Renormalization Definition and 48 Discussions

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles (e.g. collisions at different energies) shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge. Mathematics and experiments prove that positrons and more massive particles like protons exhibit precisely the same observed charge as the electron – even in the presence of much stronger interactions and more intense clouds of virtual particles.
Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. Physically, the pileup of contributions from an infinity of scales involved in a problem may then result in further infinities. When describing space-time as a continuum, certain statistical and quantum mechanical constructions are not well-defined. To define them, or make them unambiguous, a continuum limit must carefully remove "construction scaffolding" of lattices at various scales. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.
Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory. Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics.
Today, the point of view has shifted: on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and Kenneth Wilson, the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. All scales are linked in a broadly systematic way, and the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each. Wilson clarified which variables of a system are crucial and which are redundant.
Renormalization is distinct from regularization, another technique to control infinities by assuming the existence of new unknown physics at new scales.

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  1. S

    I 1-loop Fermion mass correction in toy EFT

    Where does the ##m## in ##(3.2)## come from? It doesn’t seem to enter anywhere in Feynman rules for the given diagram
  2. S

    I Expansion at first order in QCD counterterm

    What is the meaning of the expansion at first order in ##\delta_2## and ##\delta_3## at the second step in the last line? These quantities are not "small" - on the contrary, the entire point is to then take the ##\epsilon \to 0## limit and the counterterms blow up
  3. G

    A Relationship between Wilson's RG and the Callan-Symanzik Equation

    I have taken a Quantum Field Theory course recently in which we first derived the Callan-Symanzik equation and then discussed Wilson's Renormalization. However, I don't think I have a clear understanding of the procedures and how they relate to each other. For the sake of this question, let's...
  4. L

    Renormalization Group:NiemeijerVan Leeuwen Method-Ising Square Lattice

    Hello, I have to solve this problem. I will apply the Niemeijer Van Leeuwen method once I have the probability distribution proper to the renormalization group ,P(s,s'). For example, in the case of a triangular lattice, this distribution is: where I is the block index. However, it is very...
  5. K

    I Pauli-Villars regularization for Vacuum Polarization

    Hello! I am currently reading Itzykson Zuber QFT book and on Chapter 7 where for the first time loops are considered. Particular method of dealing with divergences namely Pauli-Villars regularization is considered in section 7-1-1 considering vacuum polarization diagram. I do understand physics...
  6. DaniV

    RG flow of quadrupole coupling in 6+1 dimension electrostatic problem

    I tried to do a Euler Lagrange equation to our Lagrangian: $$\frac{S_\text{eff}}{T}=\int d^6x\left[(\nabla \phi)^2+(\nabla \sigma)^2+\lambda\sigma (\nabla \phi)^2\right]+\frac{S_{p.p}}{T}$$ and then I would like to solve the equation using perturbation theory when ##Q## or somehow...
  7. M

    Mass correction in ##\phi^4##-theory

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  8. tomdodd4598

    I QED - running of coupling (beta function)

    Hey there, I am a little confused about the way most textbooks and notes I've read find the beta function for QED. They find it by looking at how the photon propagator varies with momentum ##q##, in particular in the context of a ##2\rightarrow2## scattering process which is proportional to...
  9. S

    I Wilson's RG trajectories, Lagrangians and many worlds?

    In this article [1] we can read an explanation about Wilson's approach to renormalization I have read that Kenneth G Wilson favoured the path integral/many histories interpretation of Feynman in quantum mechanics to explain it. I was wondering if he did also consider that multiple worlds...
  10. W

    I Renormalization of scalar field theory

    I was reading about the renormalization of ##\phi^4## theory and it was mentioned that in order to renormalize the 2-point function ##\Gamma^{(2)}(p)## we add the counterterm : \delta \mathcal{L}_1 = -\dfrac{gm^2}{32\pi \epsilon^2}\phi^2 to the Lagrangian, which should give rise to a...
  11. tomdodd4598

    I Renormalisation scale and running of the φ^3 coupling constant

    I am still rather new to renormalising QFT, still using the cut-off scheme with counterterms, and have only looked at the ##\varphi^4## model to one loop order (in 4D). In that case, I can renormalise with a counterterm to the one-loop four-point 1PI diagram at a certain energy scale. I can...
  12. G

    A Is renormalization the ideal solution?

