Questions tagged [regularization]

In QFT, regularization is a method of addressing divergent expressions by introducing an arbitrary regulator, such as a minimal distance *ϵ* in space, or maximal energy *Λ*. While the physical divergent result is obtained in the limit in which the regulator goes away, *ϵ* → 0 or *Λ* → ∞, the regularized result is finite, allowing comparison and combination of results as functions of *ϵ, Λ*. Use for dimensional regularization as well.

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Gauss' law in the presence of surface charges [duplicate]

Assume $V$ is a volume such that $\rho=0$ in $V$ where $\rho$ is the charge density. Assume further that we have a surface charge density $\sigma$ on the surface $S$ enclosing $V$ such that the total ...
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Charge density of Hertzian dipole

I am wanting to find the charge density of an infinitely thin Hertzian dipole, but am struggling evaluating the Dirac delta functions gradient. $$\vec{J} = I_{0}\cos(\omega t) \delta^3(r) \hat k.$$ ...
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Why does normal-ordering ensure finiteness?

I will be using Jan von Delft's rigorous construction of bosonization/refermionization as an example, but I will try to explain my question in more general terms. Consider an (countably) infinite-dim (...
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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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First example of regularization [migrated]

Background: I like to think of L'Hospital as one of the earliest authors of least-squares regression. L'Hospital, G. (1696). L'analyse des infiniment petits pour l'intelligence des lignes courbes. I'm ...
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The conditions for a shift in a loop momentum to be allowed

When we evaluate the Feynman diagram containing a loop, we commonly use the identity: \begin{align} \frac{1}{A_{1}^{m_{1}} A_{2}^{m_{2}} \cdots A_{n}^{m_{n}}}= \int_{0}^{1} d x_{1} \cdots d x_{n} \...
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Path integral with double integration involving the free particle case

Suppose we have the path integral: \begin{equation} Z=\int \mathcal{D}x\mathcal{D}y\,\exp\left[-\frac{a}{2}\int_0^1 dt\,\left(\,\dot{x}(t)^2-\,\dot{y}(t)^2\right)\right]. \end{equation} The ...
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How does one regularize Wightman function?

Can anyone share good resources for learning the process of regularisation in the context of calculating the Wightman function in the response function of an Unruh-DeWitt detector?
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Probability for scattering event

I am reading Schwartz QFT. On page 61 in eq (5.20) he gives an expression that describes the probability for a $2\to n$ scattering event to happen: $$dP=\frac{T}{V}\frac{1}{(2E_1)(2 E_2)}\left|\...
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Regularization of a sum

I am computing the partition function of a gluon gas and I have been incurred in the following sum $$ \sum_{k=0}^\infty (2k+1)^4\ln(2k+1). $$ It is clearly divergent. Is there any possible way to ...
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Water wave analog of Casimir effect — Does it involve the zeta function? If not, why do QED calculations involve the zeta function?

It is known that the Casimir effect has a water wave analog from classical wave theory. See also this video for a demonstration. In what ways are the calculations for the effect in QED and the effect ...
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Fourier transform of Wick rotated functions

I am learning the imaginary time formalism of thermal field theory / reviewing the Euclidean formalism of quantum field theory. One thing that appears to be left implicit in many treatments is a ...
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What is the Cauchy principal value in Sokhotski-Plemelj formula?

I met the Sokhotski-Plemelj formula in a paper: in which $P$ is the Cauchy principal value. But the principal value in wiki is this form: It is a limit of an integration. But the $P$ in the first ...
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Slicing momentum cutoff

Consider the following propagator: $$C_{\kappa}(x-y) = \frac{1}{(2\pi)^{2}}\int dp \frac{-i\not{p}+m^{2}}{p^{2}+m^{2}}\chi_{\kappa}(p)e^{ip(x-y)}$$ where $\chi_{\kappa}$ is a cutoff function. If we ...
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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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22 votes
2 answers
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I'm missing the point of renormalization in QFT

I am a qft noob studying from Quantum Field Theory: An Integrated Approach by Fradkin, and in section 13 it discusses the one loop corrections to the effective potential $$U_1[\Phi] = \sum^\infty_{N=1}...
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1 answer
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Functional derivative for $J[f]=\int [f(y)]^p \phi(y)dy$

In QFT for gifted amateur pg. 13, the functional derivative for the functional $$J[f]=\int [f(y)]^p \phi(y)dy$$ is given by $$\frac{\delta J[f]}{\delta f(x)}= \lim_{\epsilon\rightarrow0} \frac{1}{\...
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1 answer
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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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Why is it a problem if the branch point merges with the on-shell pole?

