Wedge reversion antisymmetry and 41 types of physical quantities in arbitrary dimensions

Gopalan, V.

doi:10.1107/S205327332000217X

Acta Crystallographica Section A
Acta Crystallographica
Section A FOUNDATIONS AND ADVANCES

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Figure 1
The action of wedge reversion $[{1^\dagger}]$ on (a) a scalar (red dots) and a vector (black arrows), (b) a bivector (the blue patch of area, $[V^{(1)} \wedge V^{(2)}]$ ) and a trivector (the sea-green 3D volume, $[V^{(1)} \wedge V^{(2)} \wedge V^{(3)}]$ ) and (c) a quadvector (the yellow hypervolume in 4D, $[V^{(1)} \wedge V^{(2)} \wedge V^{(3)} \wedge V^{(4)}]$ ). Panel (c) has to be imagined as a 4D object. Scalars, vectors and wedge products ( $[\wedge]$ ) between linearly independent vectors V⁽ⁱ⁾ indexed by natural numbers i are called blades and their grades are indicated above. Blades of grades 4g and 4g+1 remain invariant, while those of grades 4g+2 and 4g+3 reverse under the action of $[{1^\dagger}]$ , where g is 0, 1, 2, 3…etc.

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

Volume 76| Part 3| May 2020| Pages 318-327

https://doi.org/10.1107/S205327332000217X

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