Einstein Summation Convention: One as Upper, One as Lower?

Question

My question refers to the often specified rule defining Einstein Summation Notation in that summation is implied when an index is repeated twice in a single term, once as upper index and once as lower index.

Thus, a term that appears such as: $$ A^i B_i\quad=\quad\sum_i A^i B_i\quad=\quad A^1B_1+A^2B_2+A^3B_3 $$

According to this rule, the following repeated index term would not be summed at all. $$ A^iB^i $$ Because the repeated index does not appear as one upper and one lower in the term. Yet, I sometimes see various texts and other references invoke the Einstein Summation convention when such terms (both indexes upper or both indexes lower) exist.

Now, this aberrant use of Einstein Summation notation often appears in Math texts rather than Physics. For example, Chapter 1 of the Schaum's Outline Series on Tensor Calculus is named "The Einstein Summation Convention" and goes on to introduce the notation and never mentioning the upper and lower repeated index rule but explicitly gives an example of using repeated lower indexes: $$ a_1x_1+a_2x_2+a_3x_3+\cdots+a_nx_n = \sum_{i=1}^n a_ix_i $$

Usually I work with index notation in Physics subjects in my self-study of field theory and General Relativity and I don't recall ever running across examples of the summation convention that do not invoke the repeated index rule as one upper and one lower. But I am puzzled because in spite of this usage in Physics, I can't see any reason why one index must be upper and the other index lower. In other words, repeated index both as lower or both as upper does not seem to violate anything (for example, this more lax approach is used throughout the above cited Schaum's book.

My question: does the correct definition of Einstein Summation Convention demand that one index must be upper and the other repeated index must be lower. Or, is this merely style convention used in Physics (e.g. General Relativity).

Answer 1

In the 'strict' sense, you should only apply the summation convention to a pair of indices if one is raised and another is lowered.

For example, consider a vector $v$ and a dual vector $f$ (i.e. a map from vectors to numbers). Then one can compute $f(v)$, the number that results from $f$ acting on $v$. In components, this would be written as $f_i v^i$, since dual vectors have lower indices.

If, instead, you have two vectors $v$ and $w$, there is generally no way to combine them into a number, and the quantity $v^i w^i$ makes no sense. But if you have a metric $g_{ij}$, you can use it to turn $w$ from a vector into a dual vector, with new components $g_{ij} w^j$. Then you can act with this dual vector on $v$, giving $g_{ij} v^i w^j$. Note that all indices are paired correctly.

That being said, there are lots of exceptions:

• A lot of field theory texts and even GR texts will write $v^i w^i$, but you're supposed to remember it really means $g_{ij} v^i w^j$. When you do explicit computations, you have to put that factor in yourself.
• If you're working in a space with a simple metric (like Euclidean space, where $g_{ij}$ is the identity), texts might omit $g_{ij}$ because it doesn't "do anything". That is, the vector $w^j$ and corresponding dual vector $g_{ij} w^j$ always have the exact same components, so they might as well identify them.
• If the previous point is true, the author might choose to use index position to store some other kind of information, so the summation convention remains 'strict'. This happens more often in non-physics texts.

There are enough possible conventions that you should just check the front of the book every time.

Answer 2

There is nothing wrong by summing up indices when both indices are either up or down. It is just a matter of convention. However the meanings can be different if you are in a Relativistic theory.

When you sum one up and one down indices in Relativity it means you have a Lorentz invariant quantity because you are combining covariant and contravariant components in such a way that the combination does not change under Lorentz Transformations. This is a (nice) rule adopted by most authors and its importance is in the fact we can immediately identify invariant quantities. A few authors though use always down indices even for relativistic theories. For instance, Rubakov's Classical Theory of Fields.

If you are in Euclidean space, there is nothing to worry about. Normally people use all indices down.