Levi Civita symbol

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 . ${\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}}$

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. ${\displaystyle n}$ ${\displaystyle i_{1}}$ ${\displaystyle i_{n}}$ ${\displaystyle n}$ ${\displaystyle \varepsilon _{i_{1}\dots i_{n}}=0}$ ${\displaystyle \varepsilon _{i_{1}\dots i_{n}}=+1}$ ${\displaystyle \varepsilon _{i_{1}\dots i_{n}}=-1}$ ${\displaystyle \varepsilon _{132}=-1}$

definition

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: ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n-1}$

• ${\displaystyle \varepsilon _{12\dots n}=1}$.
• Under interchange of two indices, it changes the sign : . ${\displaystyle \varepsilon _{ij\dots u\dots v\dots }=-\varepsilon _{ij\dots v\dots u\dots }}$

From the second property follows immediately: if two indices are equal, the value is zero: . ${\displaystyle \varepsilon _{ij\dots u\dots u\dots }=0}$

The definition is equivalent

${\displaystyle \varepsilon _{ijk\dots }={\begin{cases}+1,&{\mbox{falls }}(i,j,k,\dots ){\mbox{ eine gerade Permutation von }}(1,2,3,\dots ){\mbox{ ist,}}\\-1,&{\mbox{falls }}(i,j,k,\dots ){\mbox{ eine ungerade Permutation von }}(1,2,3,\dots ){\mbox{ ist,}}\\0,&{\mbox{wenn mindestens zwei Indizes gleich sind.}}\end{cases}}}$

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

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}=\prod _{1\leq p<q\leq n}{\frac {i_{p}-i_{q}}{p-q}}}$.

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). ${\displaystyle N=\{1,\dots ,n\}}$ ${\displaystyle n}$ ${\displaystyle \varepsilon :\{i|i:N\rightarrow N\}\rightarrow \{-1,0,+1\}\subset \mathbb {R} }$ ${\displaystyle \varepsilon (i)=0}$ ${\displaystyle i}$ ${\displaystyle \varepsilon (i)=\operatorname {sgn}(i)}$ ${\displaystyle i}$ ${\displaystyle i}$

Relationship with the determinant

The determinant of a - matrix can Civita symbol Levi-and with the summation convention will be written as follows: ${\displaystyle n\times n}$ ${\displaystyle A=\left(A_{ij}\right)}$

${\displaystyle \det A=\varepsilon _{j_{1}\dots j_{n}}A_{1j_{1}}\dots A_{nj_{n}}\;.}$

The relationship is more general

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}\det A=\varepsilon _{j_{1}\dots j_{n}}A_{i_{1}j_{1}}\dots A_{i_{n}j_{n}}}$.

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: ${\displaystyle A}$ ${\displaystyle E_{n}}$ ${\displaystyle A_{ij}}$ ${\displaystyle \delta _{ij}}$ ${\displaystyle \det E=1}$

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}=\varepsilon _{j_{1}\dots j_{n}}\delta _{i_{1}j_{1}}\dots \delta _{i_{n}j_{n}}={\begin{vmatrix}\delta _{i_{1}1}&\dots &\delta _{i_{1}n}\\\vdots &&\vdots \\\delta _{i_{n}1}&\dots &\delta _{i_{n}n}\end{vmatrix}}=\det \left({\begin{array}{ccc}-&e_{i_{1}}&-\\&\vdots &\\-&e_{i_{n}}&-\end{array}}\right)}$.

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 : ${\displaystyle \{e_{1},\dots ,e_{n}\}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\begin{pmatrix}x_{1}&x_{2}&\dots &x_{n}\end{pmatrix}}^{T}}$ ${\displaystyle {\begin{pmatrix}x_{i_{1}}&x_{i_{2}}&\dots &x_{i_{n}}\end{pmatrix}}^{T}}$

${\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}=\det \left((e_{i_{1}}\dots e_{i_{n}})^{T}\cdot (e_{j_{1}}\dots e_{j_{n}})\right)={\begin{vmatrix}\delta _{i_{1}j_{1}}&\dots &\delta _{i_{1}j_{n}}\\\vdots &&\vdots \\\delta _{i_{n}j_{1}}&\dots &\delta _{i_{n}j_{n}}\end{vmatrix}}}$.

Using Laplace's expansion theorem one obtains the following relationship if one tapered over the first indices of both tensors : ${\displaystyle k}$

${\displaystyle \varepsilon _{i_{1}\dots i_{k}i_{k+1}\dots i_{n}}\varepsilon _{i_{1}\dots i_{k}j_{k+1}\dots j_{n}}=k!{\begin{vmatrix}\delta _{i_{k+1}j_{k+1}}&\dots &\delta _{i_{k+1}j_{n}}\\\vdots &&\vdots \\\delta _{i_{n}j_{k+1}}&\dots &\delta _{i_{n}j_{n}}\end{vmatrix}}}$.

