What is generating functional and vacuum-to-vacuum boundary conditions in QFT?

In summary, the conversation discusses the generating functional Z[J] and the vacuum-to-vacuum boundary conditions in the book QFT - L. H. Ryder. The generating functional is given by Z[J]=<0|0> and can be used to obtain vacuum expectation values for operators through functional derivatives. The vacuum-to-vacuum boundary conditions are expressed as \psi(t_i) = \psi_i and \psi(t_f) = \psi_f. The operators mentioned are the field operators and the normalization for the generating functional is Z[0]=1. The conversation concludes that the vacuum is still the vacuum even without a source.
  • #1
VietBrian
4
0
Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<
 
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  • #2
I don't know that book, but if it is an introduction to QFT, it should definitely explain those terms. The generating functional is given by

[itex]Z \left[ J \right]=<0|0>[/itex],

which can be expressed in terms of a path integral. It is called "generating" functional because you can apply functional derivatives with respect to J on it in order to gain vacuum expectation values for operators. What they mean by vacuum-to-vacuum boundary conditions could depend on the context, but it is probably the normalization

[itex]Z \left[ 0 \right]=0[/itex].
 
  • #3
Thank you very much! ^^

I had carefully read the book again. The vacuum-to-vacuum boundary conditions turned out to be [itex] \psi(t_i) = \psi_i [/itex] and [itex] \psi(t_f) = \psi_f [/itex].

:D And, are the operators you talk about above the field operators?
[itex] \dfrac{\delta Z[J]}{\delta J(t_1)\ldots \delta J(t_n)} = i^n \bra 0 \lvert T(q(t_1)\ldots q(t_n)) \rvert 0 \ket [/itex]
 
  • #4
Ah, I see.

Yes, that's exactly what I meant!
 
  • #5
Oh, yeah, it's now clearer for me ^^ Thank you!
 
  • #6
[itex]Z \left[ 0 \right]=0[/itex]

It's meant to be
[itex]Z \left[ 0 \right]=1[/itex],
sorry!
 
  • #7
Does it mean vacuum is still vacuum if there is no source ?
 
  • #8
I guess you could put it like that.
 

1. What is generating functional in QFT?

The generating functional in quantum field theory (QFT) is a mathematical tool that allows us to calculate the probabilities of different particle interactions and processes. It is a functional of the external sources in the theory and is used to derive the Green's functions and correlation functions.

2. How is generating functional related to Feynman diagrams?

The generating functional is related to Feynman diagrams as it provides a way to organize and calculate the amplitudes associated with each Feynman diagram. The functional is used to sum over all possible Feynman diagrams and obtain the total amplitude for a given process.

3. What are vacuum-to-vacuum boundary conditions in QFT?

Vacuum-to-vacuum boundary conditions are a set of conditions that describe the behavior of the fields in the vacuum state in QFT. These conditions ensure that the vacuum state remains unchanged under time evolution and that the vacuum energy is zero.

4. Why are vacuum-to-vacuum boundary conditions important in QFT?

Vacuum-to-vacuum boundary conditions are important in QFT because they provide a framework for calculating physical observables. They help us understand the behavior of the fields in the vacuum state and how they evolve over time. These conditions also play a crucial role in renormalization and the cancellation of divergences in QFT calculations.

5. How are vacuum-to-vacuum boundary conditions implemented in QFT calculations?

Vacuum-to-vacuum boundary conditions are typically implemented in QFT calculations by using the path integral formalism. This involves summing over all possible field configurations that satisfy the boundary conditions and calculating the amplitude for each configuration. The total amplitude is then obtained by integrating over all possible field configurations. Alternatively, these conditions can also be imposed through the use of counterterms in perturbative calculations.

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