Four-dimensional Minkowski Space Research Articles

Spacelike intersection curve of three spacelike hypersurfaces in E41

In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space \(E_1^4\).

Relative Energy Shift of a Two-Level Atom in a Cylindrical Spacetime

We investigate the evolution dynamics of a two-level atom system interacting with the massless scalar field in a Cylindrical spacetime. We find that both the energy shifts of ground state and excited state can be separated into two parts due to the vacuum fluctuations. One is the corresponding energy shift for a rest atom in four-dimensional Minkowski space without spatial compactification, the other is just the modification of the spatial compactified periodic length. It will reveal that the influence of the presence of one spatial compactified dimension can not be neglected in Lamb shift as the relative energy level shift of an atom.

$ \mathcal{N} $ = 2 vacua in generalized geometry

Abstract We find the conditions on compactifications of type IIA to four-dimensional Minkowski space to preserve $ \mathcal{N} $ = 2 supersymmetry in the language of Exceptional Generalized Geometry (EGG) and Generalized Complex Geometry (GCG). In EGG, off-shell $ \mathcal{N} $ =2 supersymmetry requires the existence of a pair of SU(2)R singlet and triplet algebraic structures on the exceptional generalized tangent space that encode all the scalars (NS-NS and R-R) in vector and hypermultiplets respectively. We show that on shell $ \mathcal{N} $ = 2 requires, except for a single component, these structures to be closed under a derivative twisted by the NS-NS and R-R fluxes. We also derive the corresponding GCG-type equations for the two pairs of pure spinors that build up these structures.

Lagrange Anchor and Characteristic Symmetries of Free Massless Fields

A Poincar\'e covariant Lagrange anchor is found for the non-Lagrangian relativistic wave equations of Bargmann and Wigner describing free massless fields of spin $s>1/2$ in four-dimensional Minkowski space. By making use of this Lagrange anchor, we assign a symmetry to each conservation law and perform the path-integral quantization of the theory.

Invariants and quasi-umbilicity of timelike surfaces in Minkowski space [formula omitted

We first describe the numerical invariants and the curvature hyperbola attached to the second fundamental form of a timelike surface in four-dimensional Minkowski space: beside the four natural invariants, a new invariant appears at some special points of the surface, which are said to be quasi-umbilic; at such quasi-umbilic points, the curvature hyperbola degenerates to a line with one point removed. We then study the asymptotic lines on a timelike surface, and characterize the quasi-umbilic points of the surface as the points where the asymptotic directions degenerate to a double lightlike line. We also give an interpretation of the new invariant at a quasi-umbilic point, using the Gauss map of the surface. We finally describe the timelike surfaces which are quasi-umbilic at every point.

An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our main theorem states that these seven invariants determine the surface up to a motion in Minkowski space. We introduce meridian surfaces as one-parameter systems of meridians of a rotational hypersurface in the four-dimensional Minkowski space. We find all marginally trapped meridian surfaces.

Conformal hydrodynamics in Minkowski and de Sitter spacetimes

We show how to generate non-trivial solutions to the conformally invariant, relativistic fluid dynamic equations by appealing to the Weyl covariance of the stress tensor. We use this technique to show that a recently studied solution of the relativistic conformally invariant Navier–Stokes equations in four-dimensional Minkowski space can be recast as a static flow in three-dimensional de Sitter space times a line. The simplicity of the de Sitter form of the flow enables us to consider several generalizations of it, including flows in other spacetime dimensions, second order viscous corrections, and linearized perturbations. We also construct the anti-de Sitter dual of the original four-dimensional flow. Finally, we discuss possible applications to nuclear physics.

Geometric invariants and principal configurations on spacelike surfaces immersed in ℝ3,1

We describe the numerical invariants and the curvature ellipse attached to the second fundamental form of a spacelike surface in a four-dimensional Minkowski space. We then study the configuration of the V-principal curvature lines on a spacelike surface when the normal field V is lightlike (the lightcone configuration). We end with some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces.

An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.

