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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158 views

Contributions to final states in the optical theorem

Consider the use of the optical theorem for computing the imaginary part of the total forward scattering amplitude $\mathcal M(AB \rightarrow AB)$. Then the theorem tells us to compute $$2 \text{Im} \...
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Gauge invariance, symmetries, and regularization

When regularizing integrals in a QFT with a gauge symmetry, many people state that it is important that the regulator also enjoys gauge invariance. Why is this true? What goes wrong when you use a ...
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Spinor vacuum energy

I'm reading the calculation in the book Quantum field theory in a nutshell of A. Zee of chaoter II.5 In this chapter the vacuum energy is calculated through the path integral approach. At some point ...
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342 views

How should we think of local counterterms in the context of anomalies?

Short version: effective actions, particularly ones obtained after integrating chiral fermions, are ambiguous up to the addition of local counterterms. Should we think of the counterterms as part of ...
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220 views

Can we obtain the Feynman diagrams using infinite series representation of a path integral?

While evaluating quantum amplitude of a particle using path integral approach, we deal with infinite number of paths that can usually lead to a divergent infinite series. We can then also obtain a ...
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187 views

Why do not renormalization group equations explicitly depend on cutoff?

Suppose $g$ is the parameter set and $\Lambda\equiv\Lambda_0e^{-t}$ the momentum cutoff, then usually one finds the renormalization group equations to take the form $$\frac{dg(t)}{dt}=\beta(g).$$ My ...
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Pauli-Villars Regularization

I'm considering a real scalar field theory with Lagrangian density $$-L=\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m_{0}^{2}\phi^{2}+\frac{\lambda_{0}\phi^{4}}{4!}$$ Let $m$ be the physical mass. I am ...
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308 views

Calculate the determinant of the Dirac operator

Suppose the Dirac operator $$ D = a_{\mu}\gamma^{\mu} + b_{\mu}\gamma^{\mu}\gamma_{5} - m $$ How to calculate the logarithm of its determinant $$ \text{ln}\left[\text{det}D\right]? $$ I think that for ...
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236 views

Symmetric limit in Peskin and Schroeder

I have a question on page 655 of Peskin and Schroeder. The second equation of (19.23) is discussed here. But the first equation of (19.23) is still a mystery. $$ \underset{\epsilon \to 0}{\text{symm ...
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159 views

Is there a physical reason behind zeta regularization?

Does there exist a reason why zeta regularization always works? I mean for the Casimir effect the series $ 1+8+27+64+\dots=1/120 $ gives the correct result. The same holds for $ 1+2+3+4+5+6+7+\dots =-...
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177 views

Integral from Lancaster's QFT

In Chapter 32 of Lancaster's Quantum Field Theory for the Gifted Amateur, renormalization is discussed. The amplitudes of various one-loop Feynman diagrams which are corrections to the vertex in a $\...
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Calculating the transition amplitude

Lagrangian of the Lee-Yang model is given by: $$ L=\frac{1}{2}f(q)\dot{q}^2 $$ where $f(q)$ is some differentiable function. I am trying to derive the following expression for the transition ...
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What is a logarithmic divergence?

I am reading about renormalisation in QED and I come across the term logarithmic divergence several times. Can somebody explain to me about it in simple terms?
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295 views

Evaluation of functional determinants

Consider the evaluation of the following functional determinant: $$\text{log}\ \text{det}\ (\partial^{2}+m^{2})$$ $$=\text{Tr}\ \text{log}\ (\partial^{2}+m^{2})$$ $$= \sum\limits_{k} \text{log}\ (-...
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Radiative correction to the charge form factor $F_1$ in QED

In QED, one can calculate the correction to the form factor $F_2$. To the lowest order, $F_1=1$ and $F_2=0$. At one loop, it is found that $F_2(0)$ receives a non-zero finite correction which is ...
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What are the observable effects of finite pieces of the loop corrections in QED?

