Are infinitesimal field variations in QFT similar to coordinate components?

[Mishra]

Hello,

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.VM

 

[A. Neumaier]

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.

 

[Mishra]

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?

 

[A. Neumaier]

What is there to actually understand about this results?

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.

 

[Mishra]

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.

Thanks!

 

[haushofer]

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.