All Questions
Tagged with path-integralpartition-function
108 questions
2votes
3answers
132views
What justifies the choice of covariance $C$ in rigorous path integrals?
In physics the path integral is given, after a Wick rotation, by $$Z^{-1} \int e^{-S[\phi]/\hbar}d\phi. \tag{1}$$ where $d\phi$ is formally Lebesgue measure over fields. To make this rigorous one ...
- 4,500
0votes
1answer
51views
Differentiate inside path integral
I am learning about QFT and I was wondering if it makes sense to differentiate the partition function/ path integral with respect to one of the parameters in the Lagrangian (either renormalized or not)...
- 33
0votes
0answers
50views
Deriving the partition function for the quantum spherical $p$-spin model from first principles
In this paper, the authors start with a Hamiltonian of $N$ bosonic spins with $p$-body interactions, given by the sum over eq. (1.3): $$ \mathcal{H}=\sum_{i_{1}<\ldots<i_{p}}J_{i_{1}\ldots i_{p}}...
- 145
3votes
3answers
647views
Path integral at large time
From the path integral of a QFT: $$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} $$ where $E_0$ is the ...
1vote
0answers
81views
physical interpratation of partition function in Quantym field theory
Partition function in Statistical mechanics is given by $$ Z = \sum_ne^{-\beta E_n} $$ For QFT, it is defined in terms of a path integral: $$ Z = \int D\phi e^{-S[\phi]} $$ How can we see the relation ...
2votes
1answer
130views
The definition on vacuum-vacuum amplitude with current in chapter of External Field Method of Weinberg's QFT
I'm reading Vol. 2 of Weinberg's QFT. As what I learnt from both P&S and Weinberg, the generating function is defined as $$ Z[J] = \int \mathcal{D}\phi \exp(iS_{\text{F}}[\phi] + i\int d^4x\phi(x) ...
0votes
1answer
185views
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 $$Z =\...
- 10.4k
3votes
0answers
182views
What are exactly the loop correction to the potential? [duplicate]
I am struggling with the concept of quantum effective action, but first recall the definition : given a Wilsonian effective action $W[J]$ of our theory, the quantum effective action is just $$\Gamma[\...
- 571
4votes
2answers
886views
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 ...
- 4,500
4votes
2answers
543views
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 ...
- 4,500
2votes
1answer
250views
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?
- 1,552
1vote
0answers
165views
$(\mathcal{S}\mathcal{T})^3=\mathcal{S}^2=+1$ 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
0votes
1answer
143views
Partition function for fractional Brownian motion with $H < 1/2$
Recently I was interested in computing the logarithmic derivarivative $Z'(H)/Z(H)$ of the following partition function: $$ Z(H) = \int e^{-S_H(x)} \mathcal{D} x, \quad \text{where} \quad S_H(x) = A(H) ...
- 101
1vote
1answer
658views
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}\...
- 790
4votes
0answers
67views
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) = ...
- 2,088
