Levi-Civita tensor: Difference between revisions
imported>Meg Taylor No edit summary |
imported>Meg Taylor No edit summary |
||
| Line 8: | Line 8: | ||
</ref> | </ref> | ||
This three-dimensional three-index form can be generalized to ''n'' dimensions. In ''n'' dimensions | This three-dimensional three-index form can be generalized to ''n'' dimensions. In ''n'' dimensions the completely antisymmetric tensor with ''n'' indices in ''n'' dimensions is an invariant of the special unitary group [[SU(n)]].<ref name=Vaughn> | ||
{{cite book |title=Introduction to mathematical physics |author=Michael T. Vaughn |pages=p. 484 |url=http://books.google.com/books?id=E6_DiJDIptoC&pg=PA484 |isbn=3527406271 |publisher=Wiley-VCH |year=2007}} | {{cite book |title=Introduction to mathematical physics |author=Michael T. Vaughn |pages=p. 484 |url=http://books.google.com/books?id=E6_DiJDIptoC&pg=PA484 |isbn=3527406271 |publisher=Wiley-VCH |year=2007}} | ||
Revision as of 03:25, 12 October 2013
The Levi-Civita tensor, sometimes called the Levi-Civita form, is the completely antisymmetric tensor with three indices in three dimensions, and its components are given by the Levi-Civita symbol. Both the symbol and the tensor are named after the Italian mathematician and physicist Tullio Levi-Civita.
The Levi-Civita tensor is an invariant of the special unitary group SU(3). It flips sign under reflections, and physicists call it a pseudo-tensor.[1]
This three-dimensional three-index form can be generalized to n dimensions. In n dimensions the completely antisymmetric tensor with n indices in n dimensions is an invariant of the special unitary group SU(n).[2] It also is called the alternating tensor[3] or the completely antisymmetric tensor with n indices in n dimensions.
The completely antisymmetric tensor with n indices in n-dimensions has only one independent component, and is denoted in two, three and four dimensions as εij, εijk, εijkl.[4] Consequently, in three dimensions the completely antisymmetric tensor with three indices is entirely specified by stating ε123 = εxyz = 1 in Cartesian coordinates.
Notes
- ↑ Bjørn Felsager (1998). Geometry, particles, and fields. Springer, p. 358. ISBN 0387982671.
- ↑ Michael T. Vaughn (2007). Introduction to mathematical physics. Wiley-VCH, p. 484. ISBN 3527406271.
- ↑ Vinod K. Sharma (2009). “§9.2 Alternating tensor (or Levi-Civita symbol)”, Matrix Methods and Vector Spaces in Physics. Prentice-Hall of India Pvt.Ltd, p. 370. ISBN 8120338669.
- ↑ T. Padmanabhan (2010). Gravitation: Foundations and Frontiers. Cambridge University Press, p. 22. ISBN 0521882230.