Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M^{4}. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.

Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preservingdiffeomorphismsof an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given.

The different roles and natures of spacetime appearing in a quantum field
theory and in classical physics are analyzed implying that a quantum theory of
gravitation is not necessarily a quantum theory of curved spacetime. Developing
an alternative approach to quantum gravity starts with the postulate that
inertial energy-momentum and gravitational energy-momentum need not be the same
for virtual quantum states. Separating their roles naturally leads to the
quantum gauge field theory of volume-preserving diffeomorphisms of an inner
four-dimensional space. The classical limit of this theory coupled to a
quantized scalar field is derived for an on-shell particle where inertial energy-momentum
and gravitational energy-momentum coincide. In that process the symmetry under
volume-preserving diffeomorphisms disappears and a new symmetry group emerges:
the group of coordinate transformations of four-dimensional spacetime and with
it General Relativity coupled to a classical relativistic point particle.

Inertial and gravitational mass or energy momentum need not be the same
for virtual quantum states. Separating their roles naturally leads to the gauge
theory of volume-preserving diffeomorphisms of an inner four-dimensional space.
The gauge-fixed action and the path integral measure occurring in the
generating functional for the quantum Green functions of the theory are shown
to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting
the corresponding quantum effective action is established. This equation limits
the infinite parts of the quantum effective action to have the same form as the
gauge-fixed Lagrangian of the theory proving its spacetime renormalizability.
The inner space integrals occurring in the quantum effective action which are
divergent due to the gauge group’s infinite volume are shown to be
regularizable in a way consistent with the symmetries of the theory demonstrating
as a byproduct that viable quantum gauge field theories are not limited to
finite-dimensional compact gauge groups as is commonly assumed.

Abstract:
Viewing
gravitational energy-momentum P_{G}^{}μ as equal by
observation, but different in essence from inertial energy-momentum P_{I}^{μ} naturally
leads to the gauge theory of volume-preserving diffeomorphisms of a
four-dimensional inner space. To analyse scattering in this theory, the gauge
field is coupled to two Dirac fields with different masses. Based on a
generalized LSZ reduction formula the S-matrix
element for scattering of two Dirac particles in the gravitational limit and
the corresponding scattering cross-section are calculated to leading order in
perturbation theory. Taking the non-relativistic limit for one of the initial
particles in the rest frame of the other the Rutherford-like cross-section of a
non-relativistic particle scattering off an infinitely heavy scatterer calculated
quantum mechanically in Newtonian gravity is recovered. This provides a
non-trivial test of the gauge field theory of volume-preserving diffeomorphisms
as a quantum theory of gravity.

Abstract:
Isometrodynamics (ID), the gauge theory of the group of volume-preserving diffeomorphisms of an "inner" D-dimensional flat space, is tentatively interpreted as a fundamental theory of gravity. Dimensional analysis shows that the Planck length l_P - and through it \hbar and \Gamma - enters the gauge field action linking ID and gravity in a natural way. Noting that the ID gauge field couples solely through derivatives acting on "inner" space variables all ID fields are Taylor-expanded in "inner" space. Integrating out the "inner" space variables yields an effective field theory for the coefficient fields with l_P^2 emerging as the expansion parameter. For \hbar goint to zero only the leading order field does not vanish. This classical field couples to the matter Noether currents and charges related to the translation invariance in "inner" space. A model coupling this leading order field to a matter point source is established and solved. Interpreting the matter Noether charge in terms of gravitational mass Newton's inverse square law is finally derived for a static gauge field source and a slowly moving test particle. Gravity emerges as potentially related to field variations over "inner" space and might microscopically be described by the ID gauge field or equivalently by an infinite string of coefficient fields only the leading term of which is related to the macroscopical effects of gravity.

Abstract:
A generalization of non-Abelian gauge theories of compact Lie groups is developed by gauging the non-compact group of volume-preserving diffeomorphisms of a $D$-dimensional space R^D. This group is represented on the space of fields defined on M^4 x R^D. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws of the new gauge theory are developed. Finally, the theory's Hamiltonian in the axial gauge and its Hamiltonian field dynamics are derived.

Abstract:
Classical Isometrodynamics is quantized in the Euclidean plus axial gauge. The quantization is then generalized to a broad class of gauges and the generating functional for the Green functions of Quantum Isometrodynamics (QID) is derived. Feynman rules in covariant Euclidean gauges are determined and QID is shown to be renormalizable by power counting. Asymptotic states are discussed and new quantum numbers related to the "inner" degrees of freedom introduced. The one-loop effective action in a Euclidean background gauge is formally calculated and shown to be finite and gauge-invariant after renormalization and a consistent definition of the arising "inner" space momentum integrals. Pure QID is shown to be asymptotically free for all dimensions of "inner" space $D$ whereas QID coupled to the Standard Model fields is not asymptotically free for D <= 7. Finally nilpotent BRST transformations for Isometrodynamics are derived along with the BRST symmetry of the theory and a scetch of the general proof of renormalizability for QID is given.

Abstract:
Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energy-momentum naturally leads to the gauge theory of volume-preserving diffeormorphisms of a four-dimensional in- ner space. To analyse scattering in this theory the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S-matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity

Abstract:
The atomic-scale spin structure of individual isolated skyrmions in an ultrathin film is investigated in real space by spin-polarized scanning tunneling microscopy. Their axial symmetry as well as their unique rotational sense is revealed by using both out-of-plane and in-plane sensitive tips. The size and shape of skyrmions change as a function of magnetic field. An analytical expression for the description of skyrmions is proposed and applied to connect the experimental data to the original theoretical model describing chiral skyrmions. Thereby, the relevant material parameters responsible for skyrmion formation can be obtained.