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]]. 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.<ref name=Felsager> | |||
{{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 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}} | |||
</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}} | |||
</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}} | |||
</ref> Consequently, in three dimensions the completely antisymmetric tensor with three indices is entirely specified by stating ε<sub>123</sub> = ε<sub>xyz</sub> = 1 in [[Cartesian coordinates]]. | |||
==Notes== | |||
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Latest revision as of 11:00, 11 September 2024
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.