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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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Zee's explanation of expressing bare coupling by physical coupling

In terms of bare parameter $\lambda$, the $\phi\phi\to\phi\phi$ scattering amplitude is $\lambda\phi^4$ theory is given by $$\mathcal{M}=-i\lambda+iC\lambda^2\Big[\ln\Big(\frac{\Lambda^2}{s}\Big)+\ln\...
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UV divergence integral

Could anyone please explain how to calculate integral such as $$\frac{\Omega}{2}\int_{-\infty}^{+\infty} \frac{d^3k}{(2\pi)^3}\ln\left[{1+\frac{a^2}{k^2}}\right]=-\frac{\Omega a^3}{12\pi}+I_0~?$$ ...
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Quantum expressions for the Virasoro constraints

I am trying to derive the quantum form of the Virasoro constraints. $$ L_{m} = \frac{1}{2} \sum_{n} :\alpha_{m-n}.\alpha_{n}: $$ :...: meaans normal ordering. Using the common commutator between ...
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Dimensional regularization and expansion of gamma function

In my calculations, I used dimensional regularization, i.e. replace $d\rightarrow d-\epsilon$ and calculated the divergent integral. Then, I would like to expand the answer into seriers by $\epsilon$ ...
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Feynman $i\epsilon$-prescription for fermion propagator via path integrals

In Section 9.4 of S. Weinberg's book "The quantum theory of fields" it is shown how to get the Feynman $i\epsilon$-prescription in the propagator of a free scalar field using path integrals and ...
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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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Divergent Coulomb integrals in superfluid fluctuations

In Chapter 3 of Kardar's statistical physics of fields, in the context of lower critical dimension, he works out an example about superfluids where starting from the probablity of a particular ...
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Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...
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Should the parallel propagator appear in the point-split stress-energy tensor?

The first step in Hadamard regularization of the stress-energy tensor of a free Dirac field is to write out the point-split expression $$\langle T_{\mu \nu} \rangle \equiv \frac{1}{4} \lim_{x'\to x} \...
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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)&...
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Feynman $i\varepsilon$-prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i\varepsilon$-prescription for the field propagator. I've found many ways of showing this in the literature, but it ...
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Subtraction scheme invariance in QFT

I'm currently reading Schwartz's QFT text and I'm confused on how observables are supposed to be independent of subtraction schemes. In the text it seems that the renormalized loop diagrams are ...
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Second Order Correction to the Perturbative Approximation of the Transition Amplitudes in RQM

I am studying Relativistic Quantum Mechanics from my professor's notes. When calculating the second order perturbative correction to the transition coefficient $T_{fi}$* in a scattering process by a ...
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For dimensional regularization, why the arbitrary mass scale $\mu$ has the meaning of UV cutoff?

For sharp cut off regularization, we introduce the UV cutoff $\Lambda$. When we need to do momentum integral, we integrate the momentum ball with radius of $\Lambda$. This $\Lambda$ has the explicit ...
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Interaction term in free energy for Gaussian Fixed Point

In general in statistical field theory, the free energy $F_0$ as a function of our order parameter $\phi$ can be written as $$F_0[\phi]=F_0[\phi^-]+F_0[\phi^+]+F_I[\phi^-,\phi^+]$$ where the last ...
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Regularization: Evaluating the one-loop $\phi^4$ integral to order $\lambda^2$

I'm currently on the chapter of regularization on Zee's QFT book. For the $\phi^4$ theory, an amplitude for a single loop correction to order $\lambda^2$ is given by a diagram Following the Feynman ...
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Normal ordering in 2D thermal CFT

I am trying to understand the notion of normal ordering in thermal CFT in 2D CFT, for instance I consider a two-point function of scalar primary operator with $\Delta$ dimension at finite temperature $...
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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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Value of regularization scale

After regularising a transition amplitude, we end up with an expression which depends on regularisation scale. This means that our physical observables like cross section will be a function of ...
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Are smearing functions in QFT operator independent?

