Electromagnetic and Force Field Equations

In classical physics the electromagnetic equations are described by Maxwell's equations. Maxwell's equations proved to be invariant under gauge, or Lorentz transformations. Also, Einstein's equations of the special theory of relativity are invariant under Lorentz transformations. On the other hand classical mechanics and quantum mechanics laws are invariant under Galilean transformations. This means that, there are two different dynamical structures describing our universe. Einstein's unified field theory failled in putting our universe in one dynamical structure. New electromagnetic and force field equations are going to be derived. They have the same shape like Maxwell's equations, but with different dynamical structure. Those equations are invariant under Galilean transformations and in the density matrix formalism of quantum mechanics.     

Geodesic‐invariant equations of gravitation

Einstein's equations of gravitation are not invariant under geodesic mappings, i.e. under a certain class of mappings of the Christoffel symbols and the metric tensor which leave the geodesic equations in a given coordinate system invariant. A theory in which geodesic mappings play the role of gauge transformations is considered.     

Gravitative Selbstkräfte und Strahlungsverluste klassischer Spinteilchen (erste Näherung)

The solutions ofEinstein's linearised field equations for a pol-dipol-like massdistribution are regularised at the position of the arbitrarily moving source by means ofRiesz' method of analytic continuation. Selfforce and selftorque resulting from the spin particle's gravitational proper field are computed Lorentz-covariantly. The selfforce devides into the radiation reaction caused by lightlike emission of field momentum and the exchange-force due to fluctuation of the convective field momentum. Gravitational momentum emission is gauge invariant while the convective field momentum combines with the particle's mechanical momentum to a gauge invariant quantity. Corresponding considerations apply to the selftorque. (Gauge invariance refers to gauge transformations induced by arbitrary infinitesimal deviations from Cartesian coordinates into the Lorentz-covariant theory of gravitation.)     

Some remarks on zeta function regularization in curved space-times

The question of to what extent zeta function regularization respects the invariances of a quantum field theory in a background gravitational field is investigated. It is shown that zeta function regularization provides a generalization to curved space-time of analytic propagator regularization which is known not to respect gauge invariance. Furthermore, a study of the regularized stress tensor of a conformally invariant scalar field indicates that both conformai and general coordinate invariance are violated.     

A Universal Regulator?

By using the principle of metrical invariance which requires that all physical laws are independent of the choice of units (alternatively, all physical laws are invariant with respect to scale transformations of space-time coordinates) and Goldstone's theorem, a universal regulator is discovered. The cosmic field is the Yang-Mills field of the local scale transformations. Its physical role is as follows. Cosmon, its quantum, is a massless, spinless, and neutral particle. The cosmic field is created by inertial masses. Therefore it participates in all physical processes and if its presence is taken into account, then the quantum field theory is free from all ultraviolet infinities. From the point of view of Yang-Mills field theory, it is proved that the so-called gravitational masses are identical with inertial masses and the gravitational field is created by inertial masses moving non-inertially. This fact permits to solve satisfactorily the problem of energy-momentum complex of the gravitational field. The system of equations which defines simultaneously the cosmic and gravitational fields is established. A non-Einstein cosmology is outlined.     

$\mathfrak{D}$-Differentiation in Hilbert Space and the Structure of Quantum Mechanics Part II: Accelerated Observers and Fictitious Forces

We investigate a possible form of Schrödinger’s equation as it appears to moving observers. It is shown that, in this framework, accelerated motion requires fictitious potentials to be added to the original equation. The gauge invariance of the formulation is established. The example of accelerated Euclidean transformations is treated explicitly, which contain Galilean transformations as special cases. The relationship between an acceleration and a gravitational field is found to be compatible with the picture of the ‘Einstein elevator’. The physical effects of an acceleration are illustrated by the problem of the uniformly-accelerated harmonic oscillator.     

BRST invariance of the measure in string field theory

The BRST transformations, given by gauge-fixing Witten's string field theory in the Seigel gauge, are applied to the string measure. It is shown that the simple measure (just the product of differentials of all the fields) is BRST invariant, thus maintaining the invariance of the gauge-fixed action at the quantum level.     

