# Infinitesimal field variation

1. Jan 10, 2016

### 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.

Best,
VM

2. Jan 10, 2016

### 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.

3. Jan 10, 2016

### Mishra

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?

4. Jan 10, 2016

### A. Neumaier

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.

5. Jan 11, 2016

### Mishra

Thanks!

6. Jan 11, 2016

### 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.