    Quantum gravity theories and GUTs are nonrenormalizable theories, but does this actually mean that these theories must be flawed, or does it mean that renormalization must be a flawed concept, or is this not actually a problem? If it is impossible to produce a renormalizable quantum gravity...
  13. R

    I Young physicist in seek of guidance

    Is there anyone on here who could help me fill in my gaps in quantum field theory up to renormalization? I know how to canonically quantize a theory and how to use scalars (spin 0), vectors (spin 1) and spinors (spin 1/2) but lack more advanced knowledge like renormalization which I could...
  14. M

    Beta-function for the Gross-Neveu model

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  15. A

    A Renormalization (Electron self energy)

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  16. A

    On-shell renormalization scheme

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  17. Angel Kld

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  18. A

    I What is the point of regularization?

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  19. Ken Gallock

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  20. ohwilleke

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  21. ohwilleke

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  22. D

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  23. Urs Schreiber

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  24. hilbert2

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  25. A

    I IR divergences and total energies....

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  26. T

    A Quantum Gravity: Renormalization vs. Effective Field Theory

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  27. J

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  28. J

    A Why do we need to renormalize in QFT, really?

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  29. ohwilleke

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  30. Kfir Dolev

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  31. I

    Scalar QCD renormalization

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  32. T

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  33. unknown1111

    A Computing the pole mass from a given MS mass?

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  34. unknown1111

    Top quark mass mt at energy scales μ<mt?

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  35. O

    Renormalization of Bound States in QFT

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  36. M

    Asymptotic freedom requires perturbative renormalizability?

    I have read many times that a theory (such as gravity) that contains couplings with negative mass dimensions cannot be asymptotically free. Does anyone have a reference that proves that that's the case? The argument is usually just that the coupling grows with energy, as seen in the...
  37. Giuseppe Lacagnina

    Inverse fields?

    Possibly very silly question in QFT. Consider the Lagrangian for a scalar field theory. A term like g/φ^2 should be renormalizable on power counting arguments. The mass dimension of g should be 2 (D-1) where D is the number of space-time dimensions.Does this make sense?
  38. H

    Error in Srednicki renormalization?

    On page 164-165 of srednicki's printed version (chapter 27) on other renormalization schemes, he arrives at the equation $$m_{ph}^{2} = m^2 \left [1 \left ( +\frac{5}{12}\alpha(ln \frac{\mu^2}{m^2}) +c' \right ) + O(\alpha^2)\right]$$ But after taking a log and dividing by 2 he arrives at...
  39. H

    ##\overline{MS}## in scalar theory references

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  40. P

    Counterterms in self-energy diagram

    I guess my question is pretty basic, and following a procedure in the textbook by Lahiri and Pal. You can see the relevant pages at On eqs. (12.84)-(12.86). I don't see how to get from (12.85) to...
  41. M

    Qualitative explanation of scale dependence

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  42. D

    Why is QFT insensitive to absolute energies?

    In the canonical formulation of QFT (to which I've been exposed), it is always argued that only differences in energy are physically observable and so we can deal with the fact that the vacuum energy is infinite by redefining the vacuum such that its energy is zero and we subsequently measure...
  43. G

    Typical Momentum Invariants of a 3-Point Function

    According to Peskin, p.414, at the bottom, as part of calculating the ##\beta## functions of a theory, we need to fix the counter terms by setting the "typical invariants" built from the external leg momenta to be of order ##−M^2##. For a 4-point function, these invariants are s, t and u...
  44. G

    Trouble Finding Renormalization Conditions in Yukawa Theory

    I am trying to calculate the ##\beta## functions of the massless pseudoscalar Yukawa theory, following Peskin & Schroeder, chapter 12.2. The Lagrangian is ##{L}=\frac{1}{2}(\partial_\mu \phi)^2-\frac{\lambda}{4!}\phi^4+\bar{\psi}(i\gamma^\mu \partial_\mu)\psi-ig\bar{\psi}\gamma^5\psi\phi.##...
  45. M

    Can there by a theory that is both UV and IR stable?

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  46. nikosbak

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  47. quantatanu0

    Cut-off Regularization of Chiral Perturbation Theory

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  48. T

    Order in Renormalization Theory

    I am currently studying QFT with 'An Introduction to Quantum Field Theory' by peskin. In part 2 (renormalization) of the book, he introduces counterterms and shows how to compute scattering amplitude with them. Below are counterterms of \phi^4 theory: Then he calculates a 2-2 scattering process...