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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89 views

Why is it correct to estimate divergences by the cutoff in QFT?

Let's say we have a linear divergence in a quantum field theory. The way to deal with this infinite quantum correction is to go through the whole process of renormalization. However, quite often, ...
9
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3answers
202 views

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 ...
36
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9answers
5k views

Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
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0answers
26 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}....
103
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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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66 views

why exactly do we use regulators for infinite sums? [duplicate]

Disclaimer: I really apologize for not being educated in QFT a priori. Im trying to learn. I really really am. I am a doctor of physics but i sadly never took QFT in my program and so im now ...
2
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1answer
175 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 ...
41
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1answer
4k 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
264 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
376 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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0answers
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
97 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:                    &...
1
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2answers
119 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?
0
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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
138 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
votes
2answers
103 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 $$...
1
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1answer
84 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
435 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 ...
26
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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 ...
15
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3answers
10k views

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
72 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
votes
1answer
247 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
208 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 ...
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1answer
59 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
439 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
171 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
141 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 ...
1
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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 ...
0
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0answers
23 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
88 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
105 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
39 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
52 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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0answers
93 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, ...
1
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1answer
76 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
186 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
131 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 $\Lambda\rightarrow\infty$...
9
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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
384 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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0answers
86 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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0answers
148 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
279 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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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 ...
5
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1answer
219 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 ...
33
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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 ...
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0answers
342 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 $...