Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita.
The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:
The key defining property of the symbol is total antisymmetry in the indices.
Most authors choose ε1 2 ... n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal.
The term "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example,
The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system.
The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation.
Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry[1] and twistor theory,[2] where it appears in the context of 2-spinors.
Analogous to 2-dimensional matrices, the values of the 3-dimensional Levi-Civita symbol can be arranged into a 3 × 3 × 3 array: where i is the depth (blue: i = 1; red: i = 2; green: i = 3), j is the row and k is the column.
These values can be arranged into a 4 × 4 × 4 × 4 array, although in 4 dimensions and higher this is difficult to draw.
Using the capital pi notation Π for ordinary multiplication of numbers, an explicit expression for the symbol is:[citation needed]
where the signum function (denoted sgn) returns the sign of its argument while discarding the absolute value if nonzero.
However, the Levi-Civita symbol is a pseudotensor because under an orthogonal transformation of Jacobian determinant −1, for example, a reflection in an odd number of dimensions, it should acquire a minus sign if it were a tensor.
As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.
[5] Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix.
[5] In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.
In three dimensions, the relationship is given by the following equations (vertical lines denote the determinant):[4] A special case of this result occurs when one of the indices is repeated and summed over: In Einstein notation, the duplication of the i index implies the sum on i.
If two indices are repeated (and summed over), this further reduces to: In n dimensions, when all i1, ...,in, j1, ..., jn take values 1, 2, ..., n:[citation needed] where the exclamation mark (!)
denotes the factorial, and δα...β... is the generalized Kronecker delta.
Proof: Both sides change signs upon switching two indices, so without loss of generality assume
Since the equation is antisymmetric in ij and mn, any set of values for these can be reduced to the above case (which holds).
= 6 and for any distinct indices i, j, k taking values 1, 2, 3, we have In linear algebra, the determinant of a 3 × 3 square matrix A = [aij] can be written[6] Similarly the determinant of an n × n matrix A = [aij] can be written as[5] where each ir should be summed over 1, ..., n, or equivalently: where now each ir and each jr should be summed over 1, ..., n. More generally, we have the identity[5] Let
In any arbitrary curvilinear coordinate system and even in the absence of a metric on the manifold, the Levi-Civita symbol as defined above may be considered to be a tensor density field in two different ways.
In n dimensions using the generalized Kronecker delta,[7][8] Notice that these are numerically identical.
On a pseudo-Riemannian manifold, one may define a coordinate-invariant covariant tensor field whose coordinate representation agrees with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric and matches a selected orientation.
The covariant Levi-Civita tensor (also known as the Riemannian volume form) in any coordinate system that matches the selected orientation is where gab is the representation of the metric in that coordinate system.
We can similarly consider a contravariant Levi-Civita tensor by raising the indices with the metric as usual, but notice that if the metric signature contains an odd number of negative eigenvalues q, then the sign of the components of this tensor differ from the standard Levi-Civita symbol:[9] where sgn(det[gab]) = (−1)q,
is the usual Levi-Civita symbol discussed in the rest of this article, and we used the definition of the metric determinant in the derivation.
In Minkowski space (the four-dimensional spacetime of special relativity), the covariant Levi-Civita tensor is where the sign depends on the orientation of the basis.
The contravariant Levi-Civita tensor is The following are examples of the general identity above specialized to Minkowski space (with the negative sign arising from the odd number of negatives in the signature of the metric tensor in either sign convention): This article incorporates material from Levi-Civita permutation symbol on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
