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

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Discussion Overview

The discussion revolves around the concepts of the generating functional Z[J] and vacuum-to-vacuum boundary conditions in quantum field theory (QFT). Participants seek clarification on these terms and their implications within the framework of QFT.

Discussion Character

  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant describes the generating functional Z[J] as related to vacuum expectation values for operators and mentions its expression in terms of a path integral.
  • Another participant suggests that vacuum-to-vacuum boundary conditions may relate to normalization, initially stating Z[0] = 0.
  • A later reply corrects the normalization to Z[0] = 1, indicating a potential misunderstanding in the initial claim.
  • Participants discuss the relationship between the generating functional and field operators, with one participant confirming that the operators referred to are indeed field operators.
  • There is a question raised about whether the vacuum remains unchanged in the absence of a source, to which another participant agrees with a tentative affirmation.

Areas of Agreement / Disagreement

Participants express some agreement on the definitions and implications of the generating functional and boundary conditions, but there are corrections and clarifications that indicate some uncertainty remains, particularly regarding normalization and the interpretation of vacuum states.

Contextual Notes

There are unresolved aspects regarding the definitions and implications of the generating functional and boundary conditions, particularly in terms of normalization and the conditions under which the vacuum state is considered unchanged.

VietBrian
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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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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].
 
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]
 
Ah, I see.

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

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

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