Levi-Civita tensor: Difference between revisions
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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]]. | 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]]. It is an invariant of the special unitary group [[SU(3)]]. It flips sign under reflections, and physicists call it a ''pseudo''-tensor.<ref name=Felsager> | ||
{{cite book |title= | {{cite book |title=Geometry, particles, and fields |author=Bjørn Felsager |pages=p. 358 |url=http://books.google.com/books?id=R1XkarKY7AwC&pg=PA358 |year=1998 |isbn=0387982671 |publisher=Springer}} | ||
</ref> | |||
This three dimensional form can be generalized to ''n'' dimensions. In ''n'' dimensions the 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= | {{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}} | ||
</ref> | </ref> It also is called the ''alternating tensor''<ref name=Sharma> | ||
{{cite book |title=Matrix Methods and Vector Spaces in Physics |author=Vinod K. Sharma |url=http://books.google.com/books?id=Kg2ZjUmOB9EC&pg=PT386 |pages=p. 370|chapter=§9.2 Alternating tensor (or Levi-Civita symbol) |isbn=8120338669 |publisher=Prentice-Hall of India Pvt.Ltd |year=2009}} | {{cite book |title=Matrix Methods and Vector Spaces in Physics |author=Vinod K. Sharma |url=http://books.google.com/books?id=Kg2ZjUmOB9EC&pg=PT386 |pages=p. 370|chapter=§9.2 Alternating tensor (or Levi-Civita symbol) |isbn=8120338669 |publisher=Prentice-Hall of India Pvt.Ltd |year=2009}} | ||
</ref> 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 ε<sub>ij</sub>, ε<sub>ijk</sub>, ε<sub>ijkl</sub>.<ref name=Padmanabhan> | </ref> 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 ε<sub>ij</sub>, ε<sub>ijk</sub>, ε<sub>ijkl</sub>.<ref name=Padmanabhan> | |||
{{cite book |title=Gravitation: Foundations and Frontiers |author=T. Padmanabhan |url=http://books.google.com/books?id=BSfe2MjbQ3gC&pg=PA22 |pages=p. 22 |isbn=0521882230 |publisher=Cambridge University Press |year=2010}} | {{cite book |title=Gravitation: Foundations and Frontiers |author=T. Padmanabhan |url=http://books.google.com/books?id=BSfe2MjbQ3gC&pg=PA22 |pages=p. 22 |isbn=0521882230 |publisher=Cambridge University Press |year=2010}} | ||
Revision as of 21:26, 2 January 2011
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. It 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 form can be generalized to n dimensions. In n dimensions the 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.