QFT AS PILOT-WAVE THEORY OF PARTICLE CREATION AND DESTRUCTION

States in quantum field theory (QFT) are represented by many-particle wave functions, such that a state describing n particles depends on n space–time positions. Since a general state is a superposition of states with different numbers of particles, the wave function lives in the configuration space identified with a product of an infinite number of four-dimensional Minkowski space–times. The squared absolute value of the wave function is interpreted as the probability density in the configuration space, from which the standard probabilistic predictions of QFT can be recovered. Such a formulation and probabilistic interpretation of QFT allows one to interpret the wave function as a pilot wave that describes deterministic particle trajectories, which automatically includes a deterministic and continuous description of particle creation and destruction. In particular, when the conditional wave function associated with a quantum measurement ceases to depend on one of the space–time coordinates, then the 4-velocity of the corresponding particle vanishes, describing a trajectory that stops at a particular point in space–time. In a more general situation a dependence on this space–time coordinate is negligibly small but not strictly zero, in which case the trajectory does not stop but the measuring apparatus still behaves as if this particle has been destroyed.

Gravitation and inertia; a rearrangement of vacuum in gravity

We address the gravitation and inertia in the framework of 'general gauge principle', which accounts for 'gravitation gauge group' generated by hidden local internal symmetry implemented on the flat space. We connect this group to nonlinear realization of the Lie group of 'distortion' of local internal properties of six-dimensional flat space, which is assumed as a toy model underlying four-dimensional Minkowski space. The agreement between proposed gravitational theory and available observational verifications is satisfactory. We construct relativistic field theory of inertia and derive the relativistic law of inertia. This theory furnishes justification for introduction of the Principle of Equivalence. We address the rearrangement of vacuum state in gravity resulting from these ideas.

Supersymmetry breaking, heterotic strings and fluxes

In this paper we consider compactifications of heterotic strings in the presence of background flux. The background metric is a T 2 fibration over a K3 base times four-dimensional Minkowski space. Depending on the choice of three-form flux different amounts of supersymmetry are preserved ( N = 2 , 1 , 0 ). For supersymmetric solutions unbroken space–time supersymmetry determines all background fields except one scalar function which is related to the dilaton. The heterotic Bianchi identity gives rise to a differential equation for the dilaton which we discuss in detail for solutions preserving an N = 2 supersymmetry. In this case the differential equation is of Laplace type and as a result the solvability is guaranteed.

Classes of AdS4type IIA/IIB compactifications with SU(3) × SU(3) structure

We introduce an ansatz which allows us to solve the supersymmetry equations for warped = 1 AdS4 type II supergravity compactifications of general SU(3) × SU(3) structure. As a byproduct we obtain a set of necessary conditions which every supersymmetric AdS4 vacuum should obey. The case of AdS4 compactifications of IIB on manifolds of static SU(2) structure is examined in detail. Several examples of solutions are presented. In the limit of four-dimensional Minkowski space, we present examples of supersymmetric IIB warped compactifications with partially localized NS5- and D5-branes. We also present `massive' non-supersymmetric AdS4 × 6 solutions of IIA, where 6 can be any six-dimensional Einstein-Kahler manifold.

Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension

Spatial surfaces with parallel mean curvature vector play some important roles in general relativity, theory of harmonic maps, as well as in differential geometry. Recently, spatial surfaces in four-dimensional Minkowski space time with parallel mean curvature vector were classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms,” Tohoku Math. J. 61, 1 (2009)]. In this article, we completely classify spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension. The main result states that there exist 16 families of such surfaces. As by-product, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in Minkowski space times of arbitrary dimension.

Higher spin fields in Siegel space, currents and theta functions

Dynamics of four-dimensional massless fields of all spins is formulated in the Siegel space of complex $4\times 4$ symmetric matrices. It is shown that the unfolded equations of free massless fields, that have a form of multidimensional Schrodinger equations, naturally distinguish between positive- and negative-frequency solutions of relativistic field equations, i.e. particles and antiparticles. Multidimensional Riemann theta functions are shown to solve massless field equations in the Siegel space. We establish the correspondence between conserved higher-spin currents in four-dimensional Minkowski space and those in the ten-dimensional matrix space. It is shown that global symmetry parameters of the current in the matrix space should be singular to reproduce a nonzero current in Minkowski space. The $\D$-function integral evolution formulae for 4d massless fields in the Fock-Siegel space are obtained. The generalization of the proposed scheme to higher dimensions and systems of higher ranks is considered.