I'm lost amidst the calculation of regularization and renormalization process in QED. In addition to the divergent piece in the in the self-energy correction (similarly in vacuum polarization ...
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A derivative about chiral current in Peskin's book

In Peskin's book (an introduction to QFT), Page 655, the axial vector current is defined as follows, \begin{eqnarray*} j^{\mu5} & = & \text{symm }\lim_{\epsilon\rightarrow0}\bigg\{\bar{\psi}(x+...
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Mass Renormalization: Geometric Series of One Particle Irreducible Diagrams

Pretty much everywhere I look it is stated that the full two point Green function (let's say for the Klein-Gordon field) is a geometric series in the one particle irreducible diagrams, ie. in momentum ...
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Why does a cutoff break gauge invariance?

It's been stated repeatedly that introducing a sharp momentum cutoff $\Lambda$ into a gauge theory breaks gauge invariance. Apparently, this is because momentum modes directly at the cutoff cannot be ...
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Dimension regularization “v.s.” fractional diffusion equation in wordline representation?

This is a very speculative question... Instead using dimensional regularization in QFT, for instance, $d = 4 - 2\epsilon$), one may imagine, thinking to the wordline representation of QFT (associated ...
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Inconsistency in regularization with parallel and perpendicular momenta

In deriving the axial anomaly Peskin and Schroeder use dimensional regularization, continuing loop momenta to $ 4 - \epsilon $ dimenstions. The loop momenta can now be split into pieces ``parallel'' ...
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Why are we scared of singularities? [closed]

I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner ...
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342 views

Gauge invariance of non-Abelian theories under Pauli-Villars-Regularisation

Under the ordinary Pauli -Villars Regularisation one introduces a heavy mass ($\Lambda$) term $$\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}....
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How do we know that analytic continuation agrees with UV regulators?

Consider the divergent series $$S = 1 + 1 + 1 + \ldots$$ which may appear in some calculations involving the Casimir effect. There are two main ways to evaluate this series. One can perform analytic ...
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Why Does Renormalized Perturbation Theory Work?

I've read about renormalization of $\phi^4$ theory, ie. $\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{\lambda}{4!}\phi^4\,,$ particularly from Ryder's book. But I am ...
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Divergent diagrams in QED

I was reading about how to choose divergent diagrams in QED by using the concept of Superficial degree of divergence. We have an empirical relation $$ D= 4-E_b -\frac{3}{2}E_f $$ where $E_b$ is number ...
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Quantization of free real scalar massless field in 2d

Is there a reference to literature where one explicitly constructs quantization of the free real scalar massless field in the 2-dimensional space-time? In particular, how the propagator looks like? ...
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317 views

Dependence on UV cut off of some $\phi^4$ diagrams

Consider the one loop corrections to the propagator and the vertex in $\phi^4$-theory:                    &...
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Size of box vs. discrete-ness of state of the system

From Statistical Physics, 2nd Edition by F. Mandl, pg. 36: A sufficiently large box (say 10 light-years across) will clearly not affect the properties of our system, ion plus electron sitting ...
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$i\epsilon$ in CFT correlation functions

M. Luescher in his talk on p.6 writes that the 2-point correlation function of a Hermitian local field $O_k$ of scaling dimension $d=3-k$ looks like $$ \langle 0| O_k(x) O_k(y) |0\rangle = A_k (x-y-i ...
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Why does analytic continuation as a regularization work at all?

The question is about why analytical continuation as a regularization scheme works at all, and whether there are some physical justifications. However, as this is a relatively general question, I ...
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367 views

Zeta regularization of Infinite product

I was trying to compute the product $$ P_{a,b} = \prod_{n=1}^\infty(an + b), $$ after I computed $$ P_{1,b} = \prod_{n=1}^\infty(n + b) = \frac{\sqrt{2\pi}}{\Gamma(b+1)}, $$ and the well-known $$...
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Representing propagators as Dirac delta functions [closed]

I have found online, in particular on the wolfram site, http://mathworld.wolfram.com/DeltaFunction.html, certain identities that allow one to represent a delta function as limits. Of particular ...
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Confusion with poles of single particle green's function / propagator

On p22 of "Green's Functions for Solid State Physicists" by Doniach and SondHeimer, there is the following definition: $$G^0(\omega)=\frac{1}{2M\Omega_0}\left( \frac{1}{\omega-\Omega_0+i\eta} - \frac{...
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Why is the force on the charge at the tip of a cone infinite?

Imagine a charge $q$ that is located at the top of a hollow cone with a surface charge density $\sigma.$ The slant height is $L$ and the charge $q$ sits at the vertex of angle $2\theta$. We are ...
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Can I use Pauli-Villars and dimensional regularization together?

There are at least two ways to compute the electron-self energy. You can use Pauli-Villars or dimensional regularization, for example. On Weinberg's book, it's chosen the first method, while on my ...
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Determinant of a propagator

Say I have a path integral $\int D \phi \exp(i S_0)$. $S_0$ is the usual free action $$S_0=\frac{1}{2}\int\phi (-\Box-m^2) \phi=\frac{1}{2}\int \phi G^{-1} \phi,$$ and at the moment I'm not ...
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Signs of Grandi's series $1-1+1-1+1-1+\ldots$ in the real world?

I'm talking about the convergence of the series $1-1+1-1+1-1+\ldots$ to $1/2$. I was discussing with some friends (we study physics) and I argued that Cesaro summation is a fair extension of the ...
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How to apply cutoff in path integral?

I am working on harmonic oscillator for quantum fluctuations (apart from clasical part), path may written as $$ S_q=\int_0^Tdt[(\partial_tq)^2+w^2q^2] $$ This may written as $$ S_q=\int dt(q\Delta q) ...
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218 views

Is this charge density function, from a problem in Griffiths' book, a physically valid density?

This is from the book on electrodynamics by Griffiths: A sphere of radius $R$, centered at the origin, carries charge density $$\rho(r,\theta)= k(R/r^2)(R-2r)\sin(\theta)$$ where $k$ is ...
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What is the reason/significance of using $ \sum\limits_{n=1}^{\infty}n\rightarrow-\frac{1}{12}$?

What is the reason/significance of using a trick equation in the Volume I - String Theory - Joseph Polchinsky? I have no doubts at all that the author knows extremely well the subject and that this ...
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181 views

Regularization of the 1-dimensional Laplacian

Disclaimer: this is a technical question about regularization of functional determinants which comes from a person with (relatively) strong background in QFT, string theory and path integrals, who ...
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668 views

How to handle the infrared divergence of massless $\phi^4$ in scattering

For massless $\phi^4$ theory, if exterior momentums are going to zero, then this diagram will be $$\int \frac{dk^4}{k^4}$$ will suffer from infrared divergence. Because the infrared divergence, ...
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How to choose the proper loop correction?

I review my QFT lecture notes and I am having hard times to figure out the significance of Ward identity in vacuum polarization. In class, we calculated one loop correction stated as $$ i\Pi^{\mu\...
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366 views

Two math methods apply the same loop integral lead different results! Why?

I tried to adopt the cut-off regulator to calculate a simple one-loop Feynman diagram in $\phi^4$-theory with two different math tricks. But in the end, I got two different results and was wondering ...
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Doubts with basic renormalization

When we renormalize to obtain the physical mass, the $\Lambda$ dependence of the physical mass is removed by introducing the counterterms in the Lagrangian. So whether we put $\Lambda\rightarrow\infty$...
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573 views

Why is cut-off regularization is not Lorentz invariant?

Why is it said that the cut-off regularization is not a Lorentz invariant regularization method?
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260 views

What is primitive divergence?

As in the title, what is primitive divergence? How is it distinguished from normal divergence? As a followup, what is a primitive divergent graph in a theory? Some simple examples?
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Why does regularization work in this Bessel function integral?

I encountered some days before an integral representation for a modified Bessel function and should differentiate it. But in this representation : $$K(\omega,a)=\int_0^{\infty} \frac{ds}{s} e^{-i\...
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Renormalization Using Momentum Cut-off Regularization, What Are The Subtraction Schemes Used?

In most of the books on QFT, the author talks about various methods of regularization but in the end chooses the dimensional regularization and MS-bar scheme when discussing the final renormalization, ...