For a scalar field $\hat{\phi}(x)$, the smearing is performed by convoluting the operator $\hat{\phi}(x)$ with a smooth function $f(x')$ which has support in the neighborhood of the point $x$. Is ...
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Energy differentiation with cut-off function

I am a new learner of molecular dynamics (MD) simulations methods and has a simple question regarding handling of cutoff functions. In MD, pairwise energy between two atoms is assumed to be a function,...
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Arranging coefficients in a derivation of the Casimir energy

I'm working on the derivation of the Casimir energy from quantum field theory. From the K-G equation (with $c=1$ and $\hbar=1)$ I found the vacuum energy: $$\langle 0|H|0\rangle=E_{vac}=V\int_{-\...
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In charge renormalization equation, $e=e_{0}^{2}\left[1-e_{0}^{2}A\right]$, how can an infinite $e_{0}$ and $A$ give finite $e$ in any limit?

In Griffiths elementary particle book (chapter 7, 'Quantum electrodynamics', equation 7.188), one gets the following equation for the vacuum polarization calculated to one loop correction. $$\frac{e_{...
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Trouble Understanding Computation in Weinberg Quantum Theory of Fields Vol. 2 Chapter 22

In Chapter 22 (Anomalies) of Weinberg Vol. 2, the author is evaluating the anomaly function $\mathcal{A}(x) = -2[Tr(\gamma_5 t f(-(\not{D}/M)^2))\delta(x-y)]_{y\rightarrow x}$, following Fujikawa'...
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Green's function regularization and delta distribution

I have a free Green's function which is proportional to a $2\times 2$ matrix: $$ G_0 = \frac{1}{E^2-E_k^2}\begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ The total Green's function after ...
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Schwartz's and Zee's proof of Goldstone theorem

In Refs. 1 & 2 the Goldstone theorem is proven with a rather short proof which I paraphrase as follows. Proof: Let $Q$ be a generator of the symmetry. Then $[H, Q] = 0$ and we want to consider ...
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Why can we ignore infinite constant terms that come from constant terms from the Lagrangian?

This is a follow up or better an edit to my previous question that was marked as a duplicate of this other question. I think I failed to emphasize what I really wanted. The tittle of my question was, ...
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Shall we skip explicit regularization in the process of renormalization?

In the process of renormalization, regularization is usually cited as indispensable in taming infinities encountered in quantum field theory. Is explicit regularization really necessary? Let's take ...
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Renormalization of Harmonic Oscillator

In Appendix A, Polchinski does the Euclidean path integral for the Harmonic oscillator. After he Pauli-Villars regularizes the determinant of the kinetic term, he obtains the following expression (A.1....
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Is the thermal expectation value of a square of Hermitian operator always finite?

If $\mathcal{O}$ is an hermitian operator in a system given by Hamiltonian $H$ and inverse temperature $\beta$, is $$\langle \mathcal{O} \mathcal{O} \rangle = Tr (e^{-\beta H} \mathcal{O} \mathcal{O})...
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Why are dimReg divergences power-like? Or are they?

An implicit assumption when working with dimensional regularisation is that the divergences are always of the form $\varepsilon^{-n}$ for some integer $n$ (e.g. refs.1&2). Is there any way to ...
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Divergent integrals in QFT

I am starting to learn about QFT and something that I noticed is that integrals who would diverge otherwise are assigned a value if we do it by contour integration using the residues theorem and the ...
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How can Weinberg assume that $P_b$ acts as derivative?

In QM of finitely many degrees of freedom it is well known that due to the Stone-Von Neumann theorem, the CCR $$[Q_i,P_j]=i \delta_{ij} $$ leads to a unique representation up to unitary equivalence, ...
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QFT Loop Integration: Relating Momentum-Space Cutoffs to Position-Space Cutoffs

Consider the following loop integral which appears in real, scalar $\phi^{4}$ theory: $$ \mathcal{L} = \int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2+m^2} $$ For example, it appears when computing the first-...
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How does the renormalization group justify the renomalization process?

