General Lorentz transformations and applications

Abstract

It is known that the most general proper orthochronous vector Lorentz (transformation) operator can be generated by a skew-symmetric 4×4 matrix containing an antisymmetric tensor of the second rank. The corresponding Lorentz operator for the two-component spinor is presented and, as can be expected, it contains the same tensor as the vector operator. Since the Pauli matrices of the spinor operator have very simple multiplication properties, the behavior of the tensor under multiplication of spinor operators is easily obtained. By comparison the corresponding properties of the tensor in vector operators can be obtained without multiplying 4×4 matrices. The physical meaning of the tensor contained in a Lorentz operator is discussed. Apart from the usual or regular operator a singular operator is discussed. Still other types of Lorentz operators are possible.