Quantum field theory and generating functional

In summary, the conversation discusses parts a) and b) of an attached question. It is mentioned that for part a), the integral vanishes due to a complete derivative when taking the limit of epsilon approaching 0. For part b), it is implied that epsilon is also taken to 0, but it is not explicitly stated. It is suggested that the epsilon is only needed for convergence, but for other relations like the one in part b), it is understood to be 0.
  • #1
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Homework Statement



Hi,

I am looking at the attached question, parts a) and b).
generatingfunctional.png

Homework Equations


The Attempt at a Solution



so for part a) it vanishes because in the ##lim \epsilon \to 0 ## we have a complete derivative:
## \int d\phi \frac{d}{d\phi} (Z[J]) ##

for part b) we attain part a) which we have just shown vanishes by taking the derivatives inside the integral, however, for this to be a complete derivative again we also need that ##lim \epsilon \to 0 ## don't we? but part b) imposes no condition on ##\epsilon## so isn't it asking us to show that this holds for any ##\epsilon## how would we do this instead?
 

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  • #2
bump
many thanks
 
  • #3
binbagsss said:

Homework Statement


for part b) we attain part a) which we have just shown vanishes by taking the derivatives inside the integral, however, for this to be a complete derivative again we also need that ##lim \epsilon \to 0 ## don't we? but part b) imposes no condition on ##\epsilon## so isn't it asking us to show that this holds for any ##\epsilon## how would we do this instead?

I agree that they are not being very clear. But I am sure it i implicit that epsilon is taken to zero. The epsilon is only needed to make the integral convergent so if one actually wants to integrate it, the epsilon must be kept and then set to zero after the integration is carried out. But in all other relations like the one given in b), it is understood that the epsilon is taken to be zero. To summarize, the epsilon is alway understood to be zero except when one is actually carrying out integrations.
 

What is quantum field theory?

Quantum field theory is a theoretical framework used to describe the behavior of particles and their interactions in the quantum realm. It combines the principles of quantum mechanics and special relativity to explain the dynamics of subatomic particles.

What is a generating functional in quantum field theory?

In quantum field theory, a generating functional is a mathematical tool used to calculate probabilities for various physical processes involving particles. It is a function of the fields in the theory, and its value can be interpreted as the amplitude for a particular process to occur.

How is quantum field theory used in physics?

Quantum field theory is used in many areas of physics, including particle physics, condensed matter physics, and cosmology. It provides a mathematical framework for understanding the behavior of particles and their interactions, and has been successful in making predictions and explaining various physical phenomena.

What is the role of symmetries in quantum field theory?

Symmetries play a crucial role in quantum field theory. They are used to classify particles and their interactions, and they can also be used to simplify mathematical calculations. Many fundamental principles of quantum field theory, such as the conservation of energy and momentum, are based on symmetries.

What are the challenges in studying quantum field theory?

Quantum field theory is a complex and mathematically demanding subject, making it challenging to study and understand. It also poses conceptual challenges, such as the interpretation of quantum fields and the role of virtual particles. Additionally, many open questions and problems remain in the field, making it an active area of research.

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