Levi-Civita tensor

In the case of a scalar field, the irreducible matrix D is a unit matrix, and drops out of. I1. For rotation through an angle S9t about the Cartesian axis ek, the rotational submatrix of the Lorentz matrix is given by Xkx = ()Hkekl]x], where el]k is the totally antisymmetric Levi-Civita tensor. For the one-electron Schrodinger field f, Noether s theorem defines three conserved components of a spatial axial vector,... 

Obviously the zero values of the angular variables define the unity matrix of the 50(4) group ea is the complete antisymmetric - Levi-Civita - tensor. The commutation properties of the infinitesimal operators defined in this manner are... 

In Eq. (23) the kni operator is Km = SikiotkfmfN and , / is the Levi-Civita tensor. With these definitions, the coupling tensor is given by Eq. (24), where the propagator term can be calculated at different levels of approximation. At the RPA level it must be calculated taking into account that the consistent relativistic ground state is the DHF one, i.e. 0) = DHF). The propagator takes the form of Eq. (25). 

The Levi-Civita tensor is a natural generalization of the totally antisymmetric third-rank tensor 6, as defined by Eq. (2.9). For Lorentz transformations A with det A = +1 the Levi-Civita pseudo-tensor is indeed a tensor which has... 

Viscous torques are exerted on the director of a liquid crystal during a rotation of the director and by a shear flow with a fixed director orientation. The density T] of the viscous torque is obtained by application of the Levi-Civita tensor... 

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