# Product of Two Levi-Civita Symbols in N-dimensions

• akoohpaee
In summary, the Levi-Civita symbol is defined in N-dimensions as either +1, -1, or 0 depending on the permutation of the indices. The product of two Levi-Civita symbols without dummy indices can be expressed as n! multiplied by a Kronecker delta function. The proof for this expression can be found by considering the antisymmetric behavior of the array K_{ijk} defined as \varepsilon_{ijk}. This array behaves as a tensor under coordinate transformations and is equal to its contravariant counterpart. There is no Einstein summation notation used in this statement.
akoohpaee
Dear You,

In N-dimensions Levi-Civita symbol is defined as:

\begin{align}
\varepsilon_{ijkl\dots}=
\begin{cases}
+1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\
-1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\
0 & \mbox{otherwise}
\end{cases}
\end{align}

I found the following expression for the product of two Levi-Civita symbols when there are no dummy indices (i_1,...,i_n,j_1,...,j_n are in {1,...,n}):

\begin{align}& \varepsilon_{i_1 \dots i_n} \varepsilon^{j_1 \dots j_n} = n! \delta^{j_1}_{[ i_1} \dots \delta^{j_n}_{i_n ]} &&\\& \end{align}

But I could not find its proof through literature and also I was failed to prove it!

Best Regards,
Ali

When $N=2$ we have for example $\varepsilon_{12}=+1$. Now, you didn't define $\varepsilon^{12}$ for us. Also, is there some Einstein summation convention in effect here?

Firstly, notice that the RHS is antisymmetric in exchange of any pair of the i's and any pair of the j's. So all you have to check is that you get the right answer for $i_1=1, i_2=2, \ldots$ and the same for the j's, which is straightforward.

Hi,

Assume that the array $K_{ijk}$ (i,j,k are in {1,2,3}) is defined in such a way that $K_{ijk}=\varepsilon_{ijk}$. It can be shown that, this array behave as a tensor under covariant and contravariant coordinate transformations. In other words:

\begin{align}
\varepsilon_{ijk}=\varepsilon^{ijk}
\end{align}

Also this is the case for N-dimensional:

\begin{align}
\varepsilon_{ijklm\dots}=\varepsilon^{ijklm\dots}
\end{align}

And regarding to the second part of your question: There is no Einstein summation notation in effect here. In fact there is no summation in this statement.

Ali

Dear Ali,

Thank you for your inquiry. The expression for the product of two Levi-Civita symbols in N-dimensions is indeed a well-known result in mathematics and physics. It is known as the "Generalized Kronecker Delta" and is often used in tensor calculus and index notation.

To prove this expression, we can start by considering the definition of the Levi-Civita symbol in N-dimensions:

\begin{align}
\varepsilon_{ijkl\dots}=
\begin{cases}
+1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\
-1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\
0 & \mbox{otherwise}
\end{cases}
\end{align}

Now, let's consider the product of two Levi-Civita symbols, with no dummy indices (i_1,...,i_n,j_1,...,j_n are in {1,...,n}):

\begin{align}
\varepsilon_{i_1 \dots i_n} \varepsilon^{j_1 \dots j_n} &= \varepsilon_{i_1 \dots i_n} \varepsilon_{j_1 \dots j_n} && \text{since } \varepsilon^{j_1 \dots j_n} = \varepsilon_{j_1 \dots j_n} \text{ in N-dimensions}\\
&= \varepsilon_{i_1 \dots i_n} \varepsilon_{j_1 \dots j_n} \delta^{j_1}_{[ i_1} \dots \delta^{j_n}_{i_n ]} && \text{since } \delta^{j_1}_{[ i_1} \dots \delta^{j_n}_{i_n ]} = 1 \text{ for all } i_1,...,i_n,j_1,...,j_n \\
&= \varepsilon_{i_1 \dots i_n} \varepsilon_{j_1 \dots j_n} \varepsilon^{j_1 \dots j_n} \delta^{j_1

## 1. What is the significance of the product of two Levi-Civita symbols in N-dimensions?

The product of two Levi-Civita symbols in N-dimensions is significant because it helps to define the determinant of a matrix in N-dimensional space. This is important in various mathematical and physical applications, such as calculating volumes and areas in higher dimensions.

## 2. How is the product of two Levi-Civita symbols calculated?

The product of two Levi-Civita symbols in N-dimensions is calculated by multiplying the two symbols together and then summing over all possible permutations of the indices. This can be represented by the use of the Kronecker delta symbol.

## 3. Can the product of two Levi-Civita symbols be simplified?

Yes, the product of two Levi-Civita symbols can be simplified using various identities and properties of the symbols. For example, in three dimensions, the product of two Levi-Civita symbols results in the epsilon tensor, which can be simplified using the Levi-Civita identity.

## 4. What is the geometric interpretation of the product of two Levi-Civita symbols?

The product of two Levi-Civita symbols in N-dimensions has a geometric interpretation as the determinant of a matrix. This can be visualized as the volume of a parallelepiped in N-dimensional space, where the symbols represent the orientation and direction of the sides of the parallelepiped.

## 5. How is the product of two Levi-Civita symbols used in physics?

The product of two Levi-Civita symbols is used in physics to calculate quantities such as angular momentum and electromagnetic fields in N-dimensional space. It is also useful in theories such as general relativity and quantum mechanics, where higher dimensions are often considered.

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