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.
531 questions
2
votes
0
answers
53
views
Point-splitting regularization for ghost fields
In section V of this paper, the author computes $\langle F_{\mu \nu} F^{\mu \nu} \rangle$. Using the definition of $F_{\mu \nu}$ it is not difficult to show that
$$\langle F_{\mu \nu} F^{\mu \nu}\...
1
vote
1
answer
117
views
Are Schwinger terms always $c$-numbers?
My question concerns this paper. Here, the author defines point split fermion bilinears as
$$ J_{\Gamma_A}(x,\epsilon) = \frac{1}{2}\left( \bar \psi(x+\epsilon) \Gamma_A \psi(x) + \bar \psi(x) \...
0
votes
0
answers
111
views
Is it possible to construct Smooth solutions for the unsimplified version of the viscid and incompressible Navier-Stokes eqs.? Smooth is essential
There are many exact solutions to the simplified Navier-Stokes equations. However smooth and 3d solutions do still remain elusive.
Is there a way to construct an exact 3d smooth solution generator of ...
- 1
9
votes
2
answers
1k
views
Does QFT predict that the lattice constant of the universe is zero?
The title might be confusing, so let me explain.
Planck unknowingly started the field of quantum mechanics when he described blackbody radiation spectra using a law that assumes discrete values for ...
4
votes
1
answer
265
views
Regularization of determinants in QFT
I have a question regarding regularization in quantum field theory. Hagen Kleinert talks about analytic regularization in his book "Path Integrals". In chapter 2.15 he calculates the ...
- 169
1
vote
0
answers
58
views
Dispersion relation for heavy quark correlator
In particle physics, we often encounter correlators $\Pi(q^2)$ which are functions of the squared momentum transfer $q^2$. These functions are real-valued for some $q^2$ below a threshold $M^2$, and ...
- 245
3
votes
2
answers
256
views
Understanding renormalization schemes
I'm learning renormalization in QFT, and find the following question: Use QED as example, when we do the renormalized perturbation theory, we introduce new parameters $m$ and $e$, and write the bare ...
- 274
2
votes
0
answers
111
views
Question about the mathematical treatment in P&S Chapter 11.2
The question is about the treatment of the two-point and one-point amplitudes in linear sigma model in P&S Chapter 11.2
When evaluating the one-point $\sigma$ amplitude, we encountered the diagram ...
- 747
1
vote
0
answers
34
views
More than one way of renormalization for the same Feynman loop integral? [duplicate]
My understanding for renormalization is that to deal with the divergence appeared in loop integrals, we introduce an artificial regularization parameter to make the integral converge, and when we take ...
- 1,614
0
votes
0
answers
58
views
Singularity in the propagator in non-relativistic quantum mechanics [duplicate]
The propagator in non-relativistic quantum mechanics is:
$$G(\mathbf{x}, t; \mathbf{x}', t') = \left\langle \mathbf{x} \Big| e^{-i H (t - t') / \hbar} \Big| \mathbf{x}' \right\rangle.$$
Adding the ...
3
votes
0
answers
190
views
Why is electron self-energy infrared divergent?
I'm studying QED from Mandl and Shaw's book. My question regards the regularization of the fermion self-energy correction
$$
ie_0^2\Sigma(p) = \int_{\mathbb{R}^4} \frac{\text d^4 k}{(2\pi)^4}
i\frac{(...
- 864
0
votes
0
answers
47
views
Massive Gluon Regularization resources
I am trying to learn about other regularization schemes in QFT and I came across massive gluon (MG) regularization scheme in Sections 2.5, 2.6, and 2.7 of [1]. However, the presentation is very non-...
4
votes
1
answer
174
views
Regulating certain integral
I am interested in regulating an integral of the type,
\begin{equation}
I =
\frac{1}{2\pi^2}\int_0^\infty p^2\ dp\ \frac{g(p)}{p^2 - k^2 - i\epsilon}
=
\frac{1}{4\pi^2}\int_{-\infty}^\infty p^2\ dp\ ...
- 664
2
votes
1
answer
138
views
The path integral representation of the transition amplitude between the vacua of two field theories
The background of the question: PhysRevLett.115.261602.
$
\newcommand\ket[1]{| #1 \rangle}
\newcommand\braket[1]{\left\langle{#1}\right\rangle}
\newcommand\dif{\mathrm{d}}
\newcommand\E{\mathrm{e}}
$...
- 110
5
votes
1
answer
311
views
What rational zeta series with non-integer arguments appear in physics - if any?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
- 1,073
1
vote
1
answer
801
views
Regularization of one-loop self-loop in self-energy of 4D $ \lambda \phi^4 $ theory and cutoffs
I'm learning about regularization in QFT. Sorry if this is a dumb question, but this topic is new and very confusing to me.
Say I want to calculate the 1-loop diagram in $\lambda \phi^4 $ theory. In ...
- 531
-1
votes
1
answer
136
views
Integration bounds for the canonical ensemble's partition function
As far as I understand, when calculating the partition function we are summing over all of phase space.
So, for ideal gas for example we have
$$H =\sum\limits_i\frac{p^2_i}{2m}$$
and so
$$Z=\int\...
- 79
3
votes
1
answer
159
views
Photon Mass Regulator in IR divergences
On Schwartz's QFT page 333, he metions that there is infrared divergence when we try to renormalized the two-point function of electron field in on-shell substraction scheme,
$$\frac{d}{d\,p_{\mu}\...
- 976
2
votes
0
answers
365
views
Konishi operator anomalous dimension [closed]
The Konishi operators are operators in the ${\cal N}=4$ SYM theory and are given by:
$$ K = \sum _{i=1}^6Tr\ (\phi^i\phi^i) $$
The 2 point function of this operator is:
$$ \big\langle K(x)K(y)\big\...
- 421
0
votes
2
answers
244
views
Regularization of black hole singularities
Hi I have a question: when dealing with the gravitational Lorentz factor from schwarzchild solution to EFE, used in defining gravitaional time dilation and one encounters singularities at $r=0$ or $r=...
3
votes
3
answers
431
views
Closed form expression of 2D CFT integral
I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is
$$ \int_{\...
- 93
3
votes
1
answer
196
views
Question about RG from QFT perspective, in particular Weinbergs book
This must be fairly basic I fail to understand. According to Weinberg, QFT Vol2, Ch.18 (The preamble)
When we replace bare couplings and fields with renormalized couplings and fields defined in terms ...
- 601
7
votes
1
answer
515
views
Wilsonian RG in QFT: what is the difference between renormalized and bare couplings?
I want to understand the relation between the Wilsonian RG and the usual QFT RG approach. Several questions have been asked, such as this and many others, yet I don't find a conceptual answer to what ...
- 2,237
12
votes
1
answer
446
views
Analytical continuation as regularization in Quantum Field Theory, the remaining questions
There is an old question posted (Regularization) which did not get an answer, about the validation of analytic continuation as regularization. It did get some discussion in the comments, referring to ...
- 6,131
5
votes
1
answer
630
views
Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?
Let $\phi$ denote the Klein-Gordon field. Then its propagator $\langle 0 \mid [\phi(x), \phi(y)] \mid 0 \rangle$ can be calculated as
$$\int \frac{\mathrm d^4p}{(2\pi)^3} \frac{-e^{-ip(x-y)}}{p^2 -m ^...
- 4,852