All Questions
69
questions
3
votes
3
answers
608
views
Path integral at large time
From the path integral of a QFT:
$$Z=\int D\phi e^{-S[\phi]}$$
What is a nice argument to say that when we study the theory at large time $T$, this behaves as:
$$ Z \to e^{-TE_0} $$
where $E_0$ is the ...
- 411
1
vote
0
answers
76
views
physical interpratation of partition function in Quantym field theory
Partition function in Statistical mechanics is given by
$$ Z = \sum_ne^{-\beta E_n} $$
For QFT, it is defined in terms of a path integral:
$$ Z = \int D\phi e^{-S[\phi]} $$
How can we see the relation ...
- 411
2
votes
1
answer
96
views
The definition on vacuum-vacuum amplitude with current in chapter of External Field Method of Weinberg's QFT
I'm reading Vol. 2 of Weinberg's QFT. As what I learnt from both P&S and Weinberg, the generating function is defined as
$$
Z[J] = \int \mathcal{D}\phi \exp(iS_{\text{F}}[\phi] + i\int d^4x\phi(x) ...
- 23
0
votes
1
answer
124
views
The definition of the path integral
I still have big conceptual questions about the path integral.
According to (24.6) of the book "QFT for the gifted amateur" from Lancaster & Blundell the path integral is equal to
$$Z =\...
- 9,778
3
votes
0
answers
103
views
What are exactly the loop correction to the potential? [duplicate]
I am struggling with the concept of quantum effective action, but first recall the definition : given a Wilsonian effective action $W[J]$ of our theory, the quantum effective action is just
$$\Gamma[\...
- 475
3
votes
2
answers
170
views
Making sense of stationary phase method for the path integral
I am trying to understand this paper/set of notes. I have already seen the following related question: Does the stationary phase approximation equal the tree-level term? but had some trouble following ...
- 3,350
4
votes
2
answers
425
views
Interpreting generating functional as sum of all diagrams
The generating functional is defined as:
$$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$
I know this object is used as a tool to generate ...
- 3,350
2
votes
1
answer
176
views
Examples of Path integral $\neq$ Partition function?
Are there any systems we know of whose partition function is not simply Wick rotation of the path integral? Does anyone know of any examples?
- 1,200
1
vote
0
answers
71
views
$(\mathcal{S}\mathcal{T})^3=\mathcal{S}^2=+1$ mistake in CFT big yellow book?
In Conformal Field Theory Philippe by Di Francesco, Pierre Mathieu David Sénéchal
Sec 10.l. Conformal Field Theory on the Torus
eq.10.9 says the modular transformation $\mathcal{T}$ and $\mathcal{S}$ ...
- 149
1
vote
1
answer
330
views
What is the gravitational path integral computing?
What is the gravitational path integral (which roughly goes like $\int [dg]e^{iS_{\text{EH}}[g]}$) computing?
Usually, path integrals arise from transition amplitudes such as these: $\lim_{T\to\infty}\...
- 742
4
votes
0
answers
60
views
Thermodynamic free energy of interacting system
This question concerns an interacting system's thermodynamic free energy $\Omega$. Generally speaking, The action $S$ for an interacting system has the following form:
\begin{equation}
S(\phi,\psi) = ...
- 2,038
2
votes
1
answer
106
views
External/Background Fields Meaning
(I'll work in the Euclidean for convenience)
In the path integral formulation of QFT given a field $\phi$, or a set of them if you want to, we have that the partition function is given by:
$$Z[J] = \...
- 1,611
1
vote
1
answer
74
views
Semiclassic limit of a QFT in Zinn-Justin
I am reading the Zinn-Justin book "Quantum Field Theory and Critical Phenomena" and i have come across a perplexing point.
Given the partition functional, in Euclidean QFT:
$$Z[J, \hbar] = \...
- 1,611
1
vote
0
answers
61
views
Is the Euclidean generating functional $Z_{E}[J]$ identified with original Minkowskian generating functional $Z[J]$?
In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to ...
- 81
3
votes
1
answer
278
views
Changing variable in path integral
Good evening,
I am learning about path integrals in QFT and I was wondering, can you simplify the path integral by shifting the fields? To make it more clear I will give you an example. Suppose that I ...
- 33