Four-dimensional Minkowski Space Research Articles

Static vacuum solution of direct Poincar� gauge theory in ten dimensions with four external

A torsion-free solution of the free gauge field equations of direct Poincare gauge theory on a ten-dimensional Minkowski space is constructed. This solution exhibits nontrivial curvature two-forms, but shaves the metric structure down to that of a four-dimensional Minkowski space. Universality of this solution with respect to the choice of the free field Lagrangian is established.

Yukawa couplings between (2, 1)-forms

The compactification of superstrings leads to an effective field theory for which the space-time manifold is the product of a four-dimensional Minkowski space with a six-dimensional Calabi-Yau space. The particles that are massless in the four-dimensional world correspond to differential forms of type (1, 1) and of type (2, 1) on the Calabi-Yau space. The Yukawa couplings between the families correspond to certain integrals involving three differential forms. For an important class of Calabi-Yau manifolds, which includes the cases for which the manifold may be realized as a complete intersection of polynomial equations in a projective space, the families correspond to (2, 1)-forms. The relation between (2, 1)-forms and the geometrical deformations of the Calabi-Yau space is explained and it is shown, for those cases for which the manifold may be realized as the complete intersection of polynomial equations in a single projective space or for many cases when the manifold may be realized as the transverse intersection of polynomial equations in a product of projective spaces, that the calculation of the Yukawa coupling reduces to a purely algebraic problem involving the defining polynomials. The generalization of this process is presented for a general Calabi-Yau manifold.

Superinvariant chiral and vector fields

This work is concerned with the characterization of field supermultiplets on four-dimensional Minkowski space that are stable under the action of subgroups of the superconformal group SU(2,2/1). The most general scalar and vector superfields whose Lie derivative with respect to a fermionic tangent vector vanishes are determined. Invariance under subgroups of SU(2,2/1) with more than one odd generator is also discussed.

Mass spectrum of the M 4×S D solution in Euler invariant type higher derivative gravity

The mass spectrum is computed in Euler invariant type higher derivative gravity theory in the case that the space-time is dimensionally reduced to the four-dimensional Minkowski space × D-dimensional sphere. It is shown at the linearized level that after the compactification there appear massless gravitons, massive spin-two particles, massless vectors and one massive scalar mode. All the vectors are massless and the masses of massive spin-two particles are proportional to the S D eigenvalues of the laplacian. Classical stability is shown to depend only on three parameters.

Stability analysis of M4 >= 2 >= 2 >= 2 in the ten-dimensional Einstein/Maxwell theory.

The compactifications of the ten-dimensional Einstein-Maxwell theory and the Einstein-Maxwell-scalar theory into ${M}_{4}\ifmmode\times\else\texttimes\fi{}{S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}$, where ${M}_{4}$ denotes four-dimensional Minkowski space, are found to be unstable.

Stability analysis of M4 >= 2 >= 2 in the Einstein-Maxwell system.

The compactifications of the eight-dimensional Einstein-Maxwell theory and the Einstein-Maxwell-scalar system into four-dimensional Minkowski space (${M}_{4}$)\ifmmode\times\else\texttimes\fi{}${S}^{2}$\ifmmode\times\else\texttimes\fi{}${S}^{2}$ are found to be unstable.

The Clifford algebra of differential forms

This paper reviews Clifford algebras in mathematics and in theoretical physics. In particular, the little-known differential form realization is constructed in detail for the four-dimensional Minkowski space. This setting is then used to describe spinors as differential forms, and to solve the Klein-Gordon and Kahler-Dirac equations. The approach of this paper, in obtaining the solutions directly in terms of differential forms, is much more elegant and concise than the traditional explicit matrix methods. A theorem given here differentiates between the two real forms of the Dirac algebra by showing that spin can be accommodated in only one of them.

Discrete symmetries in Kaluza-Klein theories

We investigate discrete symmetries in theories of higher-dimensional ( d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d − 4 dimensional internal space. If the action of pure d-dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d-dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor. A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong- CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.

Elementary proof of Zeeman's theorem

We prove that the nonaffine conformal transformations of four-dimensional Minkowski space are necessarily global causality violators, and use this fact to obtain an elementary group-theoretic proof of the theorem that global causality implies the Lorentz group.

Classical solutions of σ models

sigma models with linear realisations of U(N) and O(N) symmetries are studied without imposing a constraint on the modulus of the field vector. Exact solutions in four-dimensional Minkowski space are presented, which have the form of plane/spherical waves. The singularities of the solutions as well as those of the Lagrangian density and the energy-momentum tensor are discussed. All results hold under the assumption that space-time and internal symmetry of freedom are not mixed.

Classical solutions of a nonlinear field equation

The field equation for a complex scalar field with a cubic nonlinear term is studied in four-dimensional Minkowski space. Two classes of exact solutions corresponding to plane and spherical waves are presented.

