Levi-Civita symbol

In

property and definition in terms of permutations.

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:

where each index i₁, i₂, ..., i_n takes values 1, 2, ..., n. There are nⁿ indexed values of ε_i1i2...in, which can be arranged into an n-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:

If any two indices are equal, the symbol is zero. When all indices are unequal, we have:

where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i₁, i₂, ..., i_n into the order 1, 2, ..., n, and the factor (−1)^p is called the sign, or signature of the permutation. The value ε_{1 2 ... n} must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε_{1 2 ... n} = +1, 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 "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the

, for example, ${\displaystyle \mathbb {R} ^{3}}$ 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

, the Levi-Civita symbol is defined by: The values can be arranged into a 2 × 2 :

Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like

.

Three dimensions

In

$${\displaystyle \varepsilon _{ijk}={\begin{cases}+1&{\text{if }}(i,j,k){\text{ is }}(1,2,3),(2,3,1),{\text{ or }}(3,1,2),\\-1&{\text{if }}(i,j,k){\text{ is }}(3,2,1),(1,3,2),{\text{ or }}(2,1,3),\\\;\;\,0&{\text{if }}i=j,{\text{ or }}j=k,{\text{ or }}k=i\end{cases}}}$$

That is, ε_ijk is 1 if (i, j, k) is an

anticyclic permutations

are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of (1, 2, 3) 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:

where i is the depth (blue: i = 1; red: i = 2; green: i = 3), j is the row and k is the column.

Some examples:

Four dimensions

In four dimensions, the Levi-Civita symbol is defined by:

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:

Generalization to n dimensions

More generally, in

$${\displaystyle \varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}={\begin{cases}+1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an even permutation of }}(1,2,3,\dots ,n)\\-1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an odd permutation of }}(1,2,3,\dots ,n)\\\;\;\,0&{\text{otherwise}}\end{cases}}}$$

Thus, it is the

sign of the permutation

in the case of a permutation, and zero otherwise.

Using the capital pi notation Π for ordinary multiplication of numbers, an explicit expression for the symbol is:^{[citation needed]}

where the in only O(n log(n)) cost.

Properties

A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) 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.^[5]

Under a general

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

]

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,

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

In the following examples, Einstein notation is used.

Two dimensions

In two dimensions, when all i, j, m, n each take the values 1 and 2:^[3]

${\displaystyle \varepsilon _{ij}\varepsilon ^{mn}={\delta _{i}}^{m}{\delta _{j}}^{n}-{\delta _{i}}^{n}{\delta _{j}}^{m}}$

(1)

${\displaystyle \varepsilon _{ij}\varepsilon ^{in}={\delta _{j}}^{n}}$

(2)

${\displaystyle \varepsilon _{ij}\varepsilon ^{ij}=2.}$

(3)

Three dimensions

Index and symbol values

In three dimensions, when all i, j, k, m, n each take values 1, 2, and 3:^[3]

${\displaystyle \varepsilon _{ijk}\varepsilon ^{pqk}=\delta _{i}{}^{p}\delta _{j}{}^{q}-\delta _{i}{}^{q}\delta _{j}{}^{p}}$

(4)

${\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=2{\delta _{j}}^{i}}$

(5)

${\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=6.}$

(6)

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):^[4]

${\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\\\end{vmatrix}}\\[6pt]&=\delta _{il}\left(\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\right)-\delta _{im}\left(\delta _{jl}\delta _{kn}-\delta _{jn}\delta _{kl}\right)+\delta _{in}\left(\delta _{jl}\delta _{km}-\delta _{jm}\delta _{kl}\right).\end{aligned}}}$

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

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

In Einstein notation, the duplication of the i index implies the sum on i. The previous is then denoted ε_ijkε_imn = δ_jmδ_kn − δ_jnδ_km.

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

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

n dimensions

Index and symbol values

In n dimensions, when all i₁, ...,i_n, j₁, ..., j_n take values 1, 2, ..., n:^{[citation needed]}

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{j_{1}\dots j_{n}}=\delta _{i_{1}\dots i_{n}}^{j_{1}\dots j_{n}}}$

(7)

${\displaystyle \varepsilon _{i_{1}\dots i_{k}~i_{k+1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{k}~j_{k+1}\dots j_{n}}=\delta _{i_{1}\ldots i_{k}~i_{k+1}\ldots i_{n}}^{i_{1}\dots i_{k}~j_{k+1}\ldots j_{n}}=k!~\delta _{i_{k+1}\dots i_{n}}^{j_{k+1}\dots j_{n}}}$

(8)

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{n}}=n!}$

(9)

where the exclamation mark (!) denotes the

. For any n, the property

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

follows from the facts that

• every permutation is either even or odd,
• (+1)² = (−1)² = 1, and
• the number of permutations of any n-element set number is exactly n!.

The particular case of (8) with ${\textstyle k=n-2}$ is

Product

In general, for n dimensions, one can write the product of two Levi-Civita symbols as:

Proof: Both sides change signs upon switching two indices, so without loss of generality assume ${\displaystyle i_{1}\leq \cdots \leq i_{n},j_{1}\leq \cdots \leq j_{n}}$. If some ${\displaystyle i_{c}=i_{c+1}}$ then left side is zero, and right side is also zero since two of its rows are equal. Similarly for ${\displaystyle j_{c}=j_{c+1}}$. Finally, if ${\displaystyle i_{1}<\cdots <i_{n},j_{1}<\cdots <j_{n}}$, then both sides are 1.

