Questions tagged [functional-determinants]
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98
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Fourier transform of the Gaussian action for the real scalar bosonic field
In my current homework, we have to get familiar with quadratic theory in order to reach $\phi^4$-theory. So the starting point is
$$Z = \int Dx e^{-S[\phi]}$$
with the action for the real scalar ...
- 1
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Instantons and Spontaneous Symmetry Breaking
I'm following an introductory lecture on instantons by Hilmar Forkel. In a non-relativistic quantum mechanical setting we have the potential $$ V(x) = \dfrac{\alpha^2 m}{2 x_0^2} (x^2 - x_0^2)^2 \tag{...
- 1,611
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How do Dedekind's eta function arise while computing the partition function of a compact scalar field over circle?
I am following the book String Theory in a nutshell (From Elias Kiritsis). In chapter 4.18, it takes a theory following the action:
$$S=\frac{1}{4\pi l_s^2}\int X\square X\ d\sigma,\tag{4.18.1}$$
$$ \...
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Calculation of the Effective action - Lewis H. Ryder
I have been studying the book on Quantum Field Theory by Lewis H. Ryder and I am finding a Gaussian integration a little bit confusing. In the book, the transition amplitude (Eq. $(5.15)$) is given as ...
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How to do the Gaussian $p$ integration in path integrals?
I'm trying to solve an exercise on path integrals, in which I have to move from a path integral in phase space
$$
\int \mathcal{D}q \dfrac{\mathcal{D}p}{\hbar} \exp \left(\dfrac{i}{\hbar} \int dt\ (p\...
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How to integrate a Gaussian path integral of free particle using zeta function regularization?
I am attempting to integrate this path integral in Euclidean variable $\tau $ (but this need not be the same as the $X^0$ field):
$$Z=\int _{X(0)=x}^{X(i)=x'}DX\exp \left(-\int _0^i d\tau \left[\frac{...
- 182
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Multiplicative property of the functional determinant
If we consider two differential operators $\mathcal{D}_1$ and $\mathcal{D}_2$, we can compose them to create the differential operator $\mathcal{D}_1 \mathcal{D}_2$. Then we could consider an action (...
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How to integrate out the Goldstone phase in effective Ginzburg–Landau (GL) action for BCS?
In page 293 of Altland and Simons' "Condensed Matter Field Theory", just above equation (6.38), in the process of deriving the London equations from the BCS path integral, the authors say, &...
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How to understand this field redefinition example from path integral formalism?
I'm studying the Lagrangian
$$
\mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi+\lambda\phi\partial_\mu\phi\partial^\mu\phi~=~\frac{1}{2}(1+2\lambda \phi)\partial_\mu\phi \partial^\mu\phi.\...
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How is the quantum effective action defined in a theory with more than one field?
How is the one-loop quantum effective action derived in a theory with more than one interacting field? When looking at some books and my course notes I find that the expression for the one-loop ...
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Jacobian functional matrix for fermionic path integral
I am revisiting Srednicki's book Chapter 77 and struggling to understand how you define the change of variables in the fermionic field integral
Srednicki defines the Jacobian functional matrix for the ...
- 143
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Spinor field path quantization
Although I have asked a similar question here, here, I find that I don't totally understand it, so I arrange my new ideas to this post.
Begin with Berezin integral:
$$\left(\prod_i \int d \theta_i^* d ...
- 1,421
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Spinor functional quantization unitarily equivalent and determinant
On P&S's qft page 301 and 302, the book discussed functional quantization of spinor field.
The book define a Grassmann field $\psi(x)$ in terms of any set of orthonormal basis functions:
\begin{...
- 1,421
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Getting units right when computing the effective action
In QFT in Euclidean signature, the one-loop effective action is given by
$$\Gamma[\Phi] = S[\Phi] + \frac{1}{2} \mathrm{STr}\log S^{(2)}, \tag{1}$$
where $S[\Phi]$ is the theory's classical action, $\...
- 22.1k
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Trace and determinants in QFT's
I'm trying to understand this paper: https://doi.org/10.1103/PhysRevA.46.6490. It's about path integration with defects (theories on submanifolds). Let me here try to explain what in particular I'm ...
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