Obtaining consistent Lorentz gauging for a gravitationally coupled fermion 
Abstract  For internal gauge forces, the result of locally gauging, i.e., of performing the substitution $\partial \rightarrow D$, is physically the same whether performed on the action or on the corresponding EulerLagrange equations of motion. Rather unsettling, though, such commutativity fails for the standard way of coupling a Dirac fermion to the gravitational field in the setting of a local Lorentz gauge theory of general relativity in the vierbein formalism, the equivalence principle thus seemingly being here violated. This paper will present a formalism in which commutativity holds for the gravitational force as well, the action for the gravitational field itself being still the EinsteinHilbert one. Notably, in this formalism, the spinor field will carry a world/coordinate index, rather than a Lorentz spinor index as it does standardly. More generally, no Lorentz indices will figure, neither vector indices nor spinor indices, which from a parsimonious point of view seems quite satisfactory. 

Published in  J. Math. Phys. 60, 102503 (2019) 

Freely available at  arxiv.org/abs/1906.12200 

Spinor fields without Lorentz frames in curved spacetime using complexified quaternions 
Abstract  Using complexified quaternions, a formalism without Lorentz frames, and therefore also without vierbeins, for dealing with tensor and spinor fields in curved spacetime is presented. A local U(1) gauge symmetry, which, it is speculated, might be related to electromagnetism, emerges naturally. 

Published in  J. Math. Phys. 50, 083507 (2009) 

Freely available at  arxiv.org/abs/0811.1357 

Exponentiation of the spinor representation of the Lorentz group 
Abstract  An exact finite expression for the exponentiation of the (spin 1/2) spinor representation of the Lorentz group is obtained. From this expression an exact finite expression for the exponentiation of the vector representation of the Lorentz group is derived. The two expressions are compared with the literature in the special cases of either spatial rotations or boosts, only. 

Published in  J. Math. Phys. 42, 4497 (2001) 

Comment on “Wilson loops in fourdimensional quantum gravity” 
Abstract  As an example of a classical holonomy Modanese [Phys. Rev. D 49, 6534 (1994)] considered a class of circular curves in Schwarzschild geometry. The result obtained does not, as it should, vanish in the limit M=0. This Comment corrects this error. 

Published in  Phys. Rev. D 64, 088501 (2001) 

Obtaining consistent Lorentz gauging for a gravitationally coupled fermion 
Abstract  For internal gauge forces, the result of locally gauging, i.e., of performing the substitution 'partial derivatives go to covariant derivatives', is the same whether performed on the action or on the corresponding EulerLagrange equations of motion. Rather unsettling, though, this commutativity of these two procedures fails for the standard way of coupling a fermion to the gravitational field in the setting of a local Lorentz gauge theory of general relativity in the vierbein formalism. This paper will present a formalism in which commutativity holds for the gravitational force as well. Notably, in this formalism, the spinor field will carry a world/coordinate index, rather than a Lorentz spinor index as it does standardly. More generally, no Lorentz indices will figure, neither vector indices nor spinor indices. 

Freely available at  arxiv.org/abs/1906.12200 

Spinors and gravity without Lorentz indices 
Abstract  Coupling spinor fields to the gravitational field, in the setting of general relativity, is standardly done via the introduction of a vierbein field and the (associated minimal) spin connection field. This makes three types of indices feature in the formalism: world/coordinate indices, Lorentz vector indices, and Lorentz spinor indices, respectively. This article will show, though, that it is possible to dispense altogther with the Lorentz indices, both tensorial ones and spinorial ones, obtaining a formalism featuring only world indices. This will be possible by having both the 'Dirac operator' and the generators of 'Lorentz' transformations become spacetimedependent, although covariantly constant. The formalism is developed in the setting of complexified quaternions. 

Freely available at  arxiv.org/abs/1811.00377 

Spinor fields without Lorentz frames in curved spacetime using complexified quaternions 
Abstract  Using complexified quaternions, a formalism without Lorentz frames, and therefore also without vierbeins, for dealing with tensor and spinor fields in curved spacetime is presented. A local U(1) gauge symmetry, which, it is speculated, might be related to electromagnetism, emerges naturally. 

Freely available at  arxiv.org/abs/0811.1357 

Natural octonionic generalization of general relativity 
Abstract  An intriguingly natural generalization, using complex octonions, of general relativity is pointed out. The starting point is the vierbeinbased double dual formulation of the EinsteinHilbert action. In terms of two natural structures on the (complex) quaternions and (complex) octonions, the inner product and the cross products, respectively, this action is linked with the complex quaternionic structure constants, and subsequently generalized to an achtbeinbased 'double chidual' action in terms of the complex octonionic structure constants. 

Freely available at  arxiv.org/abs/0707.0554 

