All Questions
16
questions
4
votes
2
answers
425
views
Interpreting generating functional as sum of all diagrams
The generating functional is defined as:
$$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$
I know this object is used as a tool to generate ...
- 3,350
1
vote
2
answers
186
views
Why doesn't this diagram appear in the partition function in zero-dimensional QFT?
For the zero-dimensional QFT with action
$$S(\phi)=\frac{\alpha}{2}\phi^2+\frac{\lambda}{4!}\phi^4-J\phi,\tag{1}$$
we can perturbatively expand the partition function as
$$Z_\lambda(J)=\int_{-\infty}^{...
- 248
0
votes
1
answer
67
views
In Srednicki's quantum field theory, page 71. Why is $Z_1 = W_1$?
Srednicki defines here: https://arxiv.org/abs/hep-th/0409035 on p.71
a "Z1", which is $exp(iW_1)$ where $iW_1$ is the sum of all connected diagrams with sourceless ...
0
votes
0
answers
311
views
Free Energy vs. Partition Function in QFT
The partition function of QFT is defined as
$$Z=\int\mathcal{D}\varphi e^{iS[\varphi]}.$$
Now, it is a general fact that this formal path integral can be computed perturbatively as (sketchy)
$$Z=\sum_{...
- 854
7
votes
1
answer
383
views
Diagrammatic representations of generating functionals $Z[J]$, $W[J]$, and $\Gamma[\varphi]$
The book Boulevard of Broken Symmetries by Adriaan Schakel gives an excellent, if not exceedingly brief, overview of the path integral approach to perturbation theory. In particular, pages 47-58 give ...
2
votes
2
answers
675
views
Why do we need effective action $\Gamma$ given the connected generating functional $W$?
I have just learnt the path integral formalism in QFT, up to the point where we computed the generating functionals $\mathcal{Z}[J] := Z[J]/Z[0]$, $W[J]$, and $\Gamma[\varphi]$. Here $J(x)$ is the ...
3
votes
1
answer
186
views
Assigning proper powers of $i$ to vertices of Feynman diagram
I'm reading the chapter 9 "the path integral for interacting field theory" of the Srednicki's QFT book. The lagrangian we are dealing with here is given by
\begin{gather}
\mathcal{L} = \...
- 121
6
votes
1
answer
365
views
What is the physical meaning of $W[J]=\frac{\hbar}{i}\ln Z[J]$?
The quantity $Z[J]$ (which is the generating functional for all Green functions) physically represents the probability amplitude for a system to remain in the vacuum state. Can we find a similar ...
- 11.8k
0
votes
1
answer
531
views
Path integral zero dimensional QFT
We consider the following partition function$$
\mathcal{Z}[\lambda] = \int{dx \; \exp\left(-\frac{1}{2}x^2-\frac{\lambda}{4!}x^4\right)}
$$
Which is basically $\phi^4$ theory in 0+0 dimensions. The ...
- 2,035
1
vote
0
answers
431
views
Connected Diagrams [duplicate]
The generating functional for the connected part of the Green functions is defined
as
$$iW[j] = \log Z[j].$$
From this the four-point connected Green's function is then given by
$G_c(x_1,x_2,x_3,...
- 445
13
votes
2
answers
6k
views
Intuition behind Linked Cluster Theorem: connected vs. non-connected diagrams
Within statistical physics and quantum field theory, the linked cluster theorem is widely used to simplify things in the calculation of the partition function among other things.
My question has the ...
- 7,931
1
vote
0
answers
138
views
How to use the generating functional for connected functions to compute connected diagrams? [duplicate]
We have
$$W[J]\equiv\hbar i\log(Z[J])$$
How do we use it to compute the connected diagrams to some order in perturbative field theory?
I get that we need to take functional derivatives of ...
- 305
2
votes
0
answers
313
views
Feynman rules for a $0$-dimensional field theory
Consider the partition function $Z(\lambda)$ of the $0$-dimensional scalar $\phi^{4}$ theory
$$Z(\lambda)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}d\phi\ \exp\left(-\frac{1}{2}\phi^{2}-\frac{\...
- 3,203
4
votes
1
answer
465
views
Physical interpretations of the generating functions $Z[J]$ and $W[J]$ (or $E[J]$)
In quantum field theory, the generator of all Green's functions $Z[J]$ and that of the connected Green's functions $E[J]$ are related as $$Z[J]=\exp[-iE[J]]=\int D\phi\exp[i\int d^4x(\mathcal{L}(\phi)+...
- 26.8k
8
votes
4
answers
5k
views
Proof of Connected Diagrams
If $Z[J]$ is the generating functional for the path-integral, could any prove (or more reasonably, refer me to a proof) that $$W[J]\equiv\frac{\hbar}{i}\log\left(Z[J]\right)$$ "generates" only ...
- 2,024