# 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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### Renormalization group equation and method of characteristics

All of this question refers to ref. 1. The equation are numbered alike. The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage ...
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### Problem in calculating the one-loop contribution to the self–energy of $\phi$ field

Consider the theory $$\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi\right)^2-\frac{m^2}{2} \phi^2-\frac{g}{3 !} \phi^3-\frac{\lambda}{4 !} \phi^4 .$$ Find the expression for the self-energy and the ...
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### Pauli-Villars regularization and self-energy

In the calculation of the electron self-energy in QED (one-loop level), there is a UV and IR divergent integral that needs to be regularized. A common choice for the regularization is the Pauli-...
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### While calculating self-energy or work required to assemble a continuous charge distribution, do we take the energy of an element with itself?

while calculating the self-energy of a continuous charge distribution using the formula the potential $V$ here is due to the whole charge distribution but we need potential due complete charge ...
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### Does there exist QFT which do not yield infinities? [closed]

Does there exist a "simple", "polynomial-like" quantum field theory (unrealistic, not describing our world, in any spacetime dimension $d$) which does not yield infinities in ...
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### Is the $i\varepsilon$ prescription in the Feynman propagator just as "outrageous" as $1+2+3+... = -1/12$?

When the calculation of the Feynman propagator is introduced in QFT (or QM), the inclusion of the $i\varepsilon$ term is often described as a minor technical detail that is there just to "make ...
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### How to choose an appropriate value for the regularization $\eta$ in correlation functions in linear response for numerical calculations?

TL;DR How to choose an appropriate value for the regularization $\eta$ in correlation functions used in linear response for a discretized Brillouin zone? For more context, please see below. ...
1 vote
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### What is lattice regularization and how is it carried out? [duplicate]

I am new to QFT, and so far I have studied dimensional regularization and Pauli-Villars regularization. These seem to be the only two regularization techniques discussed in most introductory textbooks....
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### Calculating a rectangular Wilson loop for the free photon

I'm studying Creutz's Quarks, gluons and lattices, in chapter 6 on page 33, we have the following exercise Calculate a rectangular Wilson loop for the field theory of free photons. Using any ...
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### 3-point counterterm in scalar QED

I'm working on the 1-loop corrections to scalar QED. I'm using dimensional regularization and on-shell regularization. In trying to compute the counterterm for the 3-point graph I come across the ...
1 vote
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### On-shell renormalization (Schwartz Quantum Field Theory Equation (18.48))

I have a question about how, in section 18.3.2 in Schwartz's quantum field theory, he goes from equation (18.47) to (18.48) using Pauli-Villars regularization. It comes down to showing that to leading ...
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### Regularization scheme independence as a gauge redundancy?

Observables should not depend on the regularization scheme under some renormalization procedure. Is there some way to interpret this fact as a gauge redundancy? In particular, is there some group ...
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### How to show Pauli-Villars regularization introduce a momentum cut-off?

Pauli-Villars regularization instructs us to do such a replacement: $$\frac{1}{p^2-m^2+i\epsilon} \rightarrow \frac{1}{p^2-m^2+i\epsilon} - \frac{1}{p^2-\Lambda^2+i\epsilon}$$ And then claim: such a ...
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### Is it legitimate to use analytic continuation to equate a diverging series with a finite number in a physical theory of nature?

Analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite ...
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### Why is QFT not even unitary prior to renormalisation?

The Hamiltonian is Hermitian. That should've been enough to make it unitary. But infinite amplitudes mean it's not even unitary. One could say that this is because we're dealing with a crazy Hilbert ...
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### Finiteness of quantum mechanics

In most standard undergraduate treatments of quantum mechanics, there is rarely any need to treat divergences in perturbation theory, other than the Casimir energy perhaps? The subtleties of ...
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Weinberg obtains differential scattering rate for a process with soft photons, \begin{align*} d \Gamma_{\beta \alpha}^{\lambda}(\omega_1, ...\omega_N) = \Gamma^\lambda_{\beta \alpha} A(\... 1 vote 1 answer 76 views ### Fourier transform of propagator in Lehmann spectral representation I need help showing that: iG(k,t)=\int_{\mu/\hbar}^{+\infty} e^{-i\omega t}A(k,\omega)d\omega $$knowing that$$ G(k,\omega) = \int_{-\infty}^{+\infty} \frac{A(k,\omega')}{\omega -\omega'+i\eta\ ... 1 vote
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