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 ...

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169 views

Introducing cut-off in a renormalisation procedure for quantum mechanics

I've been reading a paper on renormalisation theory as applied to a simple one-particle Coulombic system with a short-range potential. In the process of renormalisation, the authors introduce an ...
101
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1answer
6k views

Superfields and the Inconsistency of regularization by dimensional reduction

Question: How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? Background and some references: ...
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1answer
3k views

Classical and quantum anomalies

I have read about anomalies in different contexts and ways. I would like to read an explanation that unified all these statements or points of view: Anomalies are due to the fact that quantum field ...
7
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0answers
260 views

Regulating the sum in Casimir Force

I am trying to evaluate the Casimir force using a Gaussian regulator (I know there are other much easier ways to do this, but I want to try this!) We then are reduced to evaluating the sum $$ \sum\...
3
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2answers
356 views

Pauli Villars Regularization

Consider the t-channel diagram of phi-4 one loop diagrams. Evaluated it is, with loop momenta p, $\frac{\lambda^2}{2}\displaystyle\int\frac{d^4p}{(2\pi)^4}\frac{1}{(p+q)^2-m^2}\frac{1}{p^2-m^2}$ If ...
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36 views

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 ...
2
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1answer
92 views

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? ...
3
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1answer
70 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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2answers
115 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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1answer
45 views

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 ...
5
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0answers
131 views

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 ...
1
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0answers
60 views

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 ...
3
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2answers
97 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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1answer
83 views

Connection between the cosmological constant $\Lambda$ and the cutoff scale $\Lambda$

I'm trying to understand the connection between the $\Lambda$ from cosmology and the $\Lambda$ from QFT. Cosmology: The cosmological constant enters the Einstein equations. In the special case of the ...
6
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1answer
430 views

Free Particle Path Integral Matsubara Frequency

I am trying to calculate $$Z = \int\limits_{\phi(\beta) = \phi(0) =0} D \phi\ e^{-\frac{1}{2} \int_0^{\beta} d\tau \dot{\phi}^2}$$ without transforming it to the Matsubara frequency space, I can ...
24
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4answers
1k views

Regularization of the Casimir effect

For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have ...
14
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3answers
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Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
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2answers
70 views

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{...
2
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1answer
230 views

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 ...
2
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3answers
204 views

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 ...
1
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1answer
54 views

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 ...
6
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2answers
433 views

The integral is zero! $\int \frac{\mathrm{d}^d k}{(2\pi)^d} = 0$

In using dimensional regularization in QFT calculations, one comes across integrals over propagators, they might look like $(d = \text{dimension of spacetime}, n = \text{a number})$ $$\tag{1}I(d,n)=\...
4
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1answer
169 views

Not satisfied with “trick” in zeta function regularization

I am not satisfied with the explanations of $$\sum_n \log \lambda_n = - \frac{d}{ds} \sum_n \lambda_n^{-s}\bigg|_{s=0}$$ "trick" used in zeta function regularization, discussed here and here, or the ...
4
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2answers
135 views

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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0answers
73 views

Signs of Grandi's series $1-1+1-1+1-1+1+\ldots$ in the real world?

I'm talking about the convergence of the series $1-1+1-1+1-1+1+\ldots$ 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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22 views

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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2answers
87 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 ...
3
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1answer
103 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 ...
0
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0answers
37 views

Lattice propagator computation

I am reading this lecture on random surface theory by Thordur Jonsson. I feel like I should describe the problem a little for those who don't want to read the lecture. We are ultimately interested in ...
0
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0answers
50 views

Cuts of a Feynman diagram and the massless limit

Consider a $j$ point all massive leg one loop polygonal Feynman diagram $P$ representing some scattering process cut on a particular mass channel $s_i$. Invoking the relevant Feynman rules and ...
4
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88 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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1answer
74 views

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\...
3
votes
1answer
185 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 ...
3
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1answer
126 views

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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2answers
1k views

Critical Dimension of Bosonic Strings and Regularization of $\sum_{n=1}^\infty n$

If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at $$ \frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0. $$ Now mathematically this is ...
5
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1answer
381 views

Branch cuts in two-point function

The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is ...
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84 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?
2
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146 views

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\...
1
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1answer
258 views

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, ...
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7answers
1k views

Why regularization?

In quantum field theory when dealing with divergent integrals, particularly in calculating corrections to scattering amplitudes, what is often done to render the integrals convergent is to add a ...
4
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1answer
203 views

Point splitting technique in Pesking and Schroeder

One of the cornerstones of point splitting technique of calculating chiral anomaly (Peskin and Schroeder 19.1, p.655) is a symmetric limit $\epsilon \rightarrow 0$. And this is the point that I don't ...
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1answer
51 views

Why IR divergences cancel by cross sections of next-to-leading diagrams?

I was reading QFT & Standard Model by Schwartz, Chapter 20 which is about IR divergences. He says that IR divergences only cancel cross sections for processes involving different initial or ...
32
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5answers
2k views

What exactly is regularization in QFT?

The question. Does there exist a mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory? If so, what is it? Motivation and ...
11
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0answers
337 views

Which values of the Riemann zeta funtion at negative arguments come up in physics?

For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. ...
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1answer
82 views

Dirac Current Spectral Representation

I'm reading Strocchi's book on The Non-Perturbative Foundations of Quantum Field Theory. In the chapter concerning point-splitting regularization, where the free Dirac current is defined as follows $...
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54 views

Dependence of finite part of loop integral on regularization

Recently I've calculated some process in which arise triangle loop with running two $W$ bosons and one massless fermion. The expression for integral is following: $$ I_{\alpha \beta}(r, q) = \int \...
2
votes
1answer
61 views

How do logarithms show up in the one loop calculation of the vacuum polarization in QED?

I am following Peskin with the computation of the vacuum polarization in QED and there is one thing I do not see. Equation (7.90) reads $$\frac{-8e^2}{(4\pi)^{d/2}}\int_0^1dx\,x(1-x)\frac{\Gamma(2-\...
7
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5answers
696 views

How to understand singularities in physics?

The question is probably two-folded and I will try not to make it too vague, but nonetheless the question remains general. First fold: In most physical laws, that we have analytic mathematical ...
7
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5answers
2k views

Laplacian of $1/r^2$ (context: electromagnetism and poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is $$\nabla^2\frac{q}{r}=...
3
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1answer
90 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} \,\dot{\...