Extended Grassmann and Clifford algebras

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, when we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q) can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended Clifford algebra. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms.Comment: 17 pages, to appear in Adv. Appl. Clifford Algebras 16 (3) (2006

Gauge Fixing in the Maxwell Like Gravitational Theory in Minkowski Spacetime and in the Equivalent Lorentzian Spacetime

In a previous paper we investigate a Lagrangian field theory for the gravitational field (which is there represented by a section g^a of the orthonormal coframe bundle over Minkowski spacetime. Such theory, under appropriate conditions, has been proved to be equivalent to a Lorentzian spacetime structure, where the metric tensor satisfies Einstein field equations. Here, we first recall that according to quantum field theory ideas gravitation is described by a Lagrangian theory of a possible massive graviton field (generated by matter fields and coupling also to itself) living in Minkowski spacetime. The graviton field is moreover supposed to be represented by a symmetric tensor field h carrying the representations of spin two and zero of the Lorentz group. Such a field, then (as it is well known), must necessarily satisfy the gauge condition given by Eq.(3) below. Next, we introduce an ansatz relating h to the 1-form fields g^a. Then, using the Clifford bundle formalism we derive, from our Lagrangian theory, the exact wave equation for the graviton and investigate the role of the gauge condition given by Eq.(3) in obtaining a reliable conservation law for the energy-momentum tensor of the gravitational plus the matter fields in Minkowski spacetime. Finally we ask the question: does Eq.(3) fix any gauge condition for the field g of the effective Lorentzian spacetime structure that represents the field h in our theory? We show that no gauge condition is fixed a priory, as is the case in General Relativity. Moreover we investigate under which conditions we may fix Logunov gauge condition.Comment: 15 pages. This version corrects some misprints of the published versio

Diffeomorphism Invariance and Local Lorentz Invariance

We show that diffeomorphism invariance of the Maxwell and the Dirac-Hestenes equations implies the equivalence among different universe models such that if one has a linear connection with non-null torsion and/or curvature the others have also. On the other hand local Lorentz invariance implies the surprising equivalence among different universe models that have in general different G-connections with different curvature and torsion tensors.Comment: 19 pages, Revtex, Plenary Talk presented at VII International Conference on Clifford Algebras and their Applications, Universite Paul Sabatier UFR MIG, Toulouse (FRANCE), to appear in "Clifford Algebras, Applications to Mathematics, Physics and Engineering", Progress in Math. Phys., Birkhauser, Berlin 200

Are sacred caves still safe havens for the endemic bats of Madagascar?

Despite conservation discourses in Madagascar increasingly emphasizing the role of customary institutions for wildlife management, we know relatively little about their effectiveness. Here, we used semi-structured interviews (n = 54 adults in 8 villages) to investigate whether sacred caves and taboos offer in situ conservation benefits for cave-dwelling bats in and around Tsimanampetsotsa National Park, Southwest Madagascar. Although some caves were described as sites of spiritual significance for the local communities, most interviewees (~76%) did not recognize their present-day sacred status. Similarly, only 22% of the interviewees recognized taboos inhibiting bat hunting and consumption. In general, legal protection of both bats and caves was often more acknowledged than customary regulations, although up to 30% of the interviewees reported bat bushmeat consumption within their communities. Guano extraction was often tolerated in sacred caves, in exchange for economic compensations. In view of these results, our study questions the extent to which sacred sites, taboos and legal frameworks offer protection for bats in Madagascar. These results align with previous studies documenting the erosion of customary institutions in Madagascar, including the loss of the spiritual values underpinning sacred sites. Guano harvesting may benefit bat conservation, although it is often performed through destructive and exploitative practices with low benefits for the local communities. Given that many Malagasy bats are cave-dwelling species and that most depend on the customary protection of these sites, it remains paramount to better understand the complex interactions between spiritual practices, taboos and protected areas in sustaining ‒or not‒ bat diversity