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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How to integrate a Gaussian path integral of free particle using zeta function regularization?
I am attempting to integrate this path integral in Euclidean variable $\tau $ (but this need not be the same as the $X^0$ field):
$$Z=\int _{X(0)=x}^{X(i)=x'}DX\exp \left(-\int _0^i d\tau \left[\frac{...
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A simple question on creation and annihilation operators
We know that the KG solution for a Spin-0 particle has the following Hamiltonian
$$\hat{H}=∫ d^{3}p\frac{ω_{p}}{2}(\hat{a}_{p}\hat{a}^{\dagger}_{p}+\hat{a}^{\dagger}_{p}\hat{a}_{p})\hspace{2cm}[\hat{a}...
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Renormalization group equation and method of characteristics
All of this question refers to ref. 1. The equation are numbered alike.
The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage ...
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Problem in calculating the one-loop contribution to the self–energy of $\phi$ field
Consider the theory
$$
\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi\right)^2-\frac{m^2}{2} \phi^2-\frac{g}{3 !} \phi^3-\frac{\lambda}{4 !} \phi^4 .
$$
Find the expression for the self-energy and the ...
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Pauli-Villars regularization and self-energy
In the calculation of the electron self-energy in QED (one-loop level), there is a UV and IR divergent integral that needs to be regularized. A common choice for the regularization is the Pauli-...
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LSZ Reduction formula: Peskin and Schroeder
On page 224 of P&S, they have the following expression (7.36),
The integral over $d^3q$ gives us all the $q \to p$, then the integral over $dx^0$ is computed. The RHS given matches when only the $...
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One-loop Integral for a Tensor Quantity
I am considering one-loop integrals in quantum field theory. I am happy with how these calculations work with scalar integrals, but I am a bit lost in the details of what happens when the quantity ...
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Why does non-perturbative QCD need to be regularized and renormalized?
The $n$-point correlation functions of QCD, which define the theory, are computed by performing functional derivatives on $Z_{QCD}[J]$, the generating functional of QCD,
$$\frac{\delta^nZ_{QCD}[J]}{\...
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Kustaanheimo-Stiefel (KS) transformation and initial conditions
I understand how Kustaanheimo-Stiefel (KS) transformations work and I have derived my equations of motion as well. However, I am struggling to find initial conditions in the KS space. How do I convert ...
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On the computation of natural logarithm in dimensional regularization
When calculating the integral
\begin{equation}
\int\frac{d^4q}{(2\pi)^4\left(q^2+\Delta+i\epsilon\right)^2}
\end{equation}
We encounter a term of $\ln\Delta$ and I am not sure how does one treat it ...
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Beta function of $\phi^4$ from RG equation
I am currently following David Tongs lecture notes on statistical field theory (Link to the script) and I have an issue with the calculation of the beta-function from the RG equations (equation (3.45))...
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Can gravity be the high-energy regularizer for QFTs?
In order for QFTs to be mathematically well-defined some sort of regularization scheme needs to be introduced, which typically looks like a discrete cutoff of all modes outside of some shell with ...
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Questioning a calculation in Kapusta / Infinite entropy of a Fermi gas?
I am going through Kapusta's calculation of the free energy of a Fermi gas, and I find one of his steps dubious (and if I'm right, it would mean the free energy of a Fermi gas is either infinite or ...
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Vacuum bubble with the $Z$-boson
I'm interested in computing the vacuum bubble diagram of a single $Z$-boson. The propagator for a massive vector is
$$
D_{\mu\nu}(k_\mu) = \frac{i}{k^2 - m_Z^2}\left(g_{\mu\nu} - \frac{k_\mu k_\nu}{...
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Explicit form of cutoff dependent bare coupling at first order in $\phi^4$ with cutoff regularization
I would like to renormalize $\phi^4$ theory with Lagrangian
\begin{equation}
\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m_0^2 \phi^2 - \frac{\lambda_0}{4!} \phi^4
\end{...
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The underlying cause of ill-defined loop-integrals in Quantum Field Theory
One of the main causes which leads to ill-defined loop integrals in Quantum Field Theory is that the variables of a Field Theory, $\varphi(x)$ for instance, are Quantum Fields which are governed by ...
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Does derivation of Casimir effect include infrared regularization?
The derivation of the Casimir effect in $1 + 1$ approximates, say, the left mirror and the asymptotic "in" region of spacetime as two fixed mirrors separated by a spacial distance of $L \to \...
