalism of ordinary superstring amplitudes in which an appropriate dimension-one vertex for the emission of a Ramond-sector state is lacking. Examples of tree scattering amplitudes are calculated in a c=5 model of the spin-4/3 chiral algebra realized in terms of free bosons on the string world-sheet. The target space of this model is three-dimensional flat Minkowski space-time and the twisted-sector physical states are fermions in space-time. Since the critical central charge of the spin-4/3 fractional superstring theory is 10, this c=5 model is not consistent at the string loop level.

Kac and new determinants for fractional superconformal algebras.

We derive the Kac and new determinant formulae for an arbitrary (integer) level $K$ fractional superconformal algebra using the BRST cohomology techniques developed in conformal field theory. In particular, we reproduce the Kac determinants for the Virasoro ($K=1$) and superconformal ($K=2$) algebras. For $K\geq3$ there always exist modules where the Kac determinant factorizes into a product of more fundamental new determinants. Using our results for general $K$, we sketch the non-unitarity proof for the $SU(2)$ minimal series; as expected, the only unitary models are those already known from the coset construction. We apply the Kac determinant formulae for the spin-4/3 parafermion current algebra ({\em i.e.}, the $K=4$ fractional superconformal algebra) to the recently constructed three-dimensional flat Minkowski space-time representation of the spin-4/3 fractional superstring. We prove the no-ghost theorem for the space-time bosonic sector of this theory; that is, its physical spectrum is free of negative-norm states.

The Clifford bundle and the nature of the gravitational field

In this paper we formulate Einstein's gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime.

Construction of space-time by gauge translations

The following procedure is described: Starting with a connection in a principal fiber bundle P(M,G), where G is either the Poincaré group or one of the de Sitter groups, a connection in the bundle of linear frames of a submanifold N of M is constructed by using the translational components of the original connection for frame identification. The dimension of N is gauge dependent, and the flat four-dimensional Minkowski space-time may appear of dimension less than 4 when certain gauges are used. It is shown that in the case of a de Sitter group, the minimum dimension to which the flat four-dimensional space-time can be reduced is 1, while the number is 0 for the Poincaré group. The gauge transformation that achieves the maximum dimension reduction in the de Sitter case is constant and leads to infinite strings as a result. Variable continuous gauge transformations that can reduce the dimension over a finite region of the base manifold are also considered.

The Schwinger terms and the gravitational anomaly

A new type of Schwinger term appearing in commutators of energy-momentum tensors is shown for chiral fermions in two-dimensional flat Minkowski space-time. Preserving the Jacobi identity, the Schwinger terms which correspond to gravitational anomalies appear in places different from the conformal anomaly. A relation to Faddeev's argument is also discussed.

The Gravitational Moment Densities of Classical Matter Fields with Spin

AbstractWe consider massive classical matter fields with spin 1/2, 1, 3/2 and 2 in the flat Minkowski spacetime. By the method of Gordon, we decompose their respective momentum and spin current densities into their convective and polarisation parts (see eqns. (10a, b)). We find the translational and rotational gravitational moment densities of these fields. Their general properties are discussed and we draw some conclusions for a gauge theoretical treatment of gravity.