One type of infrared divergence that can still happen even if one calculates the right physical quantity (integrating over soft photons and treating collinear divergences, for instance) is a ...
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Are constraint forces infinite?

A lot of authors claim that mechanical constraints are idealizations obtained by allowing enforcing forces to be infinite. But I either disagree or don't know what they mean. The only case where I ...
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Infinite time limit in two-point correlation function

I am reading the derivation of the two-point correlation function in Peskin and Schroeder (section 4.2). I don't understand the infinite time limit that is taken between eq. (4.26) and (4.27). They ...
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1 answer
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The regularization mass in Fujikawa's calculation of axial anomaly

I am reading Fujikawa's paper for axial anomaly: https://doi.org/10.1103/PhysRevD.21.2848 In equation (2.15), the anomalous part of axial transformation is regularized by $$\begin{align*} \mathcal{A}(...
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2 votes
1 answer
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What justifies regularization with a high-momentum cutoff?

Before renormalizing a perturbative series, during the regularization step when we insert a high-momentum UV cutoff, what justifies this step given that it's only formal and does not have a physical ...
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Multiplying distributions for QFT

My understanding is that UV divergences arise due to improperly handling the product of distributions. In what sense is it "improper"? And how does its proper handling relate to the notion ...
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Why doesn't the gluon fusion channel for Higgs production diverge?

The heavy quark triangle integral has a superficial degree of divergence of $D=1$, so one would naively expect it to diverge (more than logarithmically, in fact) in 4 dimensions. It happens, though, ...
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Could the divergences in loop Feynman diagrams be resolved by applying an exponentially decreasing probability eg fluctuation theorem?

Since the divergence originates from having to integrate over an infinite range of intermediate energies, surely the bigger the temporary energy delta, the less likely it will be, and the shorter the ...
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11 votes
1 answer
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What does mathematical consistency in QFT mean?

My question is more naive than Is QFT mathematically self-consistent? Just when people talk about the mathematical consistency of QFT, what does consistency mean? Do people want to fit QFT into ZFC? ...
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1 answer
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Negative potential infinite square well

A 1D finite square well is generally defined either by \begin{equation} V(x)=\begin{cases} 0\qquad -a\leqslant x\leqslant a\\ V_0\qquad \text{otherwise} \end{cases} \tag{1} \end{equation} or \begin{...
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QFT and divergences: what makes the finite part be regularization-independent?

It seems that the "finite part" of divergent loop integrals are the same, irrelevant of the regularization scheme used to regulate the integrals - why is this? Consider the following ...
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4 votes
2 answers
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Equal-time 2-point correlator and divergent (?) integral

Assume a massless scalar field in 3+1 dimensions which can be written as $$ \phi(t,\vec{x})=\int\frac{d^3k}{\sqrt{(2\pi)^32k}}\left(a(\vec{k})e^{-ikx}+a^\dagger(\vec{k})e^{ikx}\right)\, , $$ where $\...
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Question on Dirac delta and Gauss's law

Suppose you have something that looks like an inverse square law, i.e. $\textbf{v} = \frac{1}{r^2}\textbf{1}_r$. We know that it's divergence is 0 except in the origin. We also know that for any ...
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4 votes
0 answers
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Point splitting in bosonization

I was following two lecture notes on bosonization: https://arxiv.org/abs/cond-mat/9805275 and https://stanford.idm.oclc.org/login?url=https://www.worldscientific.com/doi/10.1142/9789814447027_0006 I ...
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3 votes
4 answers
902 views

Dirac delta function defined in Zee's Quantum Field Theory book

This is from Appendix 1 of the first chapter of Zee's Quantum Field Theory in a Nutshell: I am not sure whether it is correct to call this the Dirac delta function. Sure, the integral over all space ...
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3 votes
1 answer
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In what sense is the word "quantum fluctuation" used here?

