Product of Two Levi-Civita Symbols in N-dimensions

[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!

Can you please help me? Thanks!

Best Regards,
Ali

 

[g_edgar]

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?

 

[henry_m]

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.

 

[akoohpaee]

$# REPLY TO g_edgar$#

Hi,

Many thanks for your reply!

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.

Many thanks for your reply and your attention.

Ali

 

[blue_raver22]

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