Your Servant

Every mathematician develops his own preferences for notation. This is necessary because there are often (I’m tempted to say “usually”) many notations for the same concept.

Consider, for instance, something as fundamental as a vector. A statistician will probably indicate that some quantity is a vector by writing its symbol in bold-face type

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A physicist is more likely to indicate its vector nature by placing a small arrow (or half-arrow) above the symbol

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If you’re working in quantum mechanics, it’s not uncommon to use the “bra-ket” notation, in which a vector is indicated by a vertical bar, followed by the symbol, followed by a right angle bracket

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Alternatively, one can denote a vector by placing a subscript on the symbol

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The subscript is usually meant to denote which component of a vector is referred to. An n-dimensional vector has n components in some reference frame, e.g., a position vector may have x-, y-, and z-components which are denoted , , .

More savvy readers may object that is therefore not a vector — it’s one of the components of a vector, depending on the value of the index j. You’d be right … unless …

There’s yet another notation for vectors, not for its components, which happens to be my favorite. It’s to write a vector with a subscript

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but in this case the index marker “j” is not an ordinary index. It doesn’t represent some number specifying which component of the vector is referred to. Instead it’s an “abstract index,” which is merely a marker to indicate that the given symbol x refers to a vector. It’s really no different from putting a small arrow above the symbol to indicate its vector nature, but the “vector indicator symbol” is a subscript below rather than an arrow above. I learned this notation from Penrose & Rindler’s Spinors and space-time, which is possibly my favorite book.

One can even indicate vectors with a superscript rather than subscript

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Note that I’ve used a Greek letter for the index. This is common notation in tensor analysis, which is used in advanced physics (especially general relativity). If the superscript is an abstract index then this indicates a genuine vector, but if it’s a “specific index” then it indicates one of the components of the vector (depending on the value of the index). When I first learned tensor analysis (I kinda cut my teeth on it), it took some getting used to that the superscript was not an exponent, i.e., that referred to the 2nd component of the vector x rather than the square of some number x.

Just as there are many notations for vectors, so too there are multiple notations for simple vector operations like taking the inner product, or dot product, of two vectors. A statistician would indicate this simply by writing the vectors (in bold-face type) next to each other

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The physicist, however, would more likely write the symbols with little arrows and a dot between them

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unless he was doing quantum mechanics, in which case one vector would be a mirror-image “bra” vector, written next ot the other “ket” vector

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With index notation we can indicate the dot product as a simple sum

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However, there’s a beautiful shortcut notation known as the Einstein summation convention, in which if any index is repeated then it is to be summed over all possible values. The dot product is then

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which is really just a shorthand way of writing the sum indicated in the preceding equation.

When the index is an abstract index, the inner product is written in the same way

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This does not, however, indicate a sum — unless we know what the components are because we’ve chosen some basis for our vector space. Instead, in abstract-index notation a repeated index indicates the operation of transvection, which for vectors is the inner product. If all that seems unnecessarily confusing, rest assured there are situations in which the distinction is valuable.

When superscripts are used, there are generally two “versions” of each vector. One has a superscript and represents an “ordinary” vector, sometimes called a contravariant vector. The other has a subscript and is usually called a covariant vector, but might more rigorously be referred to as a dual vector. Strictly speaking, one is only allowed to take the inner product when one vector is contravariant and the other covariant (when one is a ordinary vector and the other is a dual vector), and it’s written as

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This can represent a sum of components if the index is specific, or the abstract operation of transvection if the index is abstract.

The “bra-ket” and subscript-superscript notations may seem unnecessarily complicated because there are two “forms” for each vector — “bra” and “ket”, or “covariant” (dual) and “contravariant” (ordinary). It’s usually a good idea to forego this complication except in circumstances where the distinction is important. In general relativity, for instance, the contravariant and covariant vectors will have different values for their components even with the same coordinate reference frame. In quantum mechanics the “components” of a bra vector with respect to some reference frame will be the complex conjugates of the components of its associated ket vector. This reflects the fact that in such circumstances the vector spaces themselves have added complications.

For instance in quantum mechanics, the inner product is no longer a symmetric operation, so the inner product of x and y is not the same as the inner product of y and x — they’re complex conjugates of each other. This ensures that the inner product of any vector with itself will always be real-valued and positive, so it makes a suitable norm for the vector space.

In general relativity, all vectors are usually real-valued so the complex-number thing doesn’t enter, but the vector space itself has a structure such that the inner product of a vector with itself can actually be negative. In the usual (or at least, my preferred) convention, vectors with positive “norm” are time-like vectors while vectors with negative norm are space-like. There are even vectors which are not themselves zero, but give zero “norm”, called null vectors.

I guess the point of all this is that there’s a reason for each of these notations. Each has its own usefulness, and each has its own field in which it tends to be used most often. At least, that’s the case when you publish — you should use the notation common in the field so as to communicate most clearly with your intended readers. But in your own notebook, it’s the custom to use whatever notation you damn well please. At least, that’s what I do. As I said, my favorite is the abstract-index notation but the fact is that at various times and under various circumstances I use them all.

Students often tend to adhere slavishly to the notation introduced in their textbooks or by their teachers. When I was young I did that myself, until I received the best mathematical advice I ever heard:

notation should be your servant, not your master.

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