All Questions
Tagged with quantum-field-theory partition-function
124
questions
3
votes
3
answers
190
views
Perturbative expansion and self-contractions in functional integral
Consider a one-dimensional integral
$$I(g)=\int dx\, e^{-x^2-gx^4}$$
One can formally expand it perturbatively order by order in $g$ so that
$$I(g)=\left<1\right>-g\left<x^4\right>+\frac{g^...
- 1,586
-1
votes
1
answer
379
views
Partition function in quantum field theory
Why does the partition function include current term in free scalar field
$$Z[J] = \int \mathcal{D}\phi \, e^{i \left(S[\phi] + \int d^4x \,J(x) \phi(x) \right)}~$$
- 39
1
vote
0
answers
256
views
What is the entropy and/or equation of state of a partition function such as $Z=\int D\phi \exp (i S[\phi]/\hbar)$?
At this link https://en.wikipedia.org/wiki/Partition_function_(mathematics), it is claimed that the following partition function:
$$
Z=\int D\phi \exp (-\beta H[\phi]) \tag{1}
$$
is a consequence of ...
- 1,548
6
votes
0
answers
197
views
How does anomaly inflow work in terms of the eta invariant?
I'm trying to understand the non-perturbative picture of anomaly inflow, mainly following these two articles by Witten and Yonekura:
[1] - https://arxiv.org/pdf/1909.08775.pdf ,
[2] - https://arxiv....
- 61
3
votes
0
answers
66
views
About the various ensembles in Thermodynamics
The properties of a system in thermodynamical equilibrium are described by a partition function:
$$
\mathcal{Z} = \text{Tr} \ e^{-\beta E} = \sum_n e^{-\beta E_n}
$$
This defines so called canonical ...
- 4,883
0
votes
0
answers
306
views
Path integral as a partition function (math)
I am reading the following Wikipedia page, but I am skeptical about what I am reading (it sounds too good to be true). Specifically, I am looking at the passage which states:
The number of variables $...
- 1,548
2
votes
2
answers
267
views
Measure in the Fourier Representation of the Coherent States Path Integral
The problem I have arises in the context of condensed matter physics. I am largely following chapter 4 about functional integration in the book by Altland and Simon. Consider the coherent states path ...
- 4,219
6
votes
1
answer
365
views
What is the physical meaning of $W[J]=\frac{\hbar}{i}\ln Z[J]$?
The quantity $Z[J]$ (which is the generating functional for all Green functions) physically represents the probability amplitude for a system to remain in the vacuum state. Can we find a similar ...
- 11.8k
0
votes
1
answer
531
views
Path integral zero dimensional QFT
We consider the following partition function$$
\mathcal{Z}[\lambda] = \int{dx \; \exp\left(-\frac{1}{2}x^2-\frac{\lambda}{4!}x^4\right)}
$$
Which is basically $\phi^4$ theory in 0+0 dimensions. The ...
- 2,035
1
vote
3
answers
2k
views
What is free energy in the context of a quantum field theory?
I was reading the papers Large $N$ behavior of mass deformed ABJM theory and New 3D ${\cal N}=2$ SCFT's with $N^{3/2}$ scaling. These papers talk about the free energy in the context of quantum field ...
- 1,222
0
votes
0
answers
167
views
Taking the derivative of the partition function describing a scalar QFT w.r.t. a constant
Premise: I know this question would be better suited to MathSE, but since it strongly concerns QFT, I'm confident I'll find a more exhaustive answer here.
Consider a generic partition function $\...
2
votes
1
answer
123
views
Show: $\langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \vert n \rangle \langle n \vert \psi \rangle$ [closed]
The book (Altland and Simons, Condensed Matter Field Theory, Ch. 4.2) I am reading makes use of the identity
\begin{equation}
\langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \...
- 23
0
votes
0
answers
132
views
Coupled quantum oscillator: Field theory
Consider two masses $m$ connected by a spring with a spring constant $k$. Each mass is also connected to the wall using the same springs. The Hamiltonian is
$$
H = \frac{p_1^2 + p_2^2}{2m} + \frac{k}{...
- 631
4
votes
1
answer
933
views
Determinant of d'Alembert Operator $\mathop\Box-m^{2}$
In quantum field theory, the partition function of a free scalar is
$$\mathcal{Z}=\int\mathcal{D}\phi\exp i\int d^{n}x\frac{1}{2}\left[(\partial_{\mu}\phi)(\partial^{\mu}\phi)-m^{2}\phi^{2}\right]$$
$...
- 2,923
2
votes
0
answers
148
views
Indexes in the Gaussian functional integral
This is a question spawning from a comment made to my previous question. There I was asking about taking some functional derivative in the effective action of the non-linear sigma model. The comment ...
- 6,067