This paper describes an abstract formalism for tensor analysis in Minkowski space which entails considerable notational simplicity and calculational advantages, as evidenced when compared with the usual component techniques. The need to express tensor equations in component form is eliminated, and manipulations become formally the same as those in Euclidean spece. The method is based on an extension to Minkowski space of the intrinsic concepts of vectors, differential operators, and polyadics in three‐dimensional Euclidean spece. Several examples from special relativity have been selected to illustrate the advantages of the formalism. In the first example, expressions in dyadic form for the Euler‐Lagrange equations and canonical energy‐momentum tensor are obtained and specialized to the electromagnetic field. From the electromagnetic‐field dyadic, invariants and other useful relations are derived easily and economically. The dyadic form of the field equations is also shown to be particularly amenable for a derivation of the Dirac‐like form of Maxwell's equations with the base elements of the Pauli algebra emerging in a most natural way. Further illustration of the practical utility of the method is given by considering several properties of the restricted homogeneous Lorentz transformations. Various dyadic expressions for these are obtained, and a detailed derivation of their eigenvalues and eigenvectors is given. By combining some of the results from the discussions of Lorentz transformations and the Dirac‐like form of Maxwell's equations, it is shown how an isomorphism between the three‐dimensional complex orthogonal group and the Lorentz group can be established in a simpler and different manner from other approaches appearing in the literature.

Topics
Electromagnetism, Covariant formulation, Maxwell equations, Special relativity, Lie algebras, Operator theory, Partial differential equations, Minkowski spacetime, Functions and mappings, Particle symmetry
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© 1968 The American Institute of Physics.
1968
The American Institute of Physics
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