Novel ansatzes and scalar quantities in gravito-electromagnetism

In this work, we focus on the theory of gravito-electromagnetism (GEM)—the theory that describes the dynamics of the gravitational field in terms of quantities met in electromagnetism—and we propose two novel forms of metric perturbations. The first one is a generalisation of the traditional GEM ansatz, and succeeds in reproducing the whole set of Maxwell’s equations even for a dynamical vector potential \(\mathbf {A}\). The second form, the so-called alternative ansatz, goes beyond that leading to an expression for the Lorentz force that matches the one of electromagnetism and is free of additional terms even for a dynamical scalar potential \(\varPhi \). In the context of the linearised theory, we then search for scalar invariant quantities in analogy to electromagnetism. We define three novel, 3rd-rank gravitational tensors, and demonstrate that the last two can be employed to construct scalar quantities that succeed in giving results very similar to those found in electromagnetism. Finally, the gauge invariance of the linearised gravitational theory is studied, and shown to lead to the gauge invariance of the GEM fields \(\mathbf {E}\) and \(\mathbf {B}\) for a general configuration of the arbitrary vector involved in the coordinate transformations.     

Derivation of an effective Lagrangian from a generalized scalar curvature

A spinor Lagrangian invariant under global coordinate, local Lorentz and local chiral SU(n) × SU(n) gauge transformations is presented. The invariance requirement necessitates the introduction of boson fields, and a theory for these fields is then developed by relating them to generalizations of the vector connections in general relativity and utilizing an expanded scalar curvature as a boson Lagrangian. In implementing this plan, the local Lorentz group is found to greatly facilitate the correlation of the boson fields occurring in the spinor Lagrangian with the generalized vector connections.The independent boson fields of the theory are assumed to be the inhomogeneously transforming irreducible parts of the connections. It turns out that no homogeneously transforming parts are necessary to reproduce the chiral Lagrangian usually used as a basis for phenomenological field theories. The Lagrangian in question appears when the gravitational interaction is turned off. It includes pseudoscalar, spinor, vector, and axial vector fields, and the vector fields carry mass in spite of the fact that the theory is locally gauge invariant.     

Local internal supersymmetry of the free Maxwell theory

The free Maxwell theory is shown to possess an extended gauge invariance consisting of local internal supersymmetry transformations in addition to the usual local phase transformations. The Maxwell lagrangian is derived as a particular gauge choice in the extended theory.     

Non-perturbative to perturbative QCD via the FFBRST

Recently a new type of quadratic gauge was introduced in QCD in which the degrees of freedom are suggestive of a phase of abelian dominance. In its simplest form it is also free of Gribov ambiguity. However this gauge is not suitable for usual perturbation theory. The finite field dependent BRST (FFBRST) transformation is a method established to interrelate generating functionals for different effective versions of gauge fixed field theories. In this paper we propose a FFBRST transformation suitable for transforming the theory in the new quadratic gauge into the standard Lorenz gauge Faddeev–Popov version of the effective lagrangian. The task is made interesting by the fact that the effective lagrangian is invariant under two different BRST transformations which leads to suitable extension of the previous procedures to accomplish the required result. We are thus able to identify a field redefinition to go from a non-perturbative phase of QCD to perturbative QCD.     

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

Nonperturbative BRS invariance

In the context of lattice gauge theories the standard requirement of local gauge invariance is replaced by BRS invariance. The most general BRS invariant lagrangian is shown to contain as many parameters as the gauge invariant one. This is done by explicitly solving the lattice BRS cohomology problem.     

The Sugawara model in general relativity. A. The case for three current vectors (SU (2))

A general method of solving the equations of Sugawara's field theory of currents has been developed, and illustrated by applying it to the set of three currents. These are inserted into Einstein's field equations which have been solved together with the co-variant ‘gauge’ conditions for a gravitational field involving cylindrical symmetry. A further transformation exhibits the triad formed by the current vectors and exhibits clearly the deviations of the line-element from Schwarzschild's exterior solution. In a subsequent paper the case for eight vector currents corresponding toSU (3) will be treated in similar fashion.     

An extension of Noether's theorem to transformations involving position-dependent parameters and their derivatives

Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only the independent and the dependent field variables, but also a set of position-dependent parameters together with their first derivatives. The stipulation that an action integral associated with the field variables be invariant under such transformations gives rise to a set of three conditions involving the Lagrangian and its derivatives, together with derivatives of the functions that define the transformations. These invariance identities constitute an extension of the classical theorem of Noether to general transformations of this kind. An application to the case of gauge fields demonstrates the existence of two distinct types of conservation laws for such fields.     

Lattice supergravity and graviton-gravitino doubling

We give a formulation of supergravity on a hypercubical lattice as a gauge theory of the super-Poincaré group. We work out the perturbative limit and show the occurrence of the gravitino doubling matched to the graviton doubling; this is shown to be implied by the exact invariance of the free lagrangian under linearized supersymmetry and reparametrization transformations.     

