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 Template:Math, for some positive integer Template:Mvar. 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 Template:Mvar or Template:Mvar, or less commonly the Latin lower case Template:Mvar. Index notation allows one to display permutations in a way compatible with tensor analysis: $${\epsilon }_{i_1{i}_2\dots i_n}$$ where each index Template:Math takes values Template:Math. There are Template:Math indexed values of Template:Math, which can be arranged into an Template:Mvar-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: $${\epsilon }_{\dots i_p\dots i_q\dots }=-{\epsilon }_{\dots i_q\dots i_p\dots }.$$

If any two indices are equal, the symbol is zero. When all indices are unequal, we have: $${\epsilon }_{i_1{i}_2\dots i_n}=(-1{)}^p{\epsilon }_{12\dots n},$$ where Template:Mvar (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble Template:Math into the order Template:Math, and the factor Template:Math is called the sign, or signature of the permutation. The value Template:Math must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose Template:Math, 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 "Template:Mvar-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol Template:Mvar matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, Template: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: $${\epsilon }_{ij}=\{\begin{array}{ll}+1& \text{if }(i,j)=(1,2)\\ -1& \text{if }(i,j)=(2,1)\\ 0& \text{if }i=j\end{array}$$ The values can be arranged into a 2 × 2 antisymmetric matrix: $$\left(\begin{array}{cc}{\epsilon }_{11}& {\epsilon }_{12}\\ {\epsilon }_{21}& {\epsilon }_{22}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

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

File:Permutation indices 3d numerical.svg

For the indices Template:Math in Template:Math, the values Template:Math occurring in the Template:Color box cyclic order Template:Math correspond to Template:Math, while occurring in the Template:Color box reverse cyclic order correspond to Template:Math, otherwise Template:Math.

In three dimensions, the Levi-Civita symbol is defined by: $${\epsilon }_{ijk}=\{\begin{array}{ll}+1& \text{if }(i,j,k)\text{ is }(1,2,3),(2,3,1),\text{ or }(3,1,2),\\ -1& \text{if }(i,j,k)\text{ is }(3,2,1),(1,3,2),\text{ or }(2,1,3),\\ 0& \text{if }i=j,\text{ or }j=k,\text{ or }k=i\end{array}$$

That is, Template:Math is Template:Math if Template:Math is an even permutation of Template:Math, Template:Math if it is an odd permutation, and 0 if any index is repeated. In three dimensions only, the cyclic permutations of Template:Math 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 Template:Math 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 3 × 3 × 3 array:

File:Epsilontensor.svg

where Template:Mvar is the depth (blue: Template:Math; red: Template:Math; green: Template:Math), Template:Mvar is the row and Template:Mvar is the column.

Some examples: $$\begin{array}{rl}{\epsilon }_{132}=-{\epsilon }_{123}& =-1\\ {\epsilon }_{312}=-{\epsilon }_{213}& =-(-{\epsilon }_{123})=1\\ {\epsilon }_{231}=-{\epsilon }_{132}& =-(-{\epsilon }_{123})=1\\ {\epsilon }_{232}=-{\epsilon }_{232}& =0\end{array}$$

Four dimensions

In four dimensions, the Levi-Civita symbol is defined by: $${\epsilon }_{ijkl}=\{\begin{array}{ll}+1& \text{if }(i,j,k,l)\text{ is an even permutation of }(1,2,3,4)\\ -1& \text{if }(i,j,k,l)\text{ is an odd permutation of }(1,2,3,4)\\ 0& \text{otherwise}\end{array}$$

These values can be arranged into a 4 × 4 × 4 × 4 array, although in 4 dimensions and higher this is difficult to draw.

