[NOTES/ME-14002]-Levi-Civita $\epsilon$ and Kronecker $\delta$ symbols as tensors

1. Properties of rotation matrix

The matrix \(R\) is an orthogonal matrix, i.e. the transpose of the matrix is its inverse \begin{equation} R \, R^T = I = R^T\ R \end{equation} Writing \(jk\) element of the matrix equation \(R\,R^T=I\), we get \begin{eqnarray} (R \, R^T)_{jk} &=& \delta_{jk}\Longrightarrow (R)_{j\ell}(R^T)_{\ell k} = \delta_{jk}\\ \therefore \qquad R_{j\ell} \,R_{k\ell} &=& \delta_{jk}. \end{eqnarray} Similarly, starting with \(jk\) element of the matrix equation \(R^T\, R=I\), we will get \begin{eqnarray} (R^T\, R)_{jk} &=& \delta_{jk} \Longrightarrow (R^T)_{j\ell}(R)_{\ell k} = \delta_{jk}\\ \therefore \qquad R_{\ell j }\,R_{\ell k} &=& \delta_{jk}. \end{eqnarray} \

2.Proper and Improper rotations

Recall for an orthogonal matrix \(O\), the orthogonality condition \(O^TO=I\) implies \(\det O=\pm1\). A \(3\times3\) real orthogonal matrix \(R\) with \(\det R=1\) represents a rotation which can be physically implemented on the coordinate axes. These rotations take right (left) handed coordinate system to another right (left) handed coordinate system. The \(3\times3\) real orthogonal matrices with \(\det R=-1\) represent improper rotation. These transform a right handed coordinate system to a left handed coordinate system and vice versa. Such a transformation cannot be implemented by a physical rotation.

3.Kronecker delta as tensor of rank 2

A tensor is defined by its transformation properties under rotation of coordinate axes. {\tt The Kronecker delta symbol is defined, in all frames, by } \begin{equation} \delta_{ij}=\begin{cases}1&\text{ if }i =j\\0 & \text{ if } i \ne j\end{cases} \end{equation} . We will now show that it transforms like a second rank tensor. Let \(K\) and \(K{'}\) be two sets of axis related by a rotation: \begin{equation}\label{EQ01} K \stackrel{R}{\longrightarrow} K{'}. \end{equation} We wish to prove that if we transform Kronecker delta like a second rank tensor, i.e. we assume \begin{eqnarray}\label{EQ02} (\delta{'})_{jk} = R_{j\ell} R_{km} \delta_{\ell m} \end{eqnarray} Then \((\delta{'})_{jk} =(\delta)_{jk} \) and we say that the Kronecker delta is an invariant second rank tensor.

Proof of EQ08

We start with its left hand side \begin{eqnarray} \text{L.H.S.of EQ08} &=& R_{j\ell} R_{km} \delta_{\ell m}\\ &=& R_{j \ell} R_{k\ell} = (R R^T)_{jk}\\ &=& \delta_{jk}. \end{eqnarray} Now we use \(\delta_{\ell m}=0\) if \(\ell\ne m\) and also \(\delta_{\ell m}=1\) if \(\ell = m\), and in the first step above we replace \(m \to \ell \) everywhere and drop \(\delta_{\ell m}\). Thus we have proved that under rotations, the components of Kronecker delta symbol do not change, i.e. \begin{eqnarray}\label{EQ03} \delta{'}_{jk}&=&R_{j\ell} R_{km} \delta_{\ell m} = \delta_{jk}. \end{eqnarray} We, therefore, have the result that \(\delta_{ij}\) is a numerical tensor of rank two which remains invariant under rotations. \noindent

Levi-Civita Symbol as a pseudo tensor of rank three The Levi-Civita symbol is defined,in all frames, as totally antisymmetric object of rank three with \(\epsilon_{123}=1\). We will now show that this definition implies that it is pseudo tensor of rank three. We now check the transformation property of Levi-Civita symbol under rotations. Consider \begin{equation} (\epsilon{'})_{ijk}= R_{i\ell} R_{jm} R_{kn}\epsilon_{\ell mn}. \end{equation} Recall that for any \(3\times 3\) matrix \(R\) the Levi Civita symbol \(\epsilon\) has the property that \begin{equation} R_{i\ell} R_{jm} R_{kn}\epsilon_{\ell mn}= \det R \epsilon_{ijk} \end{equation} Therefore we get \begin{equation} \epsilon{'}_{ijk} = (\det R)\, \epsilon_{ijk} \end{equation} This shows that the Levi Civita symbol \(\epsilon_{ijk}\) is a third rank (pseudo) tensor  which is invariant under proper rotations \(\det R=+1\).