Domenico Fiorenza Physics notation

Notation

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

x μy μ= i,j=1 nx ig ijy j x_{\mu}y^{\mu} = \sum_{i,j = 1}^{n} x_i g_{ij} y_j

equivalent writings are:

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

then:

g μνx μ=x ν 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 VV be a vector space over a field 𝕂\mathbb{K} with an inner product gg non degenerate. Recall that a tensor TT is a multilinear application:

T:V h𝕂 T : V^{\otimes^h } \to \mathbb{K}
v 1v hT(v 1v h). v_1 \otimes \dots \otimes v_h \to T(v_1 \otimes \dots \otimes v_h).

In index notation, TT is written as T=T l 1l hT= T^{l_1 \dots l_h} with each l il_i taking values in {1,2,Dim(V)}\{ 1,2, \dots Dim(V)\} . By saturating with v 1,,v iv_1, \dots, v_i vectors, we obtain a scalar in 𝕂\mathbb{K}. Furthermore, moving upside down an index, i.e. multiplying for the metric gg, has now the obvious meaning of substituting a term VV 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