Some examples: $$\begin{array}{rl}{\epsilon }_{1432}=-{\epsilon }_{1234}& =-1\\ {\epsilon }_{2134}=-{\epsilon }_{1234}& =-1\\ {\epsilon }_{4321}=-{\epsilon }_{1324}& =-(-{\epsilon }_{1234})=1\\ {\epsilon }_{3243}=-{\epsilon }_{3243}& =0\end{array}$$

Generalization to n dimensions

More generally, in [[n-dimensional space|Template:Math dimensions]], the Levi-Civita symbol is defined by: $${\epsilon }_{a_1{a}_2{a}_3\dots a_n}=\{\begin{array}{ll}+1& \text{if }(a_1,a_2,a_3,\dots ,a_n)\text{ is an even permutation of }(1,2,3,\dots ,n)\\ -1& \text{if }(a_1,a_2,a_3,\dots ,a_n)\text{ is an odd permutation of }(1,2,3,\dots ,n)\\ 0& \text{otherwise}\end{array}$$

Thus, it is the sign of the permutation in the case of a permutation, and zero otherwise.

Using the capital pi notation Template:Math for ordinary multiplication of numbers, an explicit expression for the symbol is: $$\begin{array}{rl}{\epsilon }_{a_1{a}_2{a}_3\dots a_n}& =\prod _{1\le i<j\le n}\mathrm{sgn}(a_j-a_i)\\ & =\mathrm{sgn}(a_2-a_1)\mathrm{sgn}(a_3-a_1)\cdots \mathrm{sgn}(a_n-a_1)\mathrm{sgn}(a_3-a_2)\mathrm{sgn}(a_4-a_2)\cdots \mathrm{sgn}(a_n-a_2)\cdots \mathrm{sgn}(a_n-a_{n-1})\end{array}$$ where the signum function (denoted Template:Math) returns the sign of its argument while discarding the absolute value if nonzero. The formula is valid for all index values, and for any Template:Mvar (when Template:Math or Template:Math, this is the empty product). However, computing the formula above naively has a time complexity of Template:Math, whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only Template:Math cost.

Properties

A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank Template:Mvar) 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.

In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.

Summation symbols can be eliminated by using Einstein notation, where an index repeated between two or more terms indicates summation over that index. For example,

${\epsilon }_{ijk}{\epsilon }^{imn}\equiv \sum _{i=1,2,3}{\epsilon }_{ijk}{\epsilon }^{imn}$.

In the following examples, Einstein notation is used.

Two dimensions

In two dimensions, when all Template:Math each take the values 1 and 2:

Three dimensions

Index and symbol values

In three dimensions, when all Template:Math each take values 1, 2, and 3:

Product

The Levi-Civita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations (vertical lines denote the determinant):

$\begin{array}{rl}{\epsilon }_{ijk}{\epsilon }_{lmn}& =\left|\begin{array}{ccc}{\delta }_{il}& {\delta }_{im}& {\delta }_{in}\\ {\delta }_{jl}& {\delta }_{jm}& {\delta }_{jn}\\ {\delta }_{kl}& {\delta }_{km}& {\delta }_{kn}\\ \end{array}\right|\\ & ={\delta }_{il}({\delta }_{jm}{\delta }_{kn}-{\delta }_{jn}{\delta }_{km})-{\delta }_{im}({\delta }_{jl}{\delta }_{kn}-{\delta }_{jn}{\delta }_{kl})+{\delta }_{in}({\delta }_{jl}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}).\end{array}$

A special case of this result occurs when one of the indices is repeated and summed over:

$\sum _{i=1}^3{\epsilon }_{ijk}{\epsilon }_{imn}={\delta }_{jm}{\delta }_{kn}-{\delta }_{jn}{\delta }_{km}$

In Einstein notation, the duplication of the Template:Mvar index implies the sum on Template:Mvar. The previous is then denoted Template:Math.

If two indices are repeated (and summed over), this further reduces to:

$\sum _{i=1}^3\sum _{j=1}^3{\epsilon }_{ijk}{\epsilon }_{ijn}=2{\delta }_{kn}$

n dimensions

Index and symbol values

In Template:Mvar dimensions, when all Template:Math take values Template:Math: Template:NumBlk Template:NumBlk Template:NumBlk

where the exclamation mark (Template:Math) denotes the factorial, and Template:Math is the generalized Kronecker delta. For any Template:Mvar, the property

$\sum _{i,j,k,\dots =1}^n{\epsilon }_{ijk\dots }{\epsilon }_{ijk\dots }=n!$

follows from the facts that

• every permutation is either even or odd,
• Template:Math, and
• the number of permutations of any Template:Mvar-element set number is exactly Template:Math.

