Levi Civita symbol

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The Levi-Civita symbol , also known as the permutation symbol , (a little carelessly) totally antisymmetric tensor or epsilon tensor , is a symbol that is useful in physics for vector and tensor calculations . It is named after the Italian mathematician Tullio Levi-Civita . If one generally considers permutations in mathematics , one speaks instead of the sign of the corresponding permutation. In differential geometry , one considers the antisymmetrization mapping and the Hodge star independently of coordinates .

The indices to have values ​​from 1 to . If two or more indices have the same value, then is . If the values ​​of the indices are different, the symbol indicates whether an even ( ) or an odd ( ) number of swaps of the indices is necessary in order to arrange the values ​​in ascending order. For example , it takes a single swap to get 132 into the order 123.


The Levi-Civita symbol in dimensions has indices that usually run from 1 to (0 to for some applications ). It is defined by the following properties:

  • .
  • Under interchange of two indices, it changes the sign : .

From the second property follows immediately: if two indices are equal, the value is zero: .

The definition is equivalent

An alternative definition uses a formula that is also used to represent the sign of a permutation :


Let it denote the set of natural numbers from 1 to . The Levi-Civita symbol can be understood as a mapping with , if is not bijective , and otherwise (i.e. the sign of if is a permutation).

Relationship with the determinant

The determinant of a - matrix can Civita symbol Levi-and with the summation convention will be written as follows:

The relationship is more general


If you insert into this relationship for the identity matrix , i.e. for the Kronecker delta , you get the following representation of the Levi-Civita symbol:


The rows of the matrix are the unit vectors from the standard basis of the . This matrix is ​​therefore the permutation matrix that maps the vector to . From this one obtains an expression for the following tensor product with the help of the product rule for determinants :


Using Laplace's expansion theorem one obtains the following relationship if one tapered over the first indices of both tensors :


As an application of these formulas one obtains for the entries of the adjuncts of a matrix:


Especially in three dimensions

The Levi-Civita symbol can be represented as the spatial product of three orthogonal unit vectors :

The product of two epsilon tensors makes use of the fact that the product of two determinants can be written as the determinant of the matrix product . In addition, one uses the identity of the determinant of a matrix and the determinant of the transposed matrix :

Thus, the product of two epsilon tensors can be written as a determinant of Kronecker deltas:

As components of a pseudotensor density

If one defines a -fold covariant pseudotensor density of weight -1 by specifying the and all its components by for a given ordered basis , the components of this pseudotensor density do not change when the basis is changed. In this sense, the Levi-Civita symbol represents the components of a pseudotensor density with respect to any base. It follows in particular that the symbol describes the components of a tensor if only orthonormal bases of positive orientation are considered.

In a similar way, in or more generally on a -dimensional orientable semi-Riemannian manifold, the Levi-Civita symbol can be used to define the components of a covariant totally skew-symmetric tensor field -th level, a so-called differential form. Such a differential form is only determined up to a scalar factor. The choice of the prefactor fixes the volume unit and defines the differential form as the volume form . In Euclidean space , the Levi-Civita symbol stands for the components of the standard volume in the standard base . With respect to another basis , the same tensor obviously has the components , where and is the inverse matrix . If the basis is not orthonormal with respect to the standard scalar product , then the co- and contravariant components of the tensor differ accordingly . The prefactor depends on the coordinates if curvilinear coordinates are used or the underlying base space is a ( orientable ) manifold . For a semi-Riemannian manifold with a metric tensor and the associated Riemannian volume form (see Hodge star operator ) the prefactor is given by . The sign depends on the selected orientation . The connection between Levi-Civita-Symbol and Kronecker-Delta becomes generalized to


The Levi-Civita symbol in general relativity

In the general theory of relativity, the notation is also used . It usually marks the Levi-Civita symbol in flat space and is accompanied by the definition (here conventionally in 3D)

with the metric determinants for the Levi-Civita (pseudo) tensor. The metric usually gives an orthonormal basis. The Levi-Civita tensor then transforms like a tensor. For this reason, the cross product in a three-dimensional space-like hypersurface (as used in the 3 + 1 Cauchy Initial Value formulation, see ADM mass ) is generally not clearly defined.


