# 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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### Regularization scheme independence in QFT

I know there are a few similar questions on the topic (1,2) , but I still feel they do not fully answer my questions (correct me if I am wrong!). What I am asking is a clarification on the commonly ...
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### What does the cut-off $\Lambda$ stand for in the theory of QED?

The bare electron mass $m_0$, in QED, changes as $$m_0\to m=m_0+\delta m\Big(\frac{\Lambda}{E}\Big)$$ where high momentum modes from $E$ to $\Lambda$ has been integrated out. What scale does the cut-...
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### Are the field renormalization factors infinite or finite?

We know that in quantum field theory we include infinities at each order of the perturbative expansion of the renormalization $Z$ factors about the coupling constant $\lambda$ to absorb the ...
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### How to remove the divergent part of loop integrals when employing the cuttoff procedure?

On page 130 of "A Modern Introduction to Quantum Field Theory" by Michele Maggiore they evaluate a divergent four dimensional integral writing: "We introduce a cuttoff stating that we integrate ...
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In Section 7.1 in Peskin and Schroeder, (pp. 270), the first order correction to the electron mass is calculated. They define (eq. 7.24) the physical mass $m$ of the electron as the solution to, $$[p\!... 1answer 456 views ### What is the logic behind box normalization and periodic boundary condition? Free particle energy eigenfunctions are A\exp{[i(Et-\textbf{p}\cdot\textbf{r})/\hbar]} are non-normalizable. To normalize them one introduces a procedure called 'box normalization' where one imposes ... 0answers 145 views ### Are there fundamental differences between finite and infinite systems? Most sources on classical field theory introduce classical fields as a limit of a system with N particles constrained in some way in a lattice where a continuum limit involving N, lattice size and ... 1answer 612 views ### Point splitting regularization for polynomials of operators Point-splitting regularization in quantum field theory uses the fact, that UV-divergences occurring in expressions of the type \left< \phi \left( x \right) \phi \left( x \right) \right> can be ... 1answer 384 views ### What is a point-split? I encountered the term point-split [1] several times and would like to know what this concept is all about. From my understanding, a point is splitted by adding ε and -ε to a local point x ... 2answers 836 views ### Is the fact that the sum of all natural numbers \sum_{n=1}^\infty n = -\frac{1}{12} essential to the understanding of the Casimir Force In QED? Apparently this result is used in many areas physics including the extra dimensions of string theory, which is not the scope of the question. The result is apparently also used to understand the ... 2answers 531 views ### Possible divergence structures of a renormalizable and non-renormalizable theory If a theory has a coupling with negative mass dimension, it will require an infinite number of counterterms. This is because the theory will have infinitely many divergence structures. To be concrete,... 1answer 101 views ### Physical understanding of the regularization of benign infinities in free field theories Any continuum quantum field theory (QFT), free or interacting, has uncountably infinite number of degrees of freedom in spacetime. Does it have anything to do with the appearance of infinities in ... 1answer 336 views ### Why can consistent QFTs only arise from CFTs? This is claimed by Jared Kaplan in his Lectures on AdS/CFT from the Bottom Up. He writes: It seems that all QFTs can be viewed as points along an Renormalization Flow (or RG flow, this is the ... 0answers 219 views ### How to properly truncate an infinite-dimensional Hilbert space for quantum optics simulations? In order to solve numerically a master equation in which the Hamiltonian and the jump operators are defined in terms of the infinite-dimensional annihilation and creation operators, we have to ... 0answers 205 views ### Renormalization and Infinite Series In the renormalization of QED, one subtracts infinities to get finite results. This is not possible in gravity because of infinitely many divergences, and infinitely many counter terms. The general ... 2answers 540 views ### Why is there no anomaly when particle mechanics is quantized? We know that if one or more symmetries of the action of a classical field theory is violated in its quantized version the corresponding quantum theory is said to have anomaly. Is this a sole feature ... 1answer 956 views ### Dimensional Regularization and Massless Integrals Consider the following integral: $$\int \frac{d^3 k}{(2\pi)^3} \frac{1}{k^3}$$ In dim-reg, such integrals evaluate to 0. However, if we instead consider \begin{... 0answers 70 views ### Taylor expansion of Ei(x) I'm reading a note on regularization by Muruyama, link http://hitoshi.berkeley.edu/230A/regularization.pdf On the bottom of page 2, Muruyama Taylor expanded$$ -\frac{e^{m^2/\Lambda^2}}{4\pi} \...
In light of the not so well defined integral $\int_a^b \delta(x-a) dx$ and from David Z's comment at the end of this Math.SE post, consider the following equation, which I've come across in many ...