Questions tagged [functional-determinants]

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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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1answer
75 views

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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1answer
42 views

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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1answer
36 views

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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23 views

Stationary phase approximation with a non-positive definite fluctuation operator

I wish to evaluate the following path integral - to find a conditional probability. $$P(x(T) | x(0)) = \int Dx(t) \exp(S[x]) = \int Dx(t) \exp \Big(\int -\frac{1}{2}m(x)\ddot{x} + b(x)\dot{x} + V(x) ...
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229 views

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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41 views

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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0answers
23 views

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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0answers
43 views

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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1answer
156 views

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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2answers
198 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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2answers
279 views

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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1answer
86 views

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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1answer
93 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 ...
7
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1answer
265 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 ...
2
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0answers
62 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}...
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118 views

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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0answers
73 views

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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2answers
110 views

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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1answer
157 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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1answer
96 views

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 ...
2
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2answers
175 views

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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1answer
370 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 ....
2
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1answer
77 views

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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242 views

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-...
4
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1answer
280 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....
5
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1answer
224 views

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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1answer
127 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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3answers
725 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)&...
1
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1answer
296 views

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 ...
4
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1answer
454 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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0answers
100 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 ...
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0answers
81 views

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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0answers
188 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://...
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0answers
184 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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0answers
244 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 ...
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1answer
400 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: ...
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2answers
970 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 ...
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1answer
251 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. ...
2
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1answer
333 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 ...
3
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2answers
965 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 ...
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0answers
222 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, ...
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0answers
160 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 ...
5
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1answer
330 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 $\partial^2$?
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1answer
606 views

Determinant as a fermionic path integral

I understand that the determinant of a matrix can be written in terms of a fermionic path integral. The expression is: $$Z = \int D\bar{\psi}D\psi e^{-\iint d^4x' d^4x \bar{\psi}(x')B(x',x)\psi(x)}\...
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3answers
506 views

Evaluation of functional determinants

Consider the evaluation of the following functional determinant: $$\text{log}\ \text{det}\ (\partial^{2}+m^{2})$$ $$=\text{Tr}\ \text{log}\ (\partial^{2}+m^{2})$$ $$= \sum\limits_{k} \text{log}\ (-...
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1answer
530 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) \...
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1answer
347 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 ...
5
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2answers
459 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 ...
5
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2answers
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 ...