Electron and fine structure constant II

Based on the standing wave model of the macroscopically resting electron, classical electrodynamics and quantum mechanics, a theoretical relation for the fine structure constant α has been derived. It was found that . The ratio of the rest mass of the electron and its quantum mechanical fraction, m0/mm, was calculated to be 1.005 263 277, using a non-relativistic Hamiltonian function. With this new value the reciprocal of the fine structure constant α is obtained to be 137.035 999 252. At Harvard University a new experimental value of α has been derived from the measured anomalous magnetic moment of the electron and the QED calculation of this quantity (relative standard uncertainty 0.37 × 10−9). The relative deviation between our value and this one is 1.23 × 10−9, supporting the usefulness and validity of our model. According to our result α seems to be a real constant in time and space. Relativistic QED of a point particle in the four-dimensional (Minkowski) space and non-relativistic wave electrodynamics for a macroscopically resting extended electron with an internal energy flux in the ordinary (3 + 1)-dimensional space time were found to complete each other in the sense of wave–particle dualism.

Quantum energy inequalities for the non-minimally coupled scalar field

In this paper, we discuss local averages of the energy density for the non-minimally coupled scalar quantum field, extending a previous investigation of the classical field. By an explicit example, we show that such averages are unbounded from below on the class of Hadamard states. This contrasts with the minimally coupled field, which obeys a state-independent lower bound known as a quantum energy inequality (QEI). Nonetheless, we derive a generalized QEI for the non-minimally coupled scalar field, in which the lower bound is permitted to be state-dependent. This result applies to general globally hyperbolic curved spacetimes for coupling constants in the range 0 < ξ ⩽ 1/4. We analyse the state dependence of our QEI in four-dimensional Minkowski space and show that it is a non-trivial restriction on the averaged energy density in the sense that the lower bound is of lower order, in energetic terms, than the averaged energy density itself.

Collective modes of ultra-relativistic magnetoactive electron plasma

Collective modes of ultra-relativistic electron plasma embedded in an ambient weak magnetic field are derived. Employing Maxwell equations in the four-dimensional Minkowski space and using the Vlasov model, polarization tensor is calculated and dispersion relations are obtained for both parallel and perpendicular propagations in the temporal axial gauge. Some interesting limiting cases are also discussed.

Supersymmetric sources, integrability and generalized-structure compactifications

In the context of supersymmetric compactifications of type II supergravity to four dimensions, we show that orientifold sources can be compatible with a generalized SU(3) × SU(3)-structure that is neither strictly SU(3) nor static SU(2). We illustrate this with explicit examples, obtained by suitably T-dualizing known solutions on the six-torus. In addition we prove the following integrability statements, valid under certain mild assumptions: (a) for general type II supergravity backgrounds with orientifold and/or D-brane generalized-calibrated sources, the source-corrected Einstein and dilaton equations of motion follow automatically from the supersymmetry equations once the likewise source-corrected form equations of motion and Bianchi identities are imposed; (b) in the special case of supersymmetric compactifications to four-dimensional Minkowski space, the equations of motion of all fields, including the NSNS three-form, follow automatically once the supersymmetry and the Bianchi identities of the forms are imposed. Both (a) and (b) are equally valid whether the sources are smeared or localized. As a byproduct we obtain the calibration form for a space-filling NS5-brane.

Dynamics of (SUSY) AdS space isometry breaking

Actions governing the dynamics of the Nambu–Goldstone modes resulting from the spontaneous breaking of the SO(4, 2) and SU(2, 2|1) isometries of five-dimensional anti-de Sitter space (AdS5) and SUSY AdS5 × S1 spaces, respectively, due to a restriction of the motion to embedded four-dimensional AdS4 space and four-dimensional Minkowski space (M4) probe branes are presented. The dilatonic Nambu–Goldstone mode governing the motion of the M4 space probe brane into the covolume of the SUSY AdS5 × S1 space is found to be unstable. No such instability appears in the other cases. Gauging these symmetries leads to an Einstein–Hilbert action containing, in addition to the gravitational vierbein, a massive Abelian vector field coupled to gravity.