Vector calculation

For the three-dimensional case it results

whereby .

Values ​​of the Levi-Civita symbol for a right-handed coordinate system
Matrix representation of the Levi Civita symbol and ...
Corresponding representation of the Levi-Civita symbol for a left-handed coordinate system

As can be seen in the adjacent figure, only 6 of the total of 27 components are non-zero:

Or as a rule of thumb: 123123 Now +1 if you read from left to right, and -1 if you read from right to left. In this example one also recognizes an invariance under cyclic permutation of the indices, which only applies if n is odd - if this is not the case, a cyclic permutation of the indices is accompanied by a sign change.

The following numerical example demonstrates the representation as a determinant, which in the three-dimensional case can also be expressed by the late product :

The Levi-Civita symbol with three indices proves useful in vector calculus to write the components of the cross product of two vectors. It applies

Einstein's sums convention is often used for such calculations, i.e. one omits the sum symbols and stipulates that indices that appear twice in products are always added automatically:

If the -th unit vector, then this equation can also be noted as:

The following applies to the late product


In this connection, the property of the Levi-Civita symbol as a component of a volume form becomes clear, because the spatial product is equal to the volume of the spatial spanned by the three vectors .

The relationship between the Levi-Civita symbol or the Epsilon tensor and the Kronecker delta is obtained

From this it follows (again with summation convention)

These relationships are helpful in inferring identities for the cross product .

Furthermore, the epsilon tensor assigns a vector a skew-symmetric matrix with to. The cross product can thus be expressed as a matrix product . In mathematics, this assignment is known as the Hodge star operator . One example is the assignment of the magnetic field vector to the corresponding components in the electromagnetic field strength tensor . Such an assignment is also common for other axial vectors , such as the angular momentum vector .

theory of relativity

In the theory of relativity , a distinction must be made between co- and contravariant components of the epsilon tensor. In the following, the signature of the metric tensor in the four-dimensional Minkowski space is defined as (1, -1, -1, -1). The indices should have values ​​from 0 to 3. Furthermore, let us define for the quadruple contravariant component . Different authors use different conventions for the signs in metric and epsilon tensor. As usual, indices are moved with the metric tensor. Then we get, for example, the four-fold covariant component .

The epsilon tensor remains invariant under an actual Lorentz transformation :

This follows directly from the fact that the determinant of equals 1. The epsilon tensor can be used to define the dual electromagnetic field strength tensor , with the help of which in turn the homogeneous Maxwell equations can be compactly noted.

One application of the two-stage epsilon tensor in relativity theory arises when the Minkowski space on the vector space of hermitian mapping matrices: . Here are for the Pauli matrices and the negative identity matrix . The allocation of tensors then takes place accordingly. The metric tensor is imaged onto the product of two epsilon tensors . In this formalism objects with an index spinors , and the epsilon tensor plays on the translation of cost-in contravariant components the same role as the metric tensor in ordinary Minkowski space: . This formalism is known as the Van der Waerden notation . The signature (-1,1,1,1) is usually chosen for the metric. The definition applies to the epsilon tensor .

Quantum mechanics

In quantum mechanics , the Levi-Civita symbol is used in the formulation of angular momentum algebra. In mathematical terms, the symbol agrees with the structural constants of the Lie algebras . The following example illustrates the application of the Levi Civita symbol in this context. The Lie algebra can be used as the subalgebra of the skew-symmetric matrices in , that is the real , matrices shown are. The generators (a basis) is given by the matrices , with the components . The commutators of the generators are then .

Individual evidence

  1. Éric Gourgoulhon : The 3 + 1 Formalism in General Relativity. Springer, 2012, ISBN 978-3-642-24524-4 .
  2. ^ John David Jackson: Classical Electrodynamics . 3. Edition. John Wiley & Sons, 1999, ISBN 0-471-30932-X .
  3. ^ Julius Wess, Jonathan Bagger: Supersymmetry and Supergravity . Princeton University Press, 1983, ISBN 9971-950-67-7 .