Questions tagged [functional-determinants]
110 questions
26
votes
1 answer
What is the status of Witten's and Vafa's argument that the QCD vacuum energy is a minimum for zero $\theta$ angle?
The argument, which I reproduce here from Ramond's `Journies BSM', is originally by Witten and Vafa in ($\it{Phys}$. $\it{Rev}$. $\it{Lett}$. 53, 535(1984)). The argument is that for $\theta = 0 $ (mod $2\pi$) the QCD vacuum energy $E(\theta)$ is…
DJBunk
- 3,788
23
votes
4 answers
Gelfand-Yaglom theorem for functional determinants
What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form
$Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. Here $H$ is the Hamiltonian and $z$ is a real…
Jose Javier Garcia
- 4,847
15
votes
0 answers
How to perform a derivative of a functional determinant?
Let us consider a functional determinant
$$\det G^{-1}(x,y;g_{\mu\nu})$$
where the operator $G^{-1}(x,y;g_{\mu\nu})$ reads
$$G^{-1}(x,y;g_{\mu\nu})=\delta^{(4)}(x-y)\sqrt{-g(y)}\left(g^{\mu\nu}(y)\nabla^{(y)}_\mu\nabla^{(y)}_\nu+m^2\right).$$
Such…
Wein Eld
- 3,791
14
votes
1 answer
Path integral with zero energy modes
Consider the field integral for the partition function of a free non-relativistic electron in a condensed matter setting, i.e.
$$ Z = ∫D\bar\psi D\psi \exp\left(-\sum_{k,ω} \bar\psi_{k,ω} (-iω + \frac{k^2}{2m} - \mu) \psi_{k,ω}\right) $$
where the…
Greg Graviton
- 5,397
14
votes
2 answers
Calculating $\mathrm{Tr}[\log \Delta_F]$
I am stuck with this problem for quite sometime. I have a propagator in the momentum representation (from this Phys.SE question), which looks like
$$ \widetilde\Delta_F(p) = \frac{1}{(p^0)^2-\left(\left(n\pi/L\right)^2+m^2\right)+i\epsilon} $$
I…
user35952
- 3,134
11
votes
3 answers
How does the functional measure transform under a field redefinition?
My question is: how does the path integral functional measure transform under the following field redefinitions (where $c$ is an arbitrary constant and $\phi$ is a scalar field):
\begin{align}
\phi(x)&=\theta(x)+c \,\theta^3(x)…
Luke
- 2,390
10
votes
2 answers
Computing a Gaussian path integral with a zero-mode constraint
I have the following partition function:
\begin{equation}
Z=\int_{a(0)=a(1)} \mathcal{D}a\,\delta\left(\int_0^1 d\tau \,a -\bar{\mu}\right)\exp\left(-\frac{1}{g^2}\int_0^1d\tau\, a^2\right)
\end{equation}
where $\bar{\mu}$ is a constant. How can I…
Ruben Campos Delgado
- 1,090
10
votes
1 answer
Lack of Maslov index in the path integral formalism
Introduction
Consider Feynman's famous path integral formula
\begin{equation}
K(x_a,x_b) = \int \mathcal{D}[x(t)] \exp \left[ \frac{i}{\hbar} \int_{t_a}^{t_b} dt \, \mathcal{L}(x(t),\dot{x}(t),t) \right] \, ,
\end{equation}
where the…
QuantumBrick
- 4,183
9
votes
2 answers
Determinant of Dirac operator in flat space?
How would you evaluate
\begin{equation}|iD\!\!\!\!/-m|\end{equation} Where $D_{\mu}=\partial_{\mu}-ieA_{\mu}$.
I have an idea of how to do this without the gauge field, because it's essentially…
TeeJay
- 548
9
votes
1 answer
Computing functional determinant for Dirac fermions
In the path integral formulation for quantum field theory, one often encounters functional determinants of operators, for example for a free scalar field
$\log \det (\partial^2+m^2)$. For this example, the expression can be expressed as an integral…
anon
- 91
9
votes
3 answers
Regularisation of infinite-dimensional determinants
Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM?
Edit: I failed to make myself clear. In finite dimensions, there is a function of $\lambda$ whose roots are…
joseph f. johnson
- 7,212
8
votes
2 answers
How to directly evaluate path integral for harmonic oscillator by brute force method?
It is easy to evaluate the green's function using path integral approach by evaluating classical action and using functional calculus method. Is it possible to evaluate path integral for harmonic oscillator directly by evaluating the integral for…
user135580
- 1,130
- 1
- 11
- 24
8
votes
1 answer
Path integral as a functional determinant
In Peskin and Schroeder on pg. 304, the authors call the fermionic path integral:
\begin{equation}
\int {\cal D} \bar{\psi} {\cal D} \psi \exp \left[ i \int \,d^4x \bar{\psi} ( i \gamma_\mu D^\mu - m ) \psi \right]
\end{equation}
a functional…
JeffDror
- 9,093
7
votes
1 answer
One-loop effective action for scalar field on the curved background in large potential
I hope to compute a functional integral $Z=\int \mathcal{D}\phi\,\, e^{-S[\phi]}$ with an action
$$S[\phi]=\int d^2x \sqrt{g}\Big((\nabla \phi)^2+\frac{1}{\lambda}M^2(x) \phi^2\Big)$$
The scalar field $\phi$ is defined on a two-dimensional curved…
Weather Report
- 1,624
7
votes
2 answers
Harmonic oscillator partition function via Matsubara formalism
I am trying to understand the solution to a problem in Altland & Simons, chapter 4, p. 183. As a demonstration of the finite temperature path integral, the problem asks to calculate the partition function of a single harmonic oscillator. The…
Zack
- 3,166