The particular case of (Template:EquationNote) with $k=n-2$ is $${\epsilon }_{i_1\dots i_{n-2}jk}{\epsilon }^{i_1\dots i_{n-2}lm}=(n-2)!({\delta }_j^l{\delta }_k^m-{\delta }_j^m{\delta }_k^l).$$

Product

In general, for Template:Mvar dimensions, one can write the product of two Levi-Civita symbols as: $${\epsilon }_{i_1{i}_2\dots i_n}{\epsilon }_{j_1{j}_2\dots j_n}=\left|\begin{array}{cccc}{\delta }_{i_1{j}_1}& {\delta }_{i_1{j}_2}& \dots & {\delta }_{i_1{j}_n}\\ {\delta }_{i_2{j}_1}& {\delta }_{i_2{j}_2}& \dots & {\delta }_{i_2{j}_n}\\ ⋮& ⋮& \ddots & ⋮\\ {\delta }_{i_n{j}_1}& {\delta }_{i_n{j}_2}& \dots & {\delta }_{i_n{j}_n}\\ \end{array}\right|.$$Proof: Both sides change signs upon switching two indices, so without loss of generality assume $i_1\le \cdots \le i_n,j_1\le \cdots \le j_n$. If some $i_c=i_{c+1}$ then left side is zero, and right side is also zero since two of its rows are equal. Similarly for $j_c=j_{c+1}$. Finally, if $i_1<\cdots <i_n,j_1<\cdots <j_n$, then both sides are 1.

Proofs

For (Template:EquationNote), both sides are antisymmetric with respect of Template:Mvar and Template:Mvar. We therefore only need to consider the case Template:Math and Template:Math. By substitution, we see that the equation holds for Template:Math, that is, for Template:Math and Template:Math. (Both sides are then one). Since the equation is antisymmetric in Template:Mvar and Template:Mvar, any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of Template:Mvar and Template:Mvar.

Using (Template:EquationNote), we have for (Template:EquationNote)

${\epsilon }_{ij}{\epsilon }^{in}={\delta }_i{}^i{\delta }_j{}^n-{\delta }_i{}^n{\delta }_j{}^i=2{\delta }_j{}^n-{\delta }_j{}^n={\delta }_j{}^n.$

Here we used the Einstein summation convention with Template:Mvar going from 1 to 2. Next, (Template:EquationNote) follows similarly from (Template:EquationNote).

To establish (Template:EquationNote), notice that both sides vanish when Template:Math. Indeed, if Template:Math, then one can not choose Template:Mvar and Template:Mvar such that both permutation symbols on the left are nonzero. Then, with Template:Math fixed, there are only two ways to choose Template:Mvar and Template:Mvar from the remaining two indices. For any such indices, we have

${\epsilon }_{jmn}{\epsilon }^{imn}={\left({\epsilon }^{imn}\right)}^2=1$

(no summation), and the result follows.

Then (Template:EquationNote) follows since 3! = 6 and for any distinct indices Template:Math taking values 1, 2, 3, we have

${\epsilon }_{ijk}{\epsilon }^{ijk}=1$Template:Pad(no summation, distinct Template:Math)

Applications and examples

Determinants

In linear algebra, the determinant of a 3 × 3 square matrix Template:Math can be written

$\det (\mathbf{𝐀})=\sum _{i=1}^3\sum _{j=1}^3\sum _{k=1}^3{\epsilon }_{ijk}{a}_{1i}{a}_{2j}{a}_{3k}$

Similarly the determinant of an Template:Math matrix Template:Math can be written as

$\det (\mathbf{𝐀})={\epsilon }_{i_1\dots i_n}{a}_{1i_1}\dots a_{n{i}_n},$

where each Template:Mvar should be summed over Template:Math, or equivalently:

$\det (\mathbf{𝐀})=\frac{1}{n!}{\epsilon }_{i_1\dots i_n}{\epsilon }_{j_1\dots j_n}{a}_{i_1{j}_1}\dots a_{i_n{j}_n},$

where now each Template:Mvar and each Template:Mvar should be summed over Template:Math. More generally, we have the identity

$\sum _{i_1,i_2,\dots }{\epsilon }_{i_1\dots i_n}{a}_{i_1{j}_1}\dots a_{i_n{j}_n}=\det (\mathbf{𝐀}){\epsilon }_{j_1\dots j_n}$

Vector cross product

Cross product (two vectors)

Let $({\mathbf{e}}_{\mathbf{𝟏}},{\mathbf{e}}_{\mathbf{𝟐}},{\mathbf{e}}_{\mathbf{𝟑}})$ a positively oriented orthonormal basis of a vector space. If Template:Math and Template:Math are the coordinates of the vectors Template:Math and Template:Math in this basis, then their cross product can be written as a determinant:

$\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}=\left|\begin{array}{ccc}{\mathbf{e}}_{\mathbf{𝟏}}& {\mathbf{e}}_{\mathbf{𝟐}}& {\mathbf{e}}_{\mathbf{𝟑}}\\ a^1& a^2& a^3\\ b^1& b^2& b^3\\ \end{array}\right|=\sum _{i=1}^3\sum _{j=1}^3\sum _{k=1}^3{\epsilon }_{ijk}{\mathbf{𝐞}}_i{a}^j{b}^k$

hence also using the Levi-Civita symbol, and more simply:

$(\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}{)}^i=\sum _{j=1}^3\sum _{k=1}^3{\epsilon }_{ijk}{a}^j{b}^k.$

In Einstein notation, the summation symbols may be omitted, and the Template:Mvarth component of their cross product equals

$(\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}{)}^i={\epsilon }_{ijk}{a}^j{b}^k.$

The first component is

$(\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}{)}^1=a^2{b}^3-a^3{b}^2,$

then by cyclic permutations of 1, 2, 3 the others can be derived immediately, without explicitly calculating them from the above formulae:

$\begin{array}{rl}(\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}{)}^2& =a^3{b}^1-a^1{b}^3,\\ (\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}{)}^3& =a^1{b}^2-a^2{b}^1.\end{array}$

Triple scalar product (three vectors)

From the above expression for the cross product, we have:

$\mathbf{𝐚}\mathbf{\times }\mathbf{𝐛}=-\mathbf{𝐛}\mathbf{\times }\mathbf{𝐚}$.

If Template:Math is a third vector, then the triple scalar product equals

$\mathbf{𝐚}\cdot (\mathbf{𝐛}\mathbf{\times }\mathbf{𝐜})={\epsilon }_{ijk}{a}^i{b}^j{c}^k.$

From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any pair of arguments. For example,

$\mathbf{𝐚}\cdot (\mathbf{𝐛}\mathbf{\times }\mathbf{𝐜})=-\mathbf{𝐛}\cdot (\mathbf{𝐚}\mathbf{\times }\mathbf{𝐜})$.

Curl (one vector field)

If Template:Math is a vector field defined on some open set of Template:Math as a function of position Template:Math (using Cartesian coordinates). Then the Template:Mvarth component of the curl of Template:Math equals

$(\mathrm{\nabla }\times \mathbf{𝐅}{)}^i(\mathbf{𝐱})={\epsilon }_{ijk}\frac{\partial }{\partial x^j}{F}^k(\mathbf{𝐱}),$

which follows from the cross product expression above, substituting components of the gradient vector operator (nabla).

