advanced_tools:hodge_dual
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| - | If we increase the rank of a general tensor is has more and more components. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 4x4=16 components, a rank-3 tensor has 4x4x4=64 components, and a rank-4 tensor has 4x4x4x4=256 components. | + | If we increase the rank of a general tensor it acquires more and more components. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 4x4=16 components, a rank-3 tensor has 4x4x4=64 components, and a rank-4 tensor has 4x4x4x4=256 components. |
| - | The situation changes completely if we consider (totally) antisymmetric tensors instead of general tensors. Now the number of independent components increases at first but then decreases again. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 6 components, a rank-3 tensor has 4 components, and a rank-4 tensor has just 1 component. | + | The situation changes completely if we consider (totally) antisymmetric tensors instead of general tensors. Now the number of independent components increases at first but then decreases again. For example in 4D space a rank-0 tensor has 1 component, a rank-1 tensor has 4 components, a rank-2 tensor has 6 components, a rank-3 tensor has 4 components, and a rank-4 tensor has just 1 component. Notice the pattern: 1 4 6 4 1. |
| This pattern suggests a relationship between antisymmetric rank-$p$ tensors and and antisymmetric rank-($4-p$) tensors in 4D space. This is the idea behind the Hodge dual! The diagram below shows this pattern for 2, 3, and 4-dimensional spaces. | This pattern suggests a relationship between antisymmetric rank-$p$ tensors and and antisymmetric rank-($4-p$) tensors in 4D space. This is the idea behind the Hodge dual! The diagram below shows this pattern for 2, 3, and 4-dimensional spaces. | ||
| - | For a more detailed explanation of this picture see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | + | For a more detailed explanation of this picture see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#hodge_dual|Fun with Symmetry]]. |
| [{{ :advanced_tools:hodge_dual.jpg?nolink }}] | [{{ :advanced_tools:hodge_dual.jpg?nolink }}] | ||
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