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While calculating self-energy or work required to assemble a continuous charge distribution, do we take the energy of an element with itself?
while calculating the self-energy of a continuous charge distribution using the formula
the potential $V$ here is due to the whole charge distribution but we need potential due complete charge ...
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Does there exist QFT which do not yield infinities? [closed]
Does there exist a "simple", "polynomial-like" quantum field theory (unrealistic, not describing our world, in any spacetime dimension $d$) which does not yield infinities in ...
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Is the $i\varepsilon$ prescription in the Feynman propagator just as "outrageous" as $1+2+3+... = -1/12$?
When the calculation of the Feynman propagator is introduced in QFT (or QM), the inclusion of the $i\varepsilon$ term is often described as a minor technical detail that is there just to "make ...
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How to choose an appropriate value for the regularization $\eta$ in correlation functions in linear response for numerical calculations?
TL;DR
How to choose an appropriate value for the regularization $\eta$ in correlation functions used in linear response for a discretized Brillouin zone?
For more context, please see below.
...
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What is lattice regularization and how is it carried out? [duplicate]
I am new to QFT, and so far I have studied dimensional regularization and Pauli-Villars regularization. These seem to be the only two regularization techniques discussed in most introductory textbooks....
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Why do we drop the renormalization term in momentum Klein-Gordon Field Theory?
I'm following Peskin & Schroeder's book on QFT. I managed to prove expression (2.33) which gives us the 3-momentum operator for the Klein-Gordon Theory:
$$\mathbf{P}=\int \frac{d^3p}{(2\pi)^3}\...
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Why do we Wick rotate before regularizing Feynman diagrams?
In Folland's Quantum Field Theory he mentions that we can apply Feynman's formula (Feynman parameterization) to either the Wick rotated integrals or the non-Wick rotated integrals corresponding to ...
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When should UV regulators be removed?
I have been working in QFT for a few years now, and I cannot believe I've never come across this problem. When considering an effective field theory, the allowed operators mix under renormalization:
$$...
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Calculating a rectangular Wilson loop for the free photon
I'm studying Creutz's Quarks, gluons and lattices, in chapter 6 on page 33, we have the following exercise
Calculate a rectangular Wilson loop for the field theory of free photons. Using any ...
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3-point counterterm in scalar QED
I'm working on the 1-loop corrections to scalar QED. I'm using dimensional regularization and on-shell regularization. In trying to compute the counterterm for the 3-point graph I come across the ...
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On-shell renormalization (Schwartz Quantum Field Theory Equation (18.48))
I have a question about how, in section 18.3.2 in Schwartz's quantum field theory, he goes from equation (18.47) to (18.48) using Pauli-Villars regularization. It comes down to showing that to leading ...
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Regularization scheme independence as a gauge redundancy?
Observables should not depend on the regularization scheme under some renormalization procedure. Is there some way to interpret this fact as a gauge redundancy? In particular, is there some group ...
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How to show Pauli-Villars regularization introduce a momentum cut-off?
Pauli-Villars regularization instructs us to do such a replacement:
$$\frac{1}{p^2-m^2+i\epsilon} \rightarrow \frac{1}{p^2-m^2+i\epsilon} - \frac{1}{p^2-\Lambda^2+i\epsilon}$$
And then claim: such a ...
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Is it legitimate to use analytic continuation to equate a diverging series with a finite number in a physical theory of nature?
Analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite ...
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Peskin and Schroeder's QFT eq. (9.14): Gaussian momentum field integration of phase space path integral
On Peskin and Schroeder's QFT book page 282, the book considered functional quantization of scalar field.
First, begin with
$$\left\langle\phi_b(\mathbf{x})\left|e^{-i H T}\right| \phi_a(\mathbf{x})\...
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Pretending photon has a small mass in soft bremsstrahlung
In Peskin and Schroeder chapter 6, on page 184 when discussing the infrared divergence problem in perturbative QED, the book says we can make the following equation
$$\tag{6.25} \text{Total ...
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Why is QFT not even unitary prior to renormalisation?
The Hamiltonian is Hermitian. That should've been enough to make it unitary. But infinite amplitudes mean it's not even unitary. One could say that this is because we're dealing with a crazy Hilbert ...
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Delta function squared in Weigand's QFT notes
On page 74 of Timo Weigand's QFT notes, right at the top, the following equality is used:
$$\left[(2\pi)^4\delta^{(4)}(p_f-p_i)\right]^2=(2\pi)^4\delta^{(4)}(p_f-p_i)(2\pi)^4\delta^{(4)}(0) \tag{2.167}...
