Are infinitesimal field variations in QFT similar to coordinate components?

In summary, the conversation discusses the statement ##\frac{\delta \phi(x)}{\delta \phi(y)}=\delta (x-y)## in the context of QFT and its meaning. The speaker understands the proof but is unsure of its significance. The expert summarizes that the formula is a basic one used for manipulating other formulas and that its meaning is dependent on the context. The speaker agrees and adds that the formula shows that two fields at different points are considered independent, similar to coordinate components.
  • #1
Mishra
55
1
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
 
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  • #2
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
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?
 
  • #4
Mishra said:
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.
 
  • #5
A. Neumaier said:
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!
 
  • #6
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.
 

1. What is an infinitesimal field variation?

An infinitesimal field variation refers to a small change in a continuous field, such as temperature, pressure, or electric field, over a very small distance or time interval. It is a fundamental concept in calculus and is used to describe the behavior of systems that change continuously.

2. How is infinitesimal field variation different from finite field variation?

Infinitesimal field variation is characterized by an extremely small change in the field, whereas finite field variation involves a noticeable and measurable change. Infinitesimal variations are used in mathematical models to analyze the behavior of systems at a microscopic level, while finite variations are used to analyze macroscopic changes in the field.

3. What is the importance of infinitesimal field variation in scientific research?

Infinitesimal field variation is crucial in understanding the behavior of complex systems, as it allows scientists to model and predict the behavior of these systems at a microscopic level. It is also used in the development of mathematical models and equations that accurately describe physical phenomena.

4. How is infinitesimal field variation related to the concept of limits?

The concept of infinitesimal field variation is closely related to the concept of limits in calculus. As the distance or time interval approaches zero, the infinitesimal field variation becomes more accurate and can be used to calculate the behavior of the system at that point. Limits are used to define and analyze infinitesimal variations and their effects on continuous fields.

5. Can infinitesimal field variation be observed in real-world systems?

While infinitesimal field variations cannot be directly observed, they are used in mathematical models and equations that accurately describe the behavior of real-world systems. By using these models and equations, scientists can make predictions and understand the behavior of complex systems in our world.

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