Conformal geometry and string field theory

The covariant perturbation theory rules that should arise from any gauge invariant string field theory, such as those proposed on the basis of BRST formalism, are set forth. The resulting path integral expressions naturally produce coordinate invariant densities on the moduli space of Riemann surfaces; these include the Koba-Nielsen amplitudes. The connection between string field theory and modular invariance is discussed, and it is proposed that future explorations in string field theory focus on coordinate invariant quantities on moduli space.     

Quantenphysikalischer Ursprung der Eichidee

The Quantum Physical Origin of the Gauge Idea To consider quantum physics as an interplay of creation and annihilation processes has the consequence that gauge field theories are not only possible but necessary. Since the complex conjugate phase factors of each pair of fermion creators and annihilators can be arbitrary chosen, quantum field theories must be completely phase invariant. Unfortunately, even globally the Dirac equation for systems of free fermions is not phase invariant. The Dirac matrices are namely transformed, if we multiply the spinor components by different constant phase factors. The Dirac equations before and after the transformation are however physically equivalent. We may therefore say: Systems of free fermions will be completely described, only if we consider the class of all equivalent Dirac equations. Since Dirac's commutation relations are unitarily invariant, the class equivalent Dirac equations is invariant under all transformations of the group U 4. Unitary diagonal matrices yield arbitrary phase transformations. Hence, gauge fields of the group U 4 are compatible with the postulate of general phase invariance. These gauge file are so similar to the QED that we may speak of an “extended quantum electrodynamics”, EQE. Here, we will show that EQE exists. The invariant subgroup U 1 U 4 yields QED. The complementary subgroup SU 4 includes four subgroups SU 3, there subgroups O 4, and six subgroups SU 2. The latter ones may yield three pairs of quarks and three pairs of leptons, where the quarks form a group SU 3. More than two times three pairs of elementary fermions does not exist in in EQE Probably, EQE is different from the United EQD and QCD. However, it should be a promising version of a field theory in elementary particle physics, because it follows from an existing symmetry of the empirically wel founded Dirac theory. EQE is therefore free from hypothesis in the Newtonian sense of the word. Whatever it will finally mean, it cannot be rejected, since phase invariance must be required. The invention of new symmetries and the acception of a bie number of independent spinor components is dispensable or must be postponed at least.     

Hamiltonian treatment of the spherically symmetric einstein-Yang-Mills system

A canonical formalism of the dynamics of interacting spherically symmetric Yang-Mills and gravitational fields is presented. The work is based on Dirac's technique for constrained hamiltonian systems. The gauge freedom of the Yang-Mills field is treated in the same footing with the coordinate transformation freedom of the gravitational field. In particular, the fixation of coordinates and the fixation of the internal gauge are achieved by totally similar techniques. Two classes of spherically symmetric motions are considered: (i) the class for which the Yang-Mills potentials themselves are spherically symmetric (“manifest spherical symmetry”). In this case the results are valid for an arbitrary gauge group; and (ii) the class for which, in the SO(3) gauge group, a rotation in physical space is compensated by a rotation of equal magnitude but opposite direction in isospin space (“spherical symmetry up to a gauge transformation”). For manifest spherical symmetry the problem amounts to effectively dealing with an abelian gauge group and the most general solution of the field equations turns out to be the Reissner-Nordström metric with a Coulomb field. For spherical symmetry up to a gauge transformation the problem is more interesting. the formalism contains then, besides the gravitational variables, three pairs of functions of the radial coordinate that describe the degrees of freedom of the Yang-Mills field. Two pairs of these functions can be combined into a complex field ψ and its conjugate. The hamiltonian is then invariant under r-dependent rotations in the complex ψ-plane. The third degree of freedom plays the role of a compensating field associated with this invariance under localized U(l) rotations. The compensating field can always be brought to zero by a gauge transformation. After this is done the gauge is completely fixed but the problem remains invariant under position independent rotations in the ψ plane. Static solutions of the field equations in this gauge are of the form ψ(r) = (r) exp (iΘ) with Θ independent of position. The particular case Θ = 0 corresponds to the Wu-Yang ansatz. A nontrivial static solution can be found in closed form. The Yang-Mills field is of the generalized Wu-Yang type with an extra electric term, and the metric is the Reissner-Nordström one. It is pointed out that a Higgs field can be easily introduced in the formalism. The addition of the Higgs field does not destroy the invariance of the Hamiltonian under r-dependent rotations in the ψ-plane. The conserved quantity associated with the invariance under ψ → exp (i(const))ψ coincides with the electric charge as defined by 't Hooft in a more general context.