As an application of these formulas one obtains for the entries of the adjuncts of a matrix: ${\displaystyle n\times n}$

${\displaystyle \operatorname {adj} (A)_{ij}={\dfrac {1}{(n-1)!}}\varepsilon _{i\,i_{2}\dots i_{n}}\varepsilon _{j\,j_{2}\dots j_{n}}A_{j_{2}i_{2}}\dots A_{j_{n}i_{n}}}$.

Especially in three dimensions

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

${\displaystyle \varepsilon _{ijk}={\hat {e}}_{i}\cdot ({\hat {e}}_{j}\times {\hat {e}}_{k})=\det \left({\begin{array}{ccc}-&{\hat {e}}_{i}&-\\-&{\hat {e}}_{j}&-\\-&{\hat {e}}_{k}&-\end{array}}\right)=:\det A}$

${\displaystyle \varepsilon _{lmn}={\hat {e}}_{l}\cdot ({\hat {e}}_{m}\times {\hat {e}}_{n})=\det \left({\begin{array}{ccc}-&{\hat {e}}_{l}&-\\-&{\hat {e}}_{m}&-\\-&{\hat {e}}_{n}&-\end{array}}\right)=:\det B}$

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 :

${\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&=\det A\,\det B=\det A\,\det B^{T}=\det(A\cdot B^{T})\\&=\left|\left({\begin{array}{ccc}-&{\hat {e}}_{i}&-\\-&{\hat {e}}_{j}&-\\-&{\hat {e}}_{k}&-\end{array}}\right)\cdot \left({\begin{array}{ccc}\mid &\mid &\mid \\{\hat {e}}_{l}&{\hat {e}}_{m}&{\hat {e}}_{n}\\\mid &\mid &\mid \end{array}}\right)\right|=\left|{\begin{array}{ccc}{\hat {e}}_{i}\cdot {\hat {e}}_{l}&{\hat {e}}_{i}\cdot {\hat {e}}_{m}&{\hat {e}}_{i}\cdot {\hat {e}}_{n}\\{\hat {e}}_{j}\cdot {\hat {e}}_{l}&{\hat {e}}_{j}\cdot {\hat {e}}_{m}&{\hat {e}}_{j}\cdot {\hat {e}}_{n}\\{\hat {e}}_{k}\cdot {\hat {e}}_{l}&{\hat {e}}_{k}\cdot {\hat {e}}_{m}&{\hat {e}}_{k}\cdot {\hat {e}}_{n}\end{array}}\right|\end{aligned}}}$

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

${\displaystyle \varepsilon _{ijk}\varepsilon _{lmn}=\left|{\begin{array}{ccc}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\end{array}}\right|}$

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. ${\displaystyle n}$ ${\displaystyle R^{n}}$ ${\displaystyle (i_{1},\ldots ,i_{n})\in \{1,\ldots ,n\}^{n}}$ ${\displaystyle \varepsilon _{i_{1}...i_{n}}}$

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 ${\displaystyle R^{n}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \{e_{i},\dots ,e_{n}\}}$ ${\displaystyle e'_{i}=C_{ji}e_{j}}$ ${\displaystyle (\det C^{-1})\varepsilon _{i_{1}\dots i_{n}}}$ ${\displaystyle C=(C_{ij})}$ ${\displaystyle C^{-1}}$ ${\displaystyle g}$ ${\displaystyle \pm {\sqrt {\det g}}}$

${\displaystyle (\det g)\varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}={\begin{vmatrix}g_{i_{1}j_{1}}&\dots &g_{i_{1}j_{n}}\\\vdots &&\vdots \\g_{i_{n}j_{1}}&\dots &g_{i_{n}j_{n}}\end{vmatrix}}}$.

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) ${\displaystyle [\alpha ,\beta ,\gamma ,\cdots ]}$

${\displaystyle \epsilon _{\alpha ,\beta ,\gamma }={\sqrt {g}}[\alpha ,\beta ,\gamma ,\cdots ]}$

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. ${\displaystyle g=det(g_{\mu \nu })}$ ${\displaystyle {\vec {a}}\times {\vec {b}}}$

Applications

Vector calculation

For the three-dimensional case it results

${\displaystyle \varepsilon _{ijk}={\frac {i-j}{1-2}}\cdot {\frac {i-k}{1-3}}\cdot {\frac {j-k}{2-3}}=-{\frac {1}{2}}(j-i)(k-j)(i-k)\equiv (j-i)(k-j)(i-k)\mod 3}$

whereby . ${\displaystyle i,j,k\in \lbrace 1,2,3\rbrace }$

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: ${\displaystyle \varepsilon _{ijk}}$

${\displaystyle \varepsilon _{123}=\varepsilon _{312}=\varepsilon _{231}=1,}$

${\displaystyle \varepsilon _{321}=\varepsilon _{213}=\varepsilon _{132}=-1.}$

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 :

${\displaystyle {\begin{aligned}\varepsilon _{123}&={\vec {e_{1}}}\cdot ({\vec {e_{2}}}\times {\vec {e_{3}}})\\&={\begin{pmatrix}1\\0\\0\end{pmatrix}}\cdot \left({\begin{pmatrix}0\\1\\0\end{pmatrix}}\times {\begin{pmatrix}0\\0\\1\end{pmatrix}}\right)={\begin{pmatrix}1\\0\\0\end{pmatrix}}\cdot {\begin{pmatrix}1\\0\\0\end{pmatrix}}=1\end{aligned}}}$

