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 , for some positive integer . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.
The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon or , or less commonly the Latin lower case . Index notation allows one to display permutations in a way compatible with tensor analysis: <math display="block">\varepsilon_{i_1 i_2 \dots i_n}</math> where each index takes values . There are indexed values of , which can be arranged into an -dimensional array. The key defining property of the symbol is total antisymmetry in the indices. When any two indices are interchanged, equal or not, the symbol is negated: <math display="block">\varepsilon_{\dots i_p \dots i_q \dots } = -\varepsilon_{\dots i_q \dots i_p \dots } .</math>
If any two indices are equal, the symbol is zero. When all indices are unequal, we have: <math display="block">\varepsilon_{i_1 i_2 \dots i_n} = (-1)^p \varepsilon_{1 \, 2 \, \dots n} ,</math> where (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble into the order , and the factor is called the sign, or signature of the permutation. The value must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose , which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.
The term "-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, <math>\mathbb{R}^3</math> or Minkowski space. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density.
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.
Definition
The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so these are given here before defining the general case.
Two dimensions
In two dimensions, the Levi-Civita symbol is defined by:
<math display="block"> \varepsilon_{ij} =
\begin{cases}
+1 & \text{if } (i, j) = (1, 2) \\
-1 & \text{if } (i, j) = (2, 1) \\
\;\;\,0 & \text{if } i = j
\end{cases}
</math>
The values can be arranged into a 2 × 2 antisymmetric matrix:
<math display="block">\begin{pmatrix}
\varepsilon_{11} & \varepsilon_{12} \\
\varepsilon_{21} & \varepsilon_{22}
\end{pmatrix} = \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}</math>
Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry and twistor theory, where it appears in the context of 2-spinors.
Three dimensions
thumb|210px|For the indices in , the values occurring in the cyclic order correspond to , while occurring in the reverse cyclic order correspond to , otherwise .
In three dimensions, the Levi-Civita symbol is defined by:
<math display="block"> \varepsilon_{ijk} = \begin{cases}
+1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1), \text{ or } (3,1,2), \\
-1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2), \text{ or } (2,1,3), \\
\;\;\,0 & \text{if } i = j, \text{ or } j = k, \text{ or } k = i
\end{cases}</math>
That is, is if is an even permutation of , if it is an odd permutation, and 0 if any index is repeated. In three dimensions only, the cyclic permutations of are all even permutations, similarly the anticyclic permutations are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of and easily obtain all the even or odd permutations.
Analogous to 2-dimensional matrices, the values of the 3-dimensional Levi-Civita symbol can be arranged into a array:
:200px
where is the depth (: ; : ; : ), is the row and is the column.
Some examples:
<math display="block">\begin{align}
\varepsilon_{\color{BrickRed}{1} \color{Violet}{3} \color{Orange}{2 = -\varepsilon_{\color{BrickRed}{1} \color{Orange}{2} \color{Violet}{3 &= - 1 \\
\varepsilon_{ \color{Violet}{3} \color{BrickRed}{1} \color{Orange}{2 = -\varepsilon_{ \color{Orange}{2} \color{BrickRed}{1} \color{Violet}{3 &= -(-\varepsilon_{\color{BrickRed}{1} \color{Orange}{2} \color{Violet}{3) = 1 \\
\varepsilon_{ \color{Orange}{2} \color{Violet}{3} \color{BrickRed}{1 = -\varepsilon_{\color{BrickRed}{1} \color{Violet}{3} \color{Orange}{2 &= -(-\varepsilon_{\color{BrickRed}{1} \color{Orange}{2} \color{Violet}{3) = 1 \\
\varepsilon_{ \color{Orange}{2} \color{Violet}{3} \color{Orange}{2 = -\varepsilon_{ \color{Orange}{2} \color{Violet}{3} \color{Orange}{2 &= 0
\end{align}</math>
Four dimensions
In four dimensions, the Levi-Civita symbol is defined by:
<math display="block">\varepsilon_{ijkl} = \begin{cases}
+1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\
-1 & \text{if }(i,j,k,l) \text{ is an odd permutation of } (1,2,3,4) \\
\;\;\,0 & \text{otherwise}
\end{cases}</math>
These values can be arranged into a array, although in 4 dimensions and higher this is difficult to draw.
Some examples:
<math display="block">\begin{align}
\varepsilon_{\color{BrickRed}{1} \color{RedViolet}{4}\color{Violet}{3} \color{Orange}{\color{Orange}{2} = -\varepsilon_{\color{BrickRed}{1} \color{Orange}{\color{Orange}{2 \color{Violet}{3} \color{RedViolet}{4 &= - 1\\
\varepsilon_{\color{Orange}{\color{Orange}{2 \color{BrickRed}{1} \color{Violet}{3} \color{RedViolet}{4 = -\varepsilon_{\color{BrickRed}{1} \color{Orange}{\color{Orange}{2 \color{Violet}{3} \color{RedViolet}{4 &= -1\\
\varepsilon_{\color{RedViolet}{4} \color{Violet}{3} \color{Orange}{\color{Orange}{2 \color{BrickRed}{1 = -\varepsilon_{\color{BrickRed}{1} \color{Violet}{3} \color{Orange}{\color{Orange}{2 \color{RedViolet}{4 &= -(-\varepsilon_{\color{BrickRed}{1} \color{Orange}{\color{Orange}{2 \color{Violet}{3} \color{RedViolet}{4) = 1\\
\varepsilon_{\color{Violet}{3} \color{Orange}{\color{Orange}{2 \color{RedViolet}{4} \color{Violet}{3 = -\varepsilon_{\color{Violet}{3} \color{Orange}{\color{Orange}{2 \color{RedViolet}{4}\color{Violet}{3 &= 0
\end{align}</math>
Generalization to n dimensions
More generally, in dimensions, the Levi-Civita symbol is defined by:
<math display="block">\varepsilon_{a_1 a_2 a_3 \ldots a_n} = \begin{cases}
+1 & \text{if }(a_1, a_2, a_3, \ldots, a_n) \text{ is an even permutation of } (1, 2, 3, \dots, n) \\
-1 & \text{if }(a_1, a_2, a_3, \ldots, a_n) \text{ is an odd permutation of } (1, 2, 3, \dots, n) \\
\;\;\,0 & \text{otherwise}
\end{cases}</math>
Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.
