Questions tagged [functional-determinants]

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Relationship between product integrals and functional determinants

This is in reference to the answer posted to this question. The person who answered the question claims that the functional determinant of any operator $O$ is given by a product integral $$\det O = \...
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How to perform a Gaussian functional integral?

I'm completely beginner to the quantum field theory and try to learn the basics of functional integrals. However, I could not understand clearly. Could someone please explain the idea with the help of ...
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Computation of functional determinant using Feynman diagram

The above equation is from chapter 9.5 "Functional Quantization of the Spinor Field" of Peskin's and Schroeder's book $($page $305)$. I understand that the initial determinant equal to the ...
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Second functional derivative and its units

Say I have a functional $I[\phi,g]$ with $\phi(p)$ and $g(p)$ functions from $\mathbb{R} \to \mathbb{R}$. Also say that this functional obeys the property: $$\frac{\delta I}{\delta g(p)} = -(g(p))^{-1}...
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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{...
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1 answer
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Four-point correlation function path integral for free scalars

In An Introduction to Quantum Field Theory by Peskin and Schroeder, section 9.2, they calculate the four-point correlation function for a free real scalar field $\phi(x)$ using the path integral ...
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Disappearing symmetry in gaussian functional determinant

I have the following integral $$I=\int D\varphi \; e^{-\int d^4p d^4p' \left[ -\frac{1}{2}\varphi(p) g(p) \delta(p+p') \varphi(p') \right]}.\tag{1}$$ This is the continuum limit of a gaussian matrix ...
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Boundary conditions in Gaussian path integral

The $N$-dimensional Gaussian integral $$\int \mathrm{d}^N x \, \mathrm{e}^{-\frac{1}{2}\boldsymbol{x}^\mathrm{T}A\boldsymbol{x}+\boldsymbol{b}^\text{T}\boldsymbol{x}}=\left(\frac{(2\pi)^N}{\det A}\...
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1 answer
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How are functional traces calculated?

I am trying to follow this paper concerning decay rates in QFT. In equations (E.5), (E.6), (E.7), a functional trace is calculated using Feynman diagrams. However, I am struggling to see why $$Tr[(-\...
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Is there still a Gribov ambiguity when the Faddeev-Popov determinant is treated without ghosts?

In this document (Gribov Ambiguity by Thitipat Sainapha) the setup leading to the equation $3.77$ seems to strongly depend on the treatment of the Faddeev-Popov determinants with ghosts. Indeed the ...
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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)\...
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Gelfand-Yaglom method and perturbation

I'm reading Instantons and large $N$ (https://laces.web.cern.ch/LACES10/notes/instlargen.pdf) by Marino, and I don't understand something about Gelfand–Yaglom method (from page 22), so please tell me ...
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Functional determinants

I wish to know what is the result of this Gaussian Functional Integral $$Z[\chi] = \int[\mathcal{D}\phi]~e^{-i\int d^dx ~\phi^2\chi}$$ where $\phi, \chi$ are position dependent fields. Now, my ...
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1 answer
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Functional integrations

We often see functional versions of Gaussian integrations $$ \int_{-\infty}^{\infty} d^dx e^{-x^{T}Mx} = \frac{1}{\sqrt{2\pi^d \det M}} \to \int[\mathcal{D}X] e^{-i\int X \mathcal{O}X} = (\det{O})^{-1}...
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How to deal with path integral in curved space-time for a free scalar field?

Let's say we have a complex scalar field in a curved background whose action is: \begin{equation} S=-\int d^4x \sqrt{-g}\phi^\ast(\square_g+m^2) \phi \end{equation} For some purpose I want to ...
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Units in the nonrelativistic free particle path integral

I am almost certain I have seen the answer to this question on this site before, but for the life of me I cannot find it after significant searching. If someone can located another question with the ...
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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)\...
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Determinant of differential operator as exponential of Wess-Zumino-Witten action

I am currently reading this paper (Mass Gap and Confinement in (2+1)-Dimensional Yang-Mills Theory, Dimitra Karabali) and between equations (6) and (7) the following identity is used: \begin{equation*}...
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What physical meaning “determinant” of a divergency (divergent integral or series) can have? Is there a parallel with functional determinant?

I am working on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\...
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How to find the determinant of a single derivative?

I am currently studying the path integral approach to stochastic processes. Recently I was reading Functional integral approach for multiplicative stochastic processes about the path integral ...
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1 answer
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Determinant of differential operator $( \partial^2 + m^2)$

For a scalar field in QFT the generating functional is given as: $$ Z[J] = \int \left[ d\phi \right] \exp{\left( i\ S[\phi] + i \int d^4 x\ \phi (x) J(x) \right)} $$ with $ S = \frac{1}{2} \int d^4 x\ ...
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2 votes
2 answers
274 views

Path integral identity

I am reading the Background Field Methods in the EPFL Lectures on GR as an EFT. The authors use this identity on Page 23, Equation (174): $$ \mathcal{N}^{-1}\int\mathcal{D}\phi\,\mathcal{D}\phi^*\exp\{...
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The Functional Determinants in Peskin and Schroeder (Eq.9.77)

I'm working on the Eq.9.77 in Peskin (page 304): To demonstrate this, we need only apply standard identities from linear algebra. First notice that, if a matrix $B$ has eigenvalues $b_i ,$ we can ...
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1 answer
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Regularization of functional determinant over an Instanton background