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

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

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:

${\displaystyle ({\vec {a}}\times {\vec {b}})_{i}=\varepsilon _{ijk}a_{j}b_{k}\;.}$

If the -th unit vector, then this equation can also be noted as: ${\displaystyle {\vec {e_{i}}}}$ ${\displaystyle i}$

${\displaystyle {\vec {a}}\times {\vec {b}}=\varepsilon _{ijk}a_{j}b_{k}{\vec {e_{i}}}=\varepsilon _{ijk}a_{i}b_{j}{\vec {e_{k}}}}$

The following applies to the late product

${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}=\varepsilon _{ijk}a_{i}b_{j}c_{k}}$.

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

${\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\end{vmatrix}}\\&=\delta _{il}\delta _{jm}\delta _{kn}+\delta _{im}\delta _{jn}\delta _{kl}+\delta _{in}\delta _{jl}\delta _{km}-\delta _{im}\delta _{jl}\delta _{kn}-\delta _{il}\delta _{jn}\delta _{km}-\delta _{in}\delta _{jm}\delta _{kl}{\textrm {.}}\end{aligned}}}$

From this it follows (again with summation convention)

${\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{imn}&={\begin{vmatrix}\delta _{jm}&\delta _{jn}\\\delta _{km}&\delta _{kn}\end{vmatrix}}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\\\varepsilon _{ijk}\varepsilon _{ijn}&=2\delta _{kn}\\\varepsilon _{ijk}\varepsilon _{ijk}&=3!=6\end{aligned}}}$

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 . ${\displaystyle {\vec {a}}}$ ${\displaystyle A}$ ${\displaystyle A_{ij}=\varepsilon _{ijk}a_{k}}$ ${\displaystyle {\vec {a}}\times {\vec {b}}=-A\cdot {\vec {b}}}$

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 . ${\displaystyle \,\eta _{ij}}$ ${\displaystyle \varepsilon ^{0123}=1}$ ${\displaystyle \varepsilon _{0123}=\eta _{0\mu }\eta _{1\nu }\eta _{2\varrho }\eta _{3\sigma }\varepsilon ^{\mu \nu \varrho \sigma }=\det(\eta )=-1}$

The epsilon tensor remains invariant under an actual Lorentz transformation : ${\displaystyle \Lambda }$

${\displaystyle \varepsilon ^{\prime \mu \nu \varrho \sigma }=\Lambda _{\ \mu ^{\prime }}^{\mu }\Lambda _{\ \nu ^{\prime }}^{\nu }\Lambda _{\ \varrho ^{\prime }}^{\varrho }\Lambda _{\ \sigma ^{\prime }}^{\sigma }\varepsilon ^{\mu ^{\prime }\nu ^{\prime }\varrho ^{\prime }\sigma ^{\prime }}=\varepsilon ^{\mu \nu \varrho \sigma }}$

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. ${\displaystyle \Lambda }$ ${\displaystyle {\tilde {F}}^{\mu \nu }={\tfrac {1}{2}}\varepsilon ^{\mu \nu \varrho \sigma }F_{\varrho \sigma }}$ ${\displaystyle \partial _{\mu }{\tilde {F}}^{\mu \nu }=0}$

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 . ${\displaystyle 2\times 2}$ ${\displaystyle v_{\alpha {\dot {\alpha }}}=\sigma _{\alpha {\dot {\alpha }}}^{m}v_{m}}$ ${\displaystyle \,\sigma ^{m}}$ ${\displaystyle \,m=1,2,3}$ ${\displaystyle \,\sigma _{0}=-E_{2}}$ ${\displaystyle \sigma _{\alpha {\dot {\alpha }}}^{m}\sigma _{\beta {\dot {\beta }}}^{n}\eta _{mn}=-2\varepsilon _{\alpha \beta }\varepsilon _{{\dot {\alpha }}{\dot {\beta }}}}$ ${\displaystyle \,\psi ^{\alpha }}$ ${\displaystyle \,\eta _{mn}}$ ${\displaystyle \psi _{\alpha }=\varepsilon _{\alpha \beta }\psi ^{\beta }}$ ${\displaystyle \varepsilon ^{12}=\varepsilon _{21}=1}$

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 . ${\displaystyle {\mathfrak {so}}(3,\mathbb {R} )\cong {\mathfrak {su}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {so}}(3,\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{3\times 3}}$ ${\displaystyle 3\times 3}$ ${\displaystyle {\mathfrak {so}}(3,\mathbb {R} )}$ ${\displaystyle T_{i}\in \mathbb {R} ^{3\times 3}}$ ${\displaystyle i=1,2,3}$ ${\displaystyle (T_{i})_{jk}=-\varepsilon _{ijk}}$ ${\displaystyle [T_{i},T_{j}]=\varepsilon _{ijk}T_{k}}$

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 .