# 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

${\ displaystyle \ varepsilon _ {ijk \ dots} = {\ begin {cases} +1, & {\ mbox {falls}} (i, j, k, \ dots) {\ mbox {an even permutation of}} ( 1,2,3, \ dots) {\ mbox {ist,}} \\ - 1, & {\ mbox {if}} (i, j, k, \ dots) {\ mbox {an odd permutation of}} (1,2,3, \ dots) {\ mbox {ist,}} \\ 0, & {\ mbox {if at least two indices are equal.}} \ End {cases}}}$

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 .

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

${\ displaystyle \ varepsilon _ {ijk} = {\ frac {ij} {1-2}} \ cdot {\ frac {ik} {1-3}} \ cdot {\ frac {jk} {2-3}} = - {\ frac {1} {2}} (ji) (kj) (ik) \ equiv (ji) (kj) (ik) \ mod 3}$

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 .