Proofs

For (1), both sides are antisymmetric with respect of ij and mn. We therefore only need to consider the case i ≠ j and m ≠ n. By substitution, we see that the equation holds for ε₁₂ε¹², that is, for i = m = 1 and j = n = 2. (Both sides are then one). Since the equation is antisymmetric in ij and mn, any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of ij and mn.

Using (1), we have for (2)

${\displaystyle \varepsilon _{ij}\varepsilon ^{in}=\delta _{i}{}^{i}\delta _{j}{}^{n}-\delta _{i}{}^{n}\delta _{j}{}^{i}=2\delta _{j}{}^{n}-\delta _{j}{}^{n}=\delta _{j}{}^{n}\,.}$

Here we used the

).

To establish (5), notice that both sides vanish when i ≠ j. Indeed, if i ≠ j, then one can not choose m and n such that both permutation symbols on the left are nonzero. Then, with i = j fixed, there are only two ways to choose m and n from the remaining two indices. For any such indices, we have

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

(no summation), and the result follows.

Then (6) follows since 3! = 6 and for any distinct indices i, j, k taking values 1, 2, 3, we have

${\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=1}$ (no summation, distinct i, j, k)

Applications and examples

Determinants

In linear algebra, the determinant of a 3 × 3 square matrix A = [a_ij] can be written^[6]

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

Similarly the determinant of an n × n matrix A = [a_ij] can be written as^[5]

${\displaystyle \det(\mathbf {A} )=\varepsilon _{i_{1}\dots i_{n}}a_{1i_{1}}\dots a_{ni_{n}},}$

where each i_r should be summed over 1, ..., n, or equivalently:

${\displaystyle \det(\mathbf {A} )={\frac {1}{n!}}\varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}a_{i_{1}j_{1}}\dots a_{i_{n}j_{n}},}$

where now each i_r and each j_r should be summed over 1, ..., n. More generally, we have the identity^[5]

${\displaystyle \sum _{i_{1},i_{2},\dots }\varepsilon _{i_{1}\dots i_{n}}a_{i_{1}\,j_{1}}\dots a_{i_{n}\,j_{n}}=\det(\mathbf {A} )\varepsilon _{j_{1}\dots j_{n}}}$

Vector cross product

Cross product (two vectors)

Let ${\displaystyle (\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} )}$ a

${\displaystyle \mathbf {a\times b} ={\begin{vmatrix}\mathbf {e_{1}} &\mathbf {e_{2}} &\mathbf {e_{3}} \\a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\\end{vmatrix}}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}\mathbf {e} _{i}a^{j}b^{k}}$

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

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

In Einstein notation, the summation symbols may be omitted, and the ith component of their cross product equals[4]

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

The first component is

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

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

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

Triple scalar product (three vectors)

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

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

If c = (c¹, c², c³) is a third vector, then the

equals

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

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

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

Curl (one vector field)

If F = (F¹, F², F³) is a vector field defined on some open set of ${\displaystyle \mathbb {R} ^{3}}$ as a

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

which follows from the cross product expression above, substituting components of the

operator

(nabla).

Tensor density

In any arbitrary

${\displaystyle {\begin{aligned}\varepsilon ^{\mu _{1}\dots \mu _{n}}&=\delta _{\,1\,\dots \,n}^{\mu _{1}\dots \mu _{n}}\,\\\varepsilon _{\nu _{1}\dots \nu _{n}}&=\delta _{\nu _{1}\dots \nu _{n}}^{\,1\,\dots \,n}\,.\end{aligned}}}$

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 Carroll 2004.

The covariant Levi-Civita tensor (also known as the

) in any coordinate system that matches the selected orientation is

${\displaystyle E_{a_{1}\dots a_{n}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon _{a_{1}\dots a_{n}}\,,}$

where g_ab 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,

${\displaystyle E^{a_{1}\dots a_{n}}=E_{b_{1}\dots b_{n}}\prod _{i=1}^{n}g^{a_{i}b_{i}}\,,}$

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:

${\displaystyle E^{a_{1}\dots a_{n}}={\frac {\operatorname {sgn} \left(\det[g_{ab}]\right)}{\sqrt {\left|\det[g_{ab}]\right|}}}\,\varepsilon ^{a_{1}\dots a_{n}},}$

where sgn(det[g_ab]) = (−1)^q, and ${\displaystyle \varepsilon ^{a_{1}\dots a_{n}}}$ is the usual Levi-Civita symbol discussed in the rest of this article. More explicitly, when the tensor and basis orientation are chosen such that ${\textstyle E_{01\dots n}=+{\sqrt {\left|\det[g_{ab}]\right|}}}$, we have that ${\displaystyle E^{01\dots n}={\frac {\operatorname {sgn}(\det[g_{ab}])}{\sqrt {\left|\det[g_{ab}]\right|}}}}$.

From this we can infer the identity,

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

where

${\displaystyle \delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}=(n-p)!\delta _{\beta _{1}}^{\lbrack \alpha _{1}}\dots \delta _{\beta _{n-p}}^{\alpha _{n-p}\rbrack }}$

is the generalized Kronecker delta.

Example: Minkowski space

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

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

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

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

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):

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

See also

• List of permutation topics
• Symmetric tensor

Notes

References

• Misner, C.; Thorne, K. S.; Wheeler, J. A. (1973).
  ISBN 0-7167-0344-0
  .
• Neuenschwander, D. E. (2015). Tensor Calculus for Physics. Johns Hopkins University Press. pp. 11, 29, 95.
  ISBN 978-1-4214-1565-9
  .
• ISBN 0-8053-8732-3

External links

This article incorporates material from Levi-Civita permutation symbol on

Creative Commons Attribution/Share-Alike License

.

• Weisstein, Eric W. "Permutation Tensor". MathWorld.