I found this paper: On the origin of the LIGO "mystery" noise and the high energy particle physics desert, currently only published on arXiv as far as I can tell. I do not understand any of ...
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1 vote
1 answer
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On the physical interpretation of Dirac delta distribution

The original purpose of Dirac was to make up the eigenstate of the position operator $\hat X$. Now, quantum states are complex-valued functions in the Schwartz space $\mathscr{S}(\mathbb{R}^n)$ The ...
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References for computing mass diagrams

I'm currently attempting to compute the amplitude of the shown diagram following as a guide Peskin's calculations of the electron self-energy. The problem is that in the given result of this diagram: ...
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Existence of Weyl invariant regulator for bosonic string theory

In sec $(3.4)$ Polchinksi says It is easy to preserve the diff- and Poincare invariances in the quantum theory. For example, one may define the gauge fixed path integral using a Pauli-Villars ...
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How can we show the Lorentz symmetry is not anomalous in $\phi^{4}$ theory?

how can I show in a lagrangian with scalar fields and $\phi^{4}$ interaction, the energy-momentum tensor isn't anomalous?
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Free Boson on Torus - Partition function

I am currently reading Section 10.2 of the Yellow book by Francesco et al. They have shown in Equation 10.16 that \begin{equation} Z_{bos}(\tau) = \sqrt{A} \ \prod_{n} \left (\frac {2\pi}{\lambda_n}\...
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Loop integral in $d$ dimensions

I am studying large $N$ Quantum Field Theory and I am having a hard time calculating the expansion of the simple loop integral of eq.(13.123) of Peskin and Schroeder. $$ \int\frac{d^dk}{(2\pi)^d}\frac{...
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2 votes
1 answer
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Peskin and Schroeder Page no. 95 Feynman Diagrams

From Peskin and Schroeder Page no. 95, ... First, what happened to the large time $T$ that was taken to $\infty(1- i\epsilon)$? We glossed overit completely in this section, starting with Eq. (4.43). ...
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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 ...
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Fulling-DeWitt point-splitting

I'm reading Radiation from a Moving Mirror in Two Dimensional Space-Time: Conformal Anomaly by Davies and Fulling [1]. In order to compute the energy tensor elements they used this "point ...
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Equivalence between wilsonian and non-wilsonian RGE on QFT

My current way of viewing Wilsonian RGE applied to QFT: (1) We start with a lagrangian that accurately models dynamics up to a scale $\Lambda_0$. (2) We fix $\Lambda_0$ as a cutoff to regulate the ...
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1 vote
2 answers
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How do the counterterms in QED cancel the infinities?

I'm having a hard time understanding how the cancellation of the infinities works in QED at 1 loop level. I think the aim of this procedure is to obtain a Lagrangian such that all the diagrams that ...
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Proof that different integral prescriptions lead to different scattering matrices?

So I have read that if you calculate feynman diagrams with an $i\epsilon$ prescription, that is with propagators of the form $\frac{i}{p^2-m^2+i\epsilon}$ then you get the standard $\mathcal{iM}$ ...
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1 answer
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What type of regulation is being employed?

As already mentioned in this post. In the context of QFT, the kernel of integration for the overlap of a field configuration ket, $| \Phi \rangle$ with the vacuum $|0\rangle$ in a free theory is given ...
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6 votes
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Renormalization Group and Dimensional Regularization

Currently I am learning about regularization, renormalization and the renormalization group. In particular, a lot of detail is devoted to dimensional regularization. There are a couple of things I ...
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1 vote
2 answers
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An algebra step in the Quantum Partition Function for the Harmonic Oscillator

On page 183 of Altland Simons, we are told: $$ \prod_{n = 1}^{\infty} \Big[ \Big( \frac{2\pi n}{\beta} \Big)^2 + \omega^2 \Big]^{-1} \sim \prod_{n = 1}^{\infty} \Big[ 1 + \Big( \frac{\beta \omega}{2\...
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2 votes
2 answers
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Dimensional Regularization in Three Dimensions

I am looking at the Thirring model in three dimensions, which is non-renormalizeable. I was trying to calculate the one loop self energy of the fermion to see where the infinities crop up that cannot ...
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