Is this concept the same as partial direction? If it is then the definitions of vectors, tensors, and scalars will never completely independent of the coordinate system. Einstein encountered similar problem where and when he believed he solved the principle of general covariance to complete his general relativity. He was fortunate since his theory’s principal curvatures of spacetime are always positive-definite meaning that the determinant of his symmetric metric tensor is always greater than or equal to unity. On the other hand, the infinitesimal curvatures of spacetime in a quantum field theory can be both positive and negative definite and bilinearity become meaningless. Restoring infinitesimal linearity necessitated keeping the concepts of pseudo directions and 2 other pseudo quantities: pseudoscalars and pseudovectors.
It is commonly believed that the infinitesimal vector products under rotations change sign by the process of inversion which is done by replacing all the basis vectors by their negatives. However, inversion of vector products can also be accomplished by switching the position of the base vectors such that A´B = -B´A (non-commutativity). Since volume in 3D is also defined as the triple product of vectors: (A´B)×C = volume (scalar), it gives the concept of negative volume: (B´A)×C = -volume (negative scalar). Analogous to negative length and negative time or negative energy, this is also physically meaningless. This pseudo directional difficulty can simply be removed by postulating that C is also a vector product given by D´E = C then(A´B)×(D´E)=(B´A)×(E´D) retaining positive-definiteness.


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