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Euler-Maclaurin formula for Casimir Effect
I’ve been reading this http://www.hep.caltech.edu/~phys199/lectures/lect5_6_cas.pdf for a derivation of the Casimir Effect. On page 4 we see that
$$\Delta{E} = \frac{\pi}{L}\left(\sum_{\nu = 1}^{\...
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Finiteness of quantum mechanics
In most standard undergraduate treatments of quantum mechanics, there is rarely any need to treat divergences in perturbation theory, other than the Casimir energy perhaps? The subtleties of ...
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Unitarity constraints for real soft photons (Weinberg, Section 13.3)
Weinberg obtains differential scattering rate for a process with soft photons,
$$\begin{align*}
d \Gamma_{\beta \alpha}^{\lambda}(\omega_1, ...\omega_N) = \Gamma^\lambda_{\beta \alpha} A(\...
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Fourier transform of propagator in Lehmann spectral representation
I need help showing that:
$$ iG(k,t)=\int_{\mu/\hbar}^{+\infty} e^{-i\omega t}A(k,\omega)d\omega $$
knowing that
$$ G(k,\omega) = \int_{-\infty}^{+\infty} \frac{A(k,\omega')}{\omega -\omega'+i\eta\ ...
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Significance of non-zero cross energy-momentum tensor in Fulling Davies Unruh effect
In the Fulling Davies Unruh effect We can get the outgoing particle flux from the asymptotic future region $\mathscr{I}^+$ from the mirror trajectory as the following
$$
\mathcal{F}=\int_0^\infty \...
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How to solve this type of series?
This series is from the book Quantum field theory in curved spaces by Birrell and Davies pp 95, use of Green's functions. My problem is how from 4.22 to 4.23 they are getting. We have to use some ...
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Can we do regularization by just discarding the divergent part?
Consider the one-loop $\phi^4$ integral to order $\lambda^2$
$$I=\int \frac{d^{4}k}{(2\pi)^4}\frac{1}{k^2-m^2}\frac{1}{(k-p)^2-m^2}.$$
After some transformation (refer to the answer in here)
$$I=\int_{...
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How is this change of basis performed?
In David Tong's Gauge Theory notes on page 136 eq. (3.29) he performed the following change of basis, during a momentum cut-off regularization scheme.
$$ \sum_{n} \overline{\phi}_{n}\...
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Effective Action behaviour in $SU(N)$ Gauge Theory in Solodukhin
In Solodukhins paper https://arxiv.org/abs/0802.3117
he says that the effective Action of a 4D CFT has the general structure:
$$W = \frac{a_0}{\epsilon^4} +\frac{a_1}{\epsilon^2}+a_2 \ln{\epsilon} +\...
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What is the origin of these log terms in dimensional regularization?
The following limit is implied on page 250 of Peskin and Schroeder:
$$\Gamma\left(2-\frac d2\right)\left(\frac 1 \Delta\right)^{2-\frac d2} \xrightarrow{d\rightarrow 4} \frac 2\epsilon - \log \Delta -\...
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Dimensional regularization vs. hard cutoff and their relation to the renormalization scale in 2d vs 4d to find $\beta$ functions
I would like to understand some shortcuts people are using to calculate $\beta$ functions using dim. reg. with mass scale $\mu$ and/or the hard cutoff $\Lambda$. My end goal is to use equation 12.53 ...
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Gauss' law in the presence of surface charges [duplicate]
Assume $V$ is a volume such that $\rho=0$ in $V$ where $\rho$ is the charge density. Assume further that we have a surface charge density $\sigma$ on the surface $S$ enclosing $V$ such that the total ...
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Charge density of Hertzian dipole
I am wanting to find the charge density of an infinitely thin Hertzian dipole, but am struggling evaluating the Dirac delta functions gradient.
$$\vec{J} = I_{0}\cos(\omega t) \delta^3(r) \hat k.$$
...
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Why does normal-ordering ensure finiteness?
I will be using Jan von Delft's rigorous construction of bosonization/refermionization as an example, but I will try to explain my question in more general terms.
Consider an (countably) infinite-dim (...
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Relationship between product integrals and functional determinants
This is in reference to the answer posted to this question. The person who answered the question claims that the functional determinant of any operator $O$ is given by a product integral
$$\det O = \...
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