All Questions
Tagged with quantum-field-theory partition-function
124
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
3
votes
0
answers
43
views
Physical observables in the XY/sine-Gordon duality
My question is, during the duality map, real physical quantities seem to acquire a prefactor of $i$ and become purely imaginary. And I feel uncomfortable.
Take bosonic current for example. Consider ...
- 91
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
5
votes
0
answers
78
views
How many Lagrangians can a QFT have?
I just stumbled across a presentation by Tachikawa about "What is Quantum Field Theory". He has an interesting perspective that we should think of (at least a subset of) quantum field ...
- 1,014
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:
$$Z[J, \hbar] = \...
- 1,611
1
vote
0
answers
61
views
Is the Euclidean generating functional $Z_{E}[J]$ identified with original Minkowskian generating functional $Z[J]$?
In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to ...
- 81
3
votes
1
answer
278
views
Changing variable in path integral
Good evening,
I am learning about path integrals in QFT and I was wondering, can you simplify the path integral by shifting the fields? To make it more clear I will give you an example. Suppose that I ...
- 33
1
vote
2
answers
186
views
Why doesn't this diagram appear in the partition function in zero-dimensional QFT?
For the zero-dimensional QFT with action
$$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
1
vote
0
answers
72
views
Calculating $\langle\hat{\phi_i}\rangle_t$ (Blundell's Quantum field theory) (EDITED) [closed]
I am reading Blundell's Quantum field theory for the Gifted Amateur and stuck at some calculation. In his book p.197, 21.2 Sources in statistical physics, he defined the partition function with the ...
- 323
8
votes
4
answers
2k
views
How to connect the usage of the path integral in QFT to the usage in Quantum Mechanics?
In Quantum Mechanics, path integrals are used to calculate the matrix element:
$$
\langle x_1, t_1|x_2, t_2\rangle_J=\int
e^{i(S[x(t)]+\int\!J x(t))/\hbar} d[x(t) ].\tag{1}$$
If we naively try to ...
- 6,355
7
votes
1
answer
327
views
Does a CFT need a UV regulator?
I have a very basic question about conformal field theory: Does the partition function need a UV regulator, or is it finite even without? That is, does $\int D\phi \exp(-S)$ converge, or do we need to ...
- 1,455
4
votes
0
answers
100
views
Functional determinant: linking Series, Heat-Kernel and Zeta function
I would like to express a functional determinant as a series of diagrams, using the zeta function renormalization applied to the heat-kernel method, but I don't know if it's possible. Let me explain:
...
- 2,613
6
votes
0
answers
232
views
Operator insertions vs boundary conditions in AdS/CFT
This question is motivated by AdS/CFT, but really it's just about AdS quantum gravity. Consider quantum gravity in asymptotically AdS spacetime. For simplicity, assume there are no matter fields: the ...
- 1,455
1
vote
3
answers
233
views
What is a symmetry of the generating functional, and what is the significance?
I cannot find a definition for a symmetry of the generating functional in Quantum Field Theory:
$$ Z[J] = \int \mathrm d \mu \, \exp\left\lbrace i S[J] \right\rbrace \, .$$
I know it's a simple ...
- 1,387
5
votes
1
answer
486
views
The Partition Function of $0$-Dimensional $\phi^{4}$ Theory
My question is related with this question. Several years ago, I posted an answer to the question, and the author of the reference removed the link permanently, now I have no clue what's going on.
In ...
- 2,923
1
vote
0
answers
54
views
Why does expressing the Faddeev-Popov determinant as this lead to such problems?
Background
In the following, I am interested in the Schwinger function associated with the gluon propagator when one considers the Gribov no-pole condition in the partition function. Defining $\nabla^{...
- 2,613
3
votes
0
answers
45
views
Is it OK to do this manipulation at the partition function level? (auxiliary fields in quadratic gravity)
Background
I am working with the following action in the Euclidean signature ($C^2$ is the Weyl quadratic term):
\begin{equation}
S_B = -\frac{1}{2\kappa^2}\int d^4x\sqrt{g}\left(2\Lambda_C+\zeta R-\...
- 2,613
1
vote
1
answer
83
views
Clarification regarding the terminology of Microstates
I would like to understand how microstates are defined or used in physics. Are microstates suppose to only mean eigenvalues of a given observable (or a generator of symmetry)? The reason for my ...
- 1,200
2
votes
2
answers
367
views
Partition function for bosons with path integral
In this book the partition function for bosons is defined in eq. 2.17 as:
$$Z=\mathrm{Tr}[e^{-\beta (H-\mu_i N_i)}]=\sum_a\int d\phi_a\langle\phi_a|e^{-\beta(H-\mu_i N_i)}|\phi_a\rangle$$
The ...
user255856
-2
votes
1
answer
389
views
Correlation Function and Generating Functional in QED
Peskin and Schroeder (1995, p.82 and p.292) define the two-point correlation function of a $\phi^4$ theory as
$$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle\tag{4.10}$$
and the generating ...
- 373
4
votes
0
answers
189
views
Feynman Rules from Generating Functional
For the following Lagrangian:
$$\mathcal{L}= \overline \psi \left(i \gamma^{\mu}D_{\mu} - m \right)\psi -\frac{1}{2}\left(F_{\mu\nu}\right)^2,$$
I'm trying to find the Feynman rules. I know that the ...
- 1,213
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 ...
6
votes
1
answer
199
views
Does a $d$-dimensional stat-mech theory necessarily have a $(d-1)$-dimensional quantum theory equivalence?