I am reading the paper "ABC of instantons" and meet some problems at section 8. I simplify this problem a little bit as follows. First, we have a Euclidean path integral like \begin{equation}...
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1 answer
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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 ...
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7 votes
1 answer
314 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 ...
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2 votes
0 answers
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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}...
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3 votes
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Functional determinant in scalar QED

I'm trying to integrate out the scalars from the path integral in scalar QED, but I encountered an integral I don't know how to do. The model is $S = \int_{\mathbb{R}^4}d^4x \left( -\frac{(F_{\mu\nu})...
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Propagator in massive QED/Schwinger model

I'm trying to integrate out the fermions from the path integral in the massive QED/Schwinger model $S = \int_{\mathbb{R}^d}d^{dx} \left( - \frac{(F_{\mu\nu})^2}{4} + \bar{\psi} \left( i\gamma^\mu D_\...
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3 votes
2 answers
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Why is there an difference between the exponent of the determinant of these two path integral?

When I read about Altland and Simons “Condensed matter field theory”, I came across with the path integral (3.28). $$\langle {q_f}|e^{-iHt/\hbar} |q_i\rangle = \det(\frac{i}{2\pi \hbar} \frac{\...
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4 votes
1 answer
209 views

Loop counting for determinants and anomalies

I am trying to understand an argument for why anomalies are one-loop exact, given by Bilal in Lectures on Anomalies. The relevant paragraph is reproduced here: Let us first explain why the anomaly ...
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5 votes
1 answer
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QED electron self-energy in 1PI effective action

The electron self-energy at one-loop is given by the one-particle irreducible graph I know how to calculate it using the Feynman rules but I was wondering how this diagram appears in the QED ...
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1 answer
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Convergence of the path integral

In P&S 9.3 the path integral $$ Z[J]=\int {\cal D}\phi \exp[i\int d^4x ({\cal L} + J\phi)]$$ of the (Minkowski) $\phi^4$-theory when subjected to a Wick-rotation (change of the integration path ...
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2 votes
2 answers
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Calculation of current from path integral

I would like to calculate $\langle\bar{\psi}\psi\rangle$ in free theory. I start from the following generating functional: $$Z[J]=\int\mathcal{D}[\bar{\psi},\,\psi]\exp\left(i\int d^dx\,[\bar{\psi}(i\...
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1 vote
1 answer
498 views

Peskin & Schroeder eq. 9.26 and functional methods

I have been reading chapter 9 in Peskin & Schroeder's QFT book and has been stuck in transition from equation 9.26 to 9.27. Equation 9.26 reads: $$\frac{1}{V^2} \Sigma_{m,l} \exp{[-i(k_m.x_1+k_l ....
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1 answer
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Eigenvalue counting number in Functional Integral

My question is about the calculation of a functional integral (which looks like a partition function). If we have the operator $A$ having discrete spectrum, and eigenvectors $\phi_{i}$ and ...
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5 votes
0 answers
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What is the 't Hooft determinant?

The 't Hooft vertex/determinant is somehow generated by instantons and is responsible for the generation of mass gap in pseudo-Goldstone bosons, such as an axion. For example, the complex Peccei-...
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4 votes
1 answer
330 views

Path integral measure in Chern-Simons/WZW correspondence

The relationship between 3d Chern-Simons theory on the product of the disk and the real line ($D\times \mathbb{R}$) and the chiral WZW model on $S^1\times \mathbb{R}$ was shown in Elitzur et al Nucl....
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5 votes
1 answer
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Two Questions about Path Integral from "Gauge Fields and Strings" by Polyakov

My questions are about worldline path integrals from the book Gauge Fields and Strings of Polyakov. On page 153, chapter 9, he says Let us begin with the following path integral \begin{align} &...
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3 votes
1 answer
150 views

A Question about Path Integral Measure

I want to do the following path integral. $$\mathcal{Z}=\int\mathcal{D}x e^{iS[\dot{x}]}$$ The action only denpends on $\dot{x}$. For some reason, I want to replace the integral measure $\mathcal{D}...
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11 votes
3 answers
955 views

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)&...
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1 answer
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Vacuum Energy Calculation using Path Integral

I am currently reading Zee's book on quantum field theory, and I am in the chapter where he is introducing Grassmann integrals. He re-introduces the path integral evaluated for the vacuum, i.e. no ...
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4 votes
1 answer
580 views

Determinant of d'Alembert Operator $\mathop\Box-m^{2}$

In quantum field theory, the partition function of a free scalar is $$\mathcal{Z}=\int\mathcal{D}\phi\exp i\int d^{n}x\frac{1}{2}\left[(\partial_{\mu}\phi)(\partial^{\mu}\phi)-m^{2}\phi^{2}\right]$$ $...
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1 vote
0 answers
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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 ...
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1 vote
1 answer
823 views

One-loop effective action of QED and the partition function

Given the partition function for QED $$ Z= \int \mathcal{D}A_{\mu}\mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu} + i \int \bar{\Psi} (i {\not} D-m) \...
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Finding the determinant of $(\omega^2-\partial_t^2)/2D$ in path integral? [closed]

I am looking to evaluate the following path integral: $$I=\int_{\vec x(t_0)=0}^{\vec x(t)=0}\mathcal{D}x \exp\left( -\frac{1}{2} \int^t_{t_0} d\tau \; \vec x \left\{ \frac{\omega^2-\partial_t^2}{2D} \...
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2 votes
0 answers
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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://...
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2 votes
0 answers
201 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 ...
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0 answers
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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 ...
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6 votes
2 answers
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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 ...
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