Using the capital pi notation for ordinary multiplication of numbers, an explicit expression for the symbol is:
<math display="block">\begin{align}
\varepsilon_{a_1 a_2 a_3 \ldots a_n}
& = \prod_{1 \leq i < j \leq n} \sgn (a_j - a_i) \\
& = \sgn(a_2 - a_1) \sgn(a_3 - a_1) \dotsm \sgn(a_n - a_1) \sgn(a_3 - a_2) \sgn(a_4 - a_2) \dotsm \sgn(a_n - a_2) \dotsm \sgn(a_n - a_{n-1})
\end{align}</math>
where the signum function (denoted ) returns the sign of its argument while discarding the absolute value if nonzero. The formula is valid for all index values, and for any (when or , this is the empty product). However, computing the formula above naively has a time complexity of , whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only cost.
Properties
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank ) is sometimes called a permutation tensor.
Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems related by orthogonal transformations. 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 it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor.
As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector.
Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.
:<math>\det(\mathbf{A}) = \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \varepsilon_{ijk} a_{1i} a_{2j} a_{3k}</math>
Similarly the determinant of an matrix can be written as
:<math>\begin{align}
\varepsilon^{\mu_1 \dots \mu_n} &= \delta^{\mu_1 \dots \mu_n}_{\,1 \,\dots \,n} \, \\
\varepsilon_{\nu_1 \dots \nu_n} &= \delta^{\,1 \,\dots \,n}_{\nu_1 \dots \nu_n} \,.
\end{align}</math>
Notice that these are numerically identical. In particular, the sign is the same.
Levi-Civita tensors
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. This tensor should not be confused with the tensor density field mentioned above. The presentation in this section closely follows .
The covariant Levi-Civita tensor (also known as the Riemannian volume form) in any coordinate system that matches the selected orientation is
:<math>E_{a_1\dots a_n} = \sqrt{\left|\det [g_{ab}]\right|}\, \varepsilon_{a_1\dots a_n} \,,</math>
where 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,
:<math>E^{a_1\dots a_n} = E_{b_1\dots b_n} \prod_{i=1}^n g^{a_i b_i} = \frac{1}{\sqrt{\left|\det [g_{ab}]\right| \, \varepsilon^{a_1 \dots a_n},,</math>
but notice that if the metric signature contains an odd number of negative eigenvalues , then the sign of the components of this tensor differ from the standard Levi-Civita symbol:
:<math>E^{a_1\dots a_n} = \frac{ \sgn \left( \det [g_{ab}] \right) }{ \sqrt{ \left| \det [g_{ab}] \right| \, \varepsilon^{a_1\dots a_n} ,</math>
where , <math>\varepsilon_{a_1\dots a_n}</math> 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. More explicitly, when the tensor and basis orientation are chosen such that <math display="inline">E_{01\dots n} = +\sqrt{\left|\det [g_{ab}]\right|}</math>, we have that <math>E^{01\dots n} = \frac{\sgn(\det [g_{ab}])}{\sqrt{\left|\det [g_{ab}]\right|</math>.
From this we can infer the identity,
:<math>E^{\mu_1\dots\mu_p\alpha_1\dots\alpha_{n-pE_{\mu_1\dots\mu_p\beta_1\dots\beta_{n-p = (-1)^q p!\delta^{\alpha_1\dots\alpha_{n-p_{\beta_1\dots\beta_{n-p \,,</math>
where
:<math>\delta^{\alpha_1 \dots \alpha_{n-p_{\beta_1 \dots \beta_{n-p = (n-p)! \delta^{\lbrack \alpha_1}_{\beta_1} \dots \delta^{\alpha_{n-p} \rbrack}_{\beta_{n-p</math>
is the generalized Kronecker delta.
Example: Minkowski space
In Minkowski space (the four-dimensional spacetime of special relativity), the covariant Levi-Civita tensor is
:<math>E_{\alpha \beta \gamma \delta} = \pm \sqrt{ \left| \det [g_{\mu \nu}] \right| } \, \varepsilon_{\alpha \beta \gamma \delta} \,,</math>
where the sign depends on the orientation of the basis. The contravariant Levi-Civita tensor is
:<math>E^{\alpha \beta \gamma \delta} = g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} E_{\zeta \eta \theta \iota} \,.</math>
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):
:<math>\begin{align}
E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} & = -g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \\
E^{\alpha \beta \gamma \delta} E^{\rho \sigma \mu \nu} & = -g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} \delta^{\rho \sigma \mu \nu}_{\zeta \eta \theta \iota} \\
E^{\alpha \beta \gamma \delta} E_{\alpha \beta \gamma \delta} & = - 24 \\
E^{\alpha \beta \gamma \delta} E_{\rho \beta \gamma \delta} & = - 6 \delta^{\alpha}_{\rho} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \gamma \delta} & = - \delta^{\alpha \beta}_{\rho \sigma} \\
E^{\alpha \beta \gamma \delta} E_{\rho \sigma \theta \delta} & = - \delta^{\alpha \beta \gamma}_{\rho \sigma \theta} \,.
\end{align}</math>
See also
- List of permutation topics
- Symmetric tensor