Questions tagged [functional-determinants]

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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: $$S=-\int d^4x \sqrt{-g}\phi^\ast(\square_g+m^2) \phi$$ 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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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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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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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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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 ...
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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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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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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 O =\begin{pmatrix} a(t) & \frac{d}{dt} +b(t) & c(t) \\ \frac{d}...
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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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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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I am looking to evaluate the following path integral: $$I=\int_{\vec x(t_0)=0}^{\vec x(t)=0}\mathcal{D}x \exp\left( -\frac{1}{2} \int^t_{t_0} d\tau \; \vec x \left\{ \frac{\omega^2-\partial_t^2}{2D} \... 2answers 1k views Determinant of Dirac operator in flat space? How would you evaluate $$|iD\!\!\!\!/-m|$$ Where D_{\mu}=\partial_{\mu}-ieA_{\mu}. I have an idea of how to do this without the gauge field, because it's essentially \... 2answers 1k views How to determine the trace and determinant of a differential operator? How to determine the trace and determinant of the operator like \Box or \nabla^2 etc. But first of all how to find the same for the simpler operator \frac{d}{dx}? I proceeded as follows. What ... 0answers 192 views Auxiliary field path integral in non-linear sigma models I am trying to understand the functional integral over the auxiliary field in the \mathcal{N}=(2,2) supersymmetric non-linear sigma model, or NLSM (reviewed in Chapter 13 of Mirror Symmetry http://... 1answer 942 views What is the status of Witten's and Vafa's argument that the QCD vacuum energy is a minimum for zero \theta angle? The argument, which I reproduce here from Ramond's `Journies BSM', is originally by Witten and Vafa in (\it{Phys}. \it{Rev}. \it{Lett}. 53, 535(1984)). The argument is that for \theta = 0  (... 0answers 185 views Faddeev-Popov-Determinant of Polyakov Path Integral I'm currently trying to understand the paper "Quantum Geometry of bosonic Strings" by Polyakov. I think I roughly understand the X integration, but when it comes to the integration over the metric ... 0answers 249 views A question about Gel'fand-Yaglom method of calculating functional determinants I know that the Gel'fand-Yaglom method is a way to calculate determinants of 1D differential operators. For instance, let us consider an operator -\partial_r^2+W(r). Let us define \psi_{0}(r) as ... 1answer 621 views Determinant as a fermionic path integral I understand that the determinant of a matrix can be written in terms of a fermionic path integral. The expression is:$$Z = \int D\bar{\psi}D\psi e^{-\iint d^4x' d^4x \bar{\psi}(x')B(x',x)\psi(x)}\...
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 ...