Tensor density

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. It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density of weight −1. In n dimensions using the generalized Kronecker delta,

$\begin{array}{rl}{\epsilon }^{{\mu }_1\dots {\mu }_n}& ={\delta }_{1\dots n}^{{\mu }_1\dots {\mu }_n}\\ {\epsilon }_{{\nu }_1\dots {\nu }_n}& ={\delta }_{{\nu }_1\dots {\nu }_n}^{1\dots n}.\end{array}$

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

$E_{a_1\dots a_n}=\sqrt{|\det [g_{ab}]|}{\epsilon }_{a_1\dots a_n},$

where Template:Mvar 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,

$E^{a_1\dots a_n}=E_{b_1\dots b_n}\prod _{i=1}^n{g}^{a_i{b}_i}=\frac{1}{\sqrt{|\det [g_{ab}]|}}{\epsilon }^{a_1\dots a_n},,$

but notice that if the metric signature contains an odd number of negative eigenvalues Template:Math, then the sign of the components of this tensor differ from the standard Levi-Civita symbol:

$E^{a_1\dots a_n}=\frac{\mathrm{sgn}(\det [g_{ab}])}{\sqrt{|\det [g_{ab}]|}}{\epsilon }^{a_1\dots a_n},$

where Template:Math, ${\epsilon }_{a_1\dots a_n}$ 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 $E_{01\dots n}=+\sqrt{|\det [g_{ab}]|}$, we have that $E^{01\dots n}=\frac{\mathrm{sgn}(\det [g_{ab}])}{\sqrt{|\det [g_{ab}]|}}$.

From this we can infer the identity,

$E^{{\mu }_1\dots {\mu }_p{\alpha }_1\dots {\alpha }_{n-p}}{E}_{{\mu }_1\dots {\mu }_p{\beta }_1\dots {\beta }_{n-p}}=(-1{)}^{q}p!{\delta }_{{\beta }_1\dots {\beta }_{n-p}}^{{\alpha }_1\dots {\alpha }_{n-p}},$

where

${\delta }_{{\beta }_1\dots {\beta }_{n-p}}^{{\alpha }_1\dots {\alpha }_{n-p}}=(n-p)!{\delta }_{{\beta }_1}^{[{\alpha }_1}\dots {\delta }_{{\beta }_{n-p}}^{{\alpha }_{n-p}]}$

is the generalized Kronecker delta.

Example: Minkowski space

In Minkowski space (the four-dimensional spacetime of special relativity), the covariant Levi-Civita tensor is

$E_{\alpha \beta \gamma \delta }=\pm \sqrt{|\det [g_{\mu \nu }]|}{\epsilon }_{\alpha \beta \gamma \delta },$

where the sign depends on the orientation of the basis. The contravariant Levi-Civita tensor is

$E^{\alpha \beta \gamma \delta }=g^{\alpha \zeta }{g}^{\beta \eta }{g}^{\gamma \theta }{g}^{\delta \iota }{E}_{\zeta \eta \theta \iota }.$

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):

$\begin{array}{rl}{E}_{\alpha \beta \gamma \delta }{E}_{\rho \sigma \mu \nu }& =-g_{\alpha \zeta }{g}_{\beta \eta }{g}_{\gamma \theta }{g}_{\delta \iota }{\delta }_{\rho \sigma \mu \nu }^{\zeta \eta \theta \iota }\\ E^{\alpha \beta \gamma \delta }{E}^{\rho \sigma \mu \nu }& =-g^{\alpha \zeta }{g}^{\beta \eta }{g}^{\gamma \theta }{g}^{\delta \iota }{\delta }_{\zeta \eta \theta \iota }^{\rho \sigma \mu \nu }\\ E^{\alpha \beta \gamma \delta }{E}_{\alpha \beta \gamma \delta }& =-24\\ E^{\alpha \beta \gamma \delta }{E}_{\rho \beta \gamma \delta }& =-6{\delta }_{\rho }^{\alpha }\\ E^{\alpha \beta \gamma \delta }{E}_{\rho \sigma \gamma \delta }& =-2{\delta }_{\rho \sigma }^{\alpha \beta }\\ E^{\alpha \beta \gamma \delta }{E}_{\rho \sigma \theta \delta }& =-{\delta }_{\rho \sigma \theta }^{\alpha \beta \gamma }.\end{array}$

See also

• List of permutation topics
• Symmetric tensor

External links

This article incorporates material from Levi-Civita permutation symbol on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.