Supersymmetry and Kähler manifolds

We give the supersymmetric extension of non-linear models with scalar fields taking values in a complex Kähler manifold. The supersymmetry is simple for four-dimensional Minkowski space, complex for two-dimensional space-time. The supersymmetric symmetric lagrangian is given by a simple and elegant formula in terms of the Kähler metric of the manifold. (Anti-) self- dual supersymmetric solutions are also given. The special cases of the CP n−1 models and of models with scalar fields valued in a Grassman manifold G p, q =G( p+ q)/G( p) ⊗ U( q) are discussed in some detail.

Decay of classical Yang-Mills fields

The classical Yang-Mills equations in four-dimensional Minkowski space are invariant under the conformal group. The resulting conservation laws are explicitly exhibited in terms of the Cauchy data at a fixed time. In particular, it is shown that, for any finite-energy solution of the Yang-Mills equations, the local energy tends to zero ast→∞.

Dirac spinor solitons or bags

Explicit solutions of a non-linear Dirac equation in the four-dimensional Minkowski space have been found. They are continuous eigenfunctions of spin 1 2 , energy and parity, and are confined to the interior of a sphere of definite radius a, with no tails outside the sphere. They are called solitons in the paper though the name of spinor droplets or bags may be more appropriate. Because of the Lorentz covariance of the Dirac equations, arbitrarily moving solitons can be obtained from the rest system solutions by applying proper Lorentz transformations.

Generalization of the quark-confining string

It is natural to generalize the quark-confining string model by introducing additional terms coming from the Riemann curvature tensor. For simplicity, only terms linear in the Riemann tensor are considered. In the absence of quark fields, such terms are trivial (as in two-dimensional gravity). In the presence of quark fields, embedding of the string in four-dimensional Minkowski space renders such terms nontrivial. We formulate this generalized quark-confining string in the first-order form, from which we derive its second-order form. The coupling between the quark spin and the string curvature via the connections induces a spin-spin interaction among the quark fields. This generalized version has a dimensionless parameter $\ensuremath{\kappa}$ in addition to the quark-gluon coupling $e$ and quark masses. The original version is recovered as a special case (i.e., $\ensuremath{\kappa} = 0$).

Covariant quantization of a quark-binding string

An important tool for the investigation of relativistic quantum field theories of strong interactions is the study of their strong-coupling limits. These lead to semigeometric models in which the quark-binding fields reduce to constraining volumes (bags), surfaces (bubbles), or lines (strings). These models are usually first treated classically and then quantized by canonical quantization. The present model was previously investigated by Giles and Tye and by Tye and originated from the "SLAC bubble." A two-dimensional timelike cylinder is embedded in four-dimensional Minkowski space. Massless Dirac fields (quarks) are constrained to it. These can carry any internal-symmetry label. At a fixed time the model represents a loop (closed string) on which quarks circulate. The covariant quantization is possible in a special guage and for timelike (rather than null plane) dynamics if care is taken to treat the corresponding gauge conditions in an internally consistent way. Simultaneity of relative positions of this extended structure in every instantaneous rest frame leads to additional constraints which ensure the absence of ghost states. All primary constraints are proved to be first class also for the quantized system. There are no secondary constraints. The theory is manifestly covariant; the Poincar\'e algebra is verified explicitly. There are no tachyonic states since the total momentum points into the future light cone. Linearly rising Regge trajectories result as in the free string.

There are no classical glueballs

I show that there are no finite-energy non-singular solutions of classical Yang-Mills theory in four-dimensional Minkowski space that do not radiate energy out to spatial infinity. Finite-energy non-singular solutions that are either static or periodic in time, are a fortiori non-radiant; thus this generalizes earlier theorems that state that there are no such solutions.

Are there geon analogues in sourceless gauge-field theories?

It has recently been shown that there is no finite-energy non-singular solution to the sourceless gauge-field equations in four-dimensional Minkowski space that does not radiate energy. However, this does not preclude the possibility of solutions which hold themselves together for a long time before radiating away their energy. If they existed, such objects would be analogous to the geons of general relativity. We show such objects do not exist.

Four-Dimensional Interpretation of the Internal Space Associated with the Dual-Resonance Models

An attempt is made to enlarge the internal space in which the dual-resonance models are described: To make closer the analogy with the conventional field theory, the usual planar medium is extended to the one which is realized in the four-dimensional Minkowski space. Furthermore in order to reproduce the usual success in the two-dimensional approximation, the dynamical quantities are assumed to depend on one more external four-vector which plays the role of indicator of several contractions. Discussion is given particularly in favour of the position operators represented in the two- and three-dimensional spaces.

Remarks on a class of almost trivial solutions of the gravitational field equation: K. Kraus. Institute for Theoretical Physics Universität Marburg, Marburg, Germany

The mass spectrum is computed in Euler invariant type higher derivative gravity theory in the case that the space-time is dimensionally reduced to the four-dimensional Minkowski space × D-dimensional sphere. It is shown at the linearized level that after the compactification there appear massless gravitons, massive spin-two particles, massless vectors and one massive scalar mode. All the vectors are massless and the masses of massive spin-two particles are proportional to the SD eigenvalues of the laplacian. Classical stability is shown to depend only on three parameters.