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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 ...
Johnny_T's user avatar
3 votes
2 answers
127 views

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{...
Gabriel Ybarra Marcaida's user avatar
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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: $$S=\frac{1}{4\pi l_s^2}\int X\square X\ d\sigma,\tag{4.18.1}$$ $$ \...
R. Á. Candás's user avatar
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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 ...
Jack's user avatar
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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{...
Andrew Dynneson's user avatar
2 votes
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77 views

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 (...
E. Marc.'s user avatar
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92 views

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, &...
laura_legesen's user avatar
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1 answer
265 views

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.\...
IGY's user avatar
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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 ...
Ramon's user avatar
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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 ...
Cory's user avatar
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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 ...
Daren's user avatar
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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{...
Daren's user avatar
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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, $\...
Níckolas Alves's user avatar
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168 views

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 ...
A.Dunder's user avatar
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Problem with different expressions of functional determinant

This question is a follow-up of my previous one, after having done some calculations. In this previous question I used a minimal example of my problem with $\det(\Delta) = \det(\partial^2+A(x))$, but ...
Jeanbaptiste Roux's user avatar
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1 answer
231 views

Measure of Functional Integral in Path Integral Formulation

I have a question regarding the prefactor $\sqrt{\left(\frac{m}{2\pi i \hbar \Delta t}\right)}$ in $$\left<x'|e^{-iHt}|x\right> = \int D[x] \exp(\frac{i}{\hbar}\int dt' L(x, \dot{x})),$$ where $$...
Neophyte's user avatar
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Functional determinant: linking Series, Heat-Kernel and Zeta function

I would like to express a functional determinant as a series of diagrams, using the zeta function renormalization applied to the heat-kernel method, but I don't know if it's possible. Let me explain: ...
Jeanbaptiste Roux's user avatar
1 vote
1 answer
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Computation of functional determinant of Lorenz gauge

In the Peskin and Schroeder's book P295, there is a derivation I don't quiet understand. In Lorenz gauge we have $$G(A^\alpha)=\partial^\mu A_\mu+(1/e)\partial^2 \alpha.$$ Then it says that we could ...
David Shaw's user avatar
2 votes
1 answer
174 views

Path integral in a boundary QFT

I'm trying to compute the following path integral \begin{equation} Z = \int\mathcal{D}\phi\exp\left(-\int_{\mathbb{R}^d_+}\frac{d^dx}{2}\phi(-\partial_\mu^2 + m^2)\phi \right) \propto \frac{1}{\sqrt{\...
A.Dunder's user avatar
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Computing the functional integral over the gauge and ghost fields

In Peskin&Schroeder page 517 the authors mention that the functional integration of the gauge fields and ghost fields yields the following determinants $$ (\det[-\partial^2])^{-d/2}\cdot(\det[-\...
twisted manifold's user avatar
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1 answer
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Explicit derivation that the Faddeev-Popov functional determinant is gauge invariant

I am trying to show that the Faddeev-Popov functional determinant used in the quantisation of non-Abelian gauge theory is indeed gauge invariant. As shown in my previous question when we follow the ...
twisted manifold's user avatar
2 votes
2 answers
515 views

Why is the determinant not integrated over in Faddeev-Popov?

In Peskin & Schroeder chapter 16.2 the authors go through the computation of the non-Abelian gauge boson propagator using the Faddeev-Popov procedure as is done for the QED case. The difference ...
twisted manifold's user avatar
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Why does expressing the Faddeev-Popov determinant as this lead to such problems?

Background In the following, I am interested in the Schwinger function associated with the gluon propagator when one considers the Gribov no-pole condition in the partition function. Defining $\nabla^{...
Jeanbaptiste Roux's user avatar
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0 answers
108 views

Definition of a determinant Peskin&Schroeder

In page 514 of Peskin&Schroeder we are given the definition of a determinant as $$ \det\left(\frac{1}{g}\partial_\mu D^\mu\right)=\int{\cal{D}cD\bar{c}\exp\left[i\int {d^4x\bar{c}(-\partial^\mu D_\...
twisted manifold's user avatar
6 votes
0 answers
166 views

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 = \...
Dr. user44690's user avatar
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2 answers
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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 ...
Advaita's user avatar
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2 answers
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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 ...
David Shaw's user avatar
2 votes
0 answers
93 views

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}...
Kvothe's user avatar
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10 votes
2 answers
879 views

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{...
Ruben Campos Delgado's user avatar
3 votes
1 answer
1k views

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 ...
Marcosko's user avatar
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1 vote
1 answer
67 views

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 ...
Kvothe's user avatar
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1 vote
1 answer
269 views

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}\...
Ghorbalchov's user avatar
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0 votes
1 answer
283 views

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[(-\...
awsomeguy's user avatar
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73 views

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 ...
Jeanbaptiste Roux's user avatar
1 vote
1 answer
250 views

Matsubara sum of thermal Green's function

I need to retrieve a Matsubara sum representation of the thermal Green's function $$G_{ij}(\tau)=-\frac{1}{Z}\int \mathcal{D}(\overline{\psi},\psi)\psi_i(\tau)\overline{\psi}_j(0)\exp(-\sum_k\int_0^\...
Juri V's user avatar
  • 107
2 votes
1 answer
113 views

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)\...
Valentina's user avatar
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3 votes
1 answer
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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 ...
Dr. user44690's user avatar
1 vote
1 answer
121 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}...
Dr. user44690's user avatar
1 vote
1 answer
237 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 ...
Jeanbaptiste Roux's user avatar
1 vote
1 answer
105 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 ...
Zack's user avatar
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14 votes
0 answers
437 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)\...
Wein Eld's user avatar
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1 vote
0 answers
57 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*}...
Jeanbaptiste Roux's user avatar
1 vote
0 answers
37 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. $$\...
Anixx's user avatar
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1 vote
0 answers
66 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 ...
PvpRJ's user avatar
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3 votes
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\ ...
self.grassmanian's user avatar
2 votes
2 answers
484 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\{...
Nihar Karve's user avatar
  • 8,492
2 votes
2 answers
415 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 ...
sky's user avatar
  • 169
4 votes
1 answer
174 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}...
Sven2009's user avatar
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1 vote
1 answer
466 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 ...
Stoby's user avatar
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