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]]. 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}} | {{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}} | ||
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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> | This three-dimensional three-index 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=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 10:51, 3 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. 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 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.