# Questions tagged [functional-determinants]

The tag has no usage guidance.

49 questions
Filter by
Sorted by
Tagged with
52 views

### Meaning of capital pi symbol in sum over histories integral

This question is primarily mathematical in nature. I have been reading Quantum Field Theory for the Gifted Amateur and I am reading about Feynman’s path integral approach. The definition of the “sum ...
183 views

### 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 ...
36 views

### Functional Determinant of a matrix of operators

How would we calculate the functional determinant of a matrix with both continuous and discrete indices; such as \begin{equation} O =\begin{pmatrix} a(t) & \frac{d}{dt} +b(t) & c(t) \\ \frac{d}...
80 views

46 views

### Supersymmetric localisation of 2D super YM on $S^2$

I wanna to understand how to calculate partition function for pure abelian Yang-Mills theory. To do this, I need follow some usual step's (I follow Benini, Localization in supersymmetric field ...
93 views

249 views

159 views

### Auxiliary field path integral in non-linear sigma models

I am trying to understand the functional integral over the auxiliary field in the $\mathcal{N}=(2,2)$ supersymmetric non-linear sigma model, or NLSM (reviewed in Chapter 13 of Mirror Symmetry http://...
165 views

### Faddeev-Popov-Determinant of Polyakov Path Integral

I'm currently trying to understand the paper "Quantum Geometry of bosonic Strings" by Polyakov. I think I roughly understand the X integration, but when it comes to the integration over the metric ...
218 views

### A question about Gel'fand-Yaglom method of calculating functional determinants

I know that the Gel'fand-Yaglom method is a way to calculate determinants of 1D differential operators. For instance, let us consider an operator $-\partial_r^2+W(r)$. Let us define $\psi_{0}(r)$ as ...
325 views

### Graphical determination of energy eigenvalues (symmetrical potential well)

It is about a particle with mass $m$ in a potential $V(x)$: I want to do a graphical determination(at first only the symmetrical case) of the energy eigenvalues. I will show you my previous work: ...
678 views

### 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 ...
210 views

### Integral representation of functional determinant

I'm studying the proof of a theorem and, being not very expert in QFT, I'm having problems understanding a couple of equalities that my professor said to be useful in order to understand said proof. ...
263 views

### Question on Faddeev-Popov method derivation

In the text book of Weinberg, there is a proof to show that path integral is independent of gauge fixing functional $f_a[\phi; x]$. $\phi_\Lambda$ is the result of gauge transformation on $\phi$ by an ...
736 views

### Gaussian integral formula for matrix product

I am looking for a way to prove that $$\det (M \cdot N) = \det(M)\det(N) \tag{0}$$ Where $M$ and $N$ are matrices with continuous indices, so that $\det$ is a functional determinant. A way to show ...
170 views

### The determinant of the Dirac operator in Euclidean signature

Suppose the Dirac operator determinant in Euclidean space-time with manifold $\mathbb R^{4}$: $$d = \text{det}(iD), \quad iD = i\gamma^\mu (\partial_\mu +A_{\mu})$$ The Dirac operator is elliptical, ...
159 views

### Interpretation of the chiral anomaly a-la Alvarez-Gaume

In the paper "The topological meaning of non-abelian anomalies" written by Alvares-Gaume and Ginsparg they argue the appearing of the (gauge) anomaly in a theory with chiral fermions in the following ...
309 views

### Faddeev Popov determinant for $U(1)$

The Faddeev-Popov determinant in case of $U(1)$ turns out to be ${\rm Det}(\partial^2)$. My question is: what is the determinant of $\partial^2$?
517 views

452 views

### 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) \...
297 views

### Free space propagator: reconciling two results

In quantum mechanics, the free space propagator $G(q_f=0,q_i=0;\tau)$ can be easily calculated to be $$\sqrt{\frac{m}{2\pi i \hbar \tau}}$$ by inserting an identity operator. However if we use ...
379 views

### Determinant of a propagator

Say I have a path integral $\int D \phi \exp(i S_0)$. $S_0$ is the usual free action $$S_0=\frac{1}{2}\int\phi (-\Box-m^2) \phi=\frac{1}{2}\int \phi G^{-1} \phi,$$ and at the moment I'm not ...
1k views

### How to determine the trace and determinant of a differential operator?

How to determine the trace and determinant of the operator like $\Box$ or $\nabla^2$ etc. But first of all how to find the same for the simpler operator $\frac{d}{dx}$? I proceeded as follows. What ...
230 views

### Regularization of the 1-dimensional Laplacian

Disclaimer: this is a technical question about regularization of functional determinants which comes from a person with (relatively) strong background in QFT, string theory and path integrals, who ...
284 views

### What do we take the functional determinant of in computing th effective action in the Background field method?

I have some schematic notes on computing the effective action and I would like someone to help me fill the gaps. We start with \begin{equation*} \int{}\mathcal{D}\phi\,e^{-iS[\phi]} \end{equation*} ...
974 views

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 $$\widetilde\Delta_F(p) = \frac{1}{(p^0)^2-\left(\... 1answer 1k views ### Path integral as a functional determinant In Peskin and Schroeder on pg. 304, the authors call the fermionic path integral: \begin{equation} \int {\cal D} \bar{\psi} {\cal D} \psi \exp \left[ i \int \,d^4x \bar{\psi} ( i \gamma_\mu D^\mu - m ... 2answers 943 views ### Determinant of Dirac operator in flat space? How would you evaluate \begin{equation}|iD\!\!\!\!/-m|\end{equation} Where D_{\mu}=\partial_{\mu}-ieA_{\mu}. I have an idea of how to do this without the gauge field, because it's essentially \... 1answer 866 views ### Computing functional determinant for Dirac fermions In the path integral formulation for quantum field theory, one often encounters functional determinants of operators, for example for a free scalar field \log \det (\partial^2+m^2). For this ... 1answer 879 views ### What is the status of Witten's and Vafa's argument that the QCD vacuum energy is a minimum for zero \theta angle? The argument, which I reproduce here from Ramond's `Journies BSM', is originally by Witten and Vafa in (\it{Phys}. \it{Rev}. \it{Lett}. 53, 535(1984)). The argument is that for \theta = 0  (... 0answers 161 views ### functional determinant evaluaton given a Hamiltonian and the semiclassical WKB partition function in units  \hbar =2m=1   \Theta (t) = \frac{1}{2\pi} \iint dx dp exp(-tp^{2}-tV(x))  can i use this Theta function to evaluate the ... 0answers 206 views ### Functional determinant approximation Let the Hamiltonian in one dimension be H+z, then I would like to evaluate \det(H+z). I have thought that if I know the function Z(t) = \sum_{n>0}\exp(-tE_{n}) I can use$$\sum_{n} (z+E_{n})...
Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in ...
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 $y'(0)=1$. ...