Renormalization and divergences

In summary, renormalization is a technique used in quantum field theory to deal with divergent integrals. It involves performing iterative substitutions to simplify the integrals, but this only works for renormalizable theories. Non-renormalizable theories have integrals that cannot be simplified even after these substitutions, leading to unresolved divergences.
  • #1
eljose
492
0
renormalization and divergences...

let suppose we have a formula for the mass in the form:

[tex]m=\int_{0}^{\infty}dxf(x)e^{-ax} [/tex] [tex]a=ln\epsilon [/tex]

with epsilon tending to zero so a is divergent..but if we perform the integral numerically:

[tex]m=\sum_{j}w(x_{j})c_{j}f(x_{j})e^{-ax_{j}) [/tex]

so we could express the quantity a in terms of the mass m so [tex]a=g(m)[/tex] so we could put inside the integral to calculate the m:

[tex]m=\int_{0}^{\infty}dxf(x)e^{-xg(m)} [/tex] and from this equation obtain a value for the mass m.

I Know something similar is made for renormalizable theory..but why can not be made for non-renormalizable