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Silviu
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Hello! Does ##(x_\mu)^2## actually means ##x_\mu x^\mu## in Einstein notation?
So this means they are the same, but ##x_{\mu}^2## is bad notation?dextercioby said:Drop the book using the ##x_{\mu}^2##.
Although I agree that it is really a bastard notation using this for ##g_{\mu\nu}x^\mu x^\nu##, I would not categorically advice to drop such a book. You will find that many physics papers use a similar notation in the kinetic term for a field, i.e., ##(\partial_\mu\phi)^2##. There really is only one thing that can mean and still be meaningful. Of course, the understanding of this is predicated on first having learned it properly ... In Schwartz's QFT book, he places all indices down with the initial statement that it should be subtextual that one should be considered contravariant and the other covariant.dextercioby said:Drop the book using the ##x_{\mu}^2##.
Einstein notation, also known as Einstein summation convention, is a mathematical notation commonly used in physics to simplify and condense equations involving vectors, tensors, and matrices. It uses the summation symbol, ∑, to represent repeated indices in a product of terms.
In Einstein notation, ##x_\mu x^\mu## represents the sum of products of all the components of a vector ##x##, where ##\mu## is a repeated index. This is known as the inner product or dot product of a vector with itself and is equivalent to the square of the magnitude of the vector.
Yes, ##x_\mu x^\mu## is mathematically equivalent to ##(x_\mu)^2## in Einstein notation. This is because the repeated index in the product represents a summation over all the components of the vector, resulting in the square of the magnitude of the vector.
Einstein notation is useful in simplifying and condensing equations involving vectors, tensors, and matrices in physics. It allows for concise representation of mathematical expressions and makes it easier to perform calculations, especially in higher-dimensional spaces.
Yes, there are other notations that are similar to Einstein notation, such as the index-free notation and the Penrose graphical notation. These notations also use indices to represent repeated terms, but they have different rules and conventions compared to Einstein notation.