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Expanding field derivative

  1. Jul 28, 2015 #1
    I just came across this in a textbook: ## (\partial_{\mu}\phi)^2 = (\partial_{\mu}\phi)(\partial^{\mu}\phi) ##

    Can someone explain why this makes sense? Thanks.
  2. jcsd
  3. Jul 28, 2015 #2


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    The right-hand side is a short-hand way of writing
    $$\sum_{\mu=0}^d (\partial_{\mu}\phi)(\partial^{\mu}\phi).$$
    More generally, this shorthand is called the Einstein summation convention, commonly used whenever vectors and tensors appear, where a sum on a repeated index is assumed from the context of the expression. The left-hand side is a further shorthand, where if the quantity is supposed to be a scalar from the context, then it is assumed that the index is summed over with an appropriate metric tensor to raise one of the indices in the square. This is a shorthand that is more likely to be confusing and is used much less frequently than the Einstein summation convention itself.
  4. Jul 28, 2015 #3


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    As you should know, ## \partial_\mu \phi## is a covariant first rank tensor, so you may name it ##S_\mu##. Now considering ##S^2\equiv S_\mu S^\mu##, will give you what you want.
  5. Jul 29, 2015 #4


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    Specifically, that is the Einstein summation convention for tensors, a notational convention: The same index, both as a subscript and a superscript is interpreted as a summation index.
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