Appendix A: Vectors, Tensors, and Their Duality Transformations

Abstract

Scalar and vector products of two vectors A ^μ and B ^μ are written as:

$\displaystyle A\cdot B=A_\mu B^\mu \;\;\;\;\;\;\;\;\left( A\wedge B\right) _{\mu \nu }=A_\mu B_\nu -A_\nu B_\mu$

$\displaystyle \left( \partial \wedge A\right) _{\mu \nu }=\left( \partial _\mu A_\nu \right) -\left( \partial _\nu A_\mu \right)~~~~~~{\rm (A.1)}$

If S ^μν=-S ^νμ is an antisymmetric tensor, its contractions with a vector A ^μ are written in the form:

$\displaystyle \left( A\cdot S\right) ^\mu =A_\nu S^{\nu \mu }=-S^{\mu \nu }A_\nu =-\left( S\cdot A\right) ^\mu$      

If S and T are two antisymmetric tensors, we write:

$\displaystyle S\cdot T=ST=\frac 12S_{\mu \nu }T^{\mu \nu }\;\;\;\;\;\;S\cdot S\equiv S^2=\frac 12S_{\mu \nu }S^{\mu \nu }~~~~~~{\rm (A.3)}$

The dot “⋅ ” is not necessary when it is clear which symbols are vectors, which are antisymmetric tensors, etc. In such cases we can write, for example:

$\displaystyle \left( TA\right) _\mu =T_{\mu \nu }A^\nu \;\;\;\;\;\;ASB=A_\mu S^{\mu \nu }B_\nu \;\;etc.~~~~~~{\rm (A.4)}$