A $d$-dimensional stat-mech theory on a lattice usually can be represented by a $d$-dimensional tensor network. Taking a row/slice of tensors ($M$ tensors or sites) as the transfer matrix (MPO in 2$d$ ...
- 356
4
votes
1
answer
324
views
Propagator in Path Integrals
I am reading Section I.2 in Zee's QFT in a Nutshell. The amplitude for a particle to start at position $I$ and end at $F$ is (eq. (I.2.6)):
$$
\langle q_f|e^{-iHT}|q_I\rangle=\int Dq(t)\ e^{i\int_0^T ...
- 51
3
votes
0
answers
158
views
Wouldn't a simple scalar field fix the non-renormalizability of gravity?
It is well known that quadratic gravity is renormalizable. On the other hand it is possible to transform the partition function of Einstein-Hilbert + free minimally coupled complex scalar field into a ...
- 2,613
4
votes
2
answers
2k
views
QFT generating functional and Green function and propagator
I am confused about why does the generating functional gives the propagator by differentiation, and why that propagator is the Green function.
I understand how to take the functional derivative like ...
- 232
0
votes
0
answers
311
views
Free Energy vs. Partition Function in QFT
The partition function of QFT is defined as
$$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
2
votes
1
answer
113
views
Conditions on the covariance operator in Gaussian Path Integrals
In field theory, one typically encounters integrals of the form:
$$ \mathcal{Z}[J] = \int \mathcal{D}[\phi] \exp \left( - \frac{1}{2} \int d^Dx d^Dx' \ \phi(x)A(x,x')\phi(x')+ \int d^Dx \phi(x) J(x)\...
- 539
1
vote
0
answers
248
views
Quantum to classical mapping
I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works.
Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
- 741
1
vote
0
answers
352
views
Complete the square for the generating functional of the Dirac field
Quote Peskin page 302 the Dirac generating function was
$$Z[\bar \eta ,\eta ]=\int D\bar\psi D\psi\exp[i\int dx^4 (\bar\psi (i\gamma^\mu\partial_\mu -m )\psi+\bar\eta \psi+\bar\psi \eta)]$$
could be ...
- 3,256
0
votes
1
answer
344
views
How to write the spatial photon propagator of the generating function $Z[J]$ in QED?
There was a part in the lecture one didn't quite understand. (The charge $e=0$ in this post.) The partition function for massive fields such as scalar fields or the spin-1/2 fields were quite standard....
- 3,256
4
votes
1
answer
424
views
In QFT, why are the vacuum partition function and the zero-temperature imaginary-time partition function the same?
When doing thermal field theory, one can start with the definition of the (thermal) partition function $Z = Tr[e^{-\beta H}]$, and inserting a number of completness-relations, we can arrive at (I am ...
- 577
4
votes
3
answers
523
views
The full path integral of a quantum field theory
Suppose if one is able to do a full path integral of a QFT with an action say $S[\phi]$ i.e.
$$Z = \int [\mathcal{D}\phi] e^{iS[\phi]}.$$ What can I use $Z$ for? Can I use the $Z$ like the partition ...
- 1,200
1
vote
0
answers
251
views
Free massive scalar field partition function in QFT?
Consider the (euclidean) path integral for the free massive scalar field in $d$ dimensions, giving the partition function
$$Z_m=\int\mathcal{D}\phi~e^{\int dx^d~\phi(\Delta-m^2)\phi}$$
with Laplace-...
- 1,517
1
vote
0
answers
159
views
Connected part of $S$-matrix generating functional
I am currently studying an article by A.Jevicki et. al. (https://doi.org/10.1103/PhysRevD.37.1485) and I am a little confused. They say that the generating functional of the $S$-matrix is related to ...
- 93
1
vote
3
answers
549
views
Taking functional derivatives of generating functional
I'm learning how to compute functional derivatives of generating funtionals. Suppose I have the generating functional
$$Z[J] = \exp\left\{\int{dy_1 \; dz_1\; J(y_1) \Delta(y_1 - z_1) J(z_1)}\right\}.$$...
- 373
12
votes
2
answers
505
views
How to calculate a TQFT Gaussian path integral from Seiberg's "fun with free field theory"?
In his talk "Fun with Free Field Theory", Seiberg discusses a topological quantum field theory in $d+1$ dimensions with the action
$$ S = \frac{n}{2\pi} \int \phi\, \mathrm{d} a \tag{1}$$
...
- 2,084
2
votes
2
answers
247
views
Calculating generating functional with stationary phase approximation
Let's say that I have a generating functional $Z[J]$ defined as:
\begin{equation*}
Z[J]=\int \mathcal{D}\phi\,e^{iS[\phi]+i\int d^4x\,J\phi}.\tag{1}
\end{equation*}
I want to use the stationary phase ...
- 2,613
5
votes
1
answer
609
views
When does QFT perturbation theory stop being valid?
When introduced to the concept of perturbation theory in Quantum Mechanics we split the hamiltonian $H= H_0 + \delta H$ where $\delta H$ is small in some manner, ie if say $\epsilon$ is the relevant ...
- 766
1
vote
0
answers
61
views
Derivation of fermionic partition function, how does commutation work?
When deriving the fermionic partition function with coherent states $|\psi\rangle$ we make the following step
$$
\mathcal Z=\int d(\bar\psi,\psi)\ e^{-\sum_i\bar\psi_i \psi_i}\sum_n\langle n|\psi\...
- 23