I recently learned "Renormalization" and "RG". My textbook says "RG allows us to make sense of why a renormalized quantum field theory describe Nature." To me, it sounds like "RG justifies the ...
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Mermin-Wagner and Heisenberg spin chains

The Hamiltonian for the spin 1/2 ferromagnetic Heisenberg spin chain is $H=-J\sum_i \vec \sigma_i \cdot \vec\sigma_{i+1}$ with $J>0$ and $\vec\sigma_i$ the Pauli matrices acting on ith lattice site....
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Physical significance of sum of Grandi's series [closed]

I watched a video of numberphile in which they explain that how you can get Grandi's series sum as $1/2$ ( by Cesàro summation). Then they also give one example of flipping of bulb $1$ means turn on ...
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Can cut-off regularisation cause a Poincaré anomaly?

Momentum cut-off regularisation leads to non-covariant results, i.e., it breaks the Poincaré covariance of the theory. Is there any guarantee that Poincaré covariance is always restored when we remove ...
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Is Wick rotation of loop integrals legitimate?

In Feynman diagram calculations, we seem to invariably Euclideanise loop integrals in order to exploit the resulting spherical symmetry. This Wick rotation is simply a deformation of the contour; ...
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What is the propagator replacement exactly in Pauli Villars Regularization?

The Pauli Villars regularization involves replacing every propagator in a divergent diagram by a "subtracted propagator", where we subtract a fictitious, heavy particle propagator from the original ...
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Does the angular measure matter in dimensional regularization?

In dimensional regularization, we replace a momentum integral $I= \int d^nk f(|k|)$ with the family of regularized integrals $$\mu^{n-d}\int d^dk f(|k|) = \mu^{\epsilon}\Omega_d \int p^{d-1} f(p)dp.\...
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Can dimensional regularization be viewed as a soft version of a Wilsonian cutoff?

In the Wilsonian picture of renormalization, a quantum field theory is defined to have degrees of freedom only up to an energy scale $\Lambda$. The results of low-energy experiments shouldn't change ...
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Basic cut-off regularization

I've been reading these notes on regularization by Hitoshi Murayama here, and on page 3 there's a few lines of calculations on a quick method of regularizing an integral. But I can't follow the steps ...
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Reproducing Ramond's sunset diagram calculation for $\phi^4$ theory

I am unable to reproduce the calculation of the sunset diagram for $\phi^4$ theory in Pierre Ramond's Fied Theory a Modern Primer. This is the second edition chapter 4.4. He starts with eq. (4.4.19) \...
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What cancels this tree level IR divergence?

Computing the amplitude squared for $e^-\mu^-\rightarrow e^-\mu^-$ at tree level we get \begin{equation} \frac{1}{4}\sum_\mathrm{spins}|\mathcal{M}(s,t)|^2=2e^4\frac{s^2+u^2}{t^2} \end{equation} which ...
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Help understanding loops with negative superficial degree of divergence

Consider $$\int \frac{d^4k}{(2\pi)^4} \frac{1}{k^2}\frac{1}{k^2}\frac{1}{k^2}.\tag{1}$$ We can Wick rotate $k_0 \to i k_0$: $$ i \int \frac{d^4k_E}{(2\pi)^4} \frac{1}{k_E^2}\frac{1}{k_E^2}\frac{1}...
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Free quantum field theories as fixed points of Wilsonian RG

Consider Euclidean Klein Gordon quantum field theory on the toroidal spacetime $X\simeq S^1\times \cdots\times S^1$, with action $$S(\varphi) = \int_X \varphi(\Delta+m^2)\varphi$$ and scalar field $\...
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Anomalies and short-distance divergences$.$

Let $J$ be a certain Noether current $$ J=J[\phi] $$ where $\phi$ is a field. This object is classically conserved, although in the quantum-mechanical case it may be anomalous. In the functional ...
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Integrals of the form $\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}}$ in $D=4-2\varepsilon$ dimensions?

In a massless theory we often get integrals of the form $$\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}} \tag{*}$$ where $D=4-2\varepsilon$. I have tried to calculate this in two ways in Minkowski space ...
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Wick rotations, convergence and Feynman propagators?

It is said (in e.g. Hawking, 1979, Euclidean quantum gravity) that the integral: $$ \int \mathcal{D}\phi \exp(iS[\phi])\tag{1} $$ for real fields in Minkowski space does not converge, but the Wick ...