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Infinitesimal field variation

  1. Jan 10, 2016 #1

    In the context of QFT, I do not understand the statement:

    ##\frac{\delta \phi(x)}{\delta \phi(y)}=\delta (x-y)##

    I understand the proof which arises from the definition of the functional derivative but I do not get its meaning. From what I see is generalizes ##\frac{\partial q_i}{\partial q_j}=\delta_{ij}## which I am not sure to understand either.

    Would somebody be kind enough to explain me ?

    I realize this is not a question concerning QFT only but it is where I have the better chance to find a good answer since it is fundamental in that field.

  2. jcsd
  3. Jan 10, 2016 #2

    A. Neumaier

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    Integrate both sides with an arbitrary smooth test function, and simplify both sides, and the functional derivative will make sense. In the finite-dimensional analogue, multiply by arbitrary constants ##c_i## and sum, to see the same.
  4. Jan 10, 2016 #3
    Hello and thank you for your answer.
    I understand de proof (at least for fields) of this results. What I do not understand is its meaning, I see a lot of discussion using this (interpretation) to talk about causality for example. What is there to actually understand about this results?
  5. Jan 10, 2016 #4

    A. Neumaier

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    Nothing more than what I said. It is just a basic formula useful to manipulate other formulas. If something else is to be understood it is in the context that you didn't give, and not in this formula.
  6. Jan 11, 2016 #5
  7. Jan 11, 2016 #6


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    As far as I could tell your formula tells you that two fields at different points are considered to be independent, just like two different coordinate components are independent.
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