Domenico Fiorenza Physics notation

Notation

We introduce the Einstein notation, often used by physicist. It is given a vector space, for example $\mathbb{R}^n$ and a real metric $g= \{ g_{ij} \}_{i,j=1, \dots, n}$ . Remark that a metric can be assigned once a real matrix $n \times n$ is given. In physics, it is written:

x_{\mu}y^{\mu} = \sum_{i,j = 1}^{n} x_i g_{ij} y_j

equivalent writings are:

x_{\mu}y^{\mu} = x^{\mu}y_{\mu} = g^{\mu \nu } x_{\mu}y_{\mu} = g_{\mu \nu }x^{\mu}y^{\mu},

then:

g^{\mu \nu} x_{\mu} = x^{\nu}

We say that repeated indeces are “saturated”. Notice that for being saturated, the repeated indeces cannot be both up or down. This notation is particolary usefull for writing tensors. Let $V$ be a vector space over a field $\mathbb{K}$ with an inner product $g$ non degenerate. Recall that a tensor $T$ is a multilinear application:

T : V^{\otimes^h } \to \mathbb{K}

v_1 \otimes \dots \otimes v_h \to T(v_1 \otimes \dots \otimes v_h).

In index notation, $T$ is written as $T= T^{l_1 \dots l_h}$ with each $l_i$ taking values in $\{ 1,2, \dots Dim(V)\}$ . By saturating with $v_1, \dots, v_i$ vectors, we obtain a scalar in $\mathbb{K}$ . Furthermore, moving upside down an index, i.e. multiplying for the metric $g$ , has now the obvious meaning of substituting a term $V$ in the tensor product with the dual, since a non degenerate inner product makes the vector space isomorphic to its dual.

Created on November 24, 2010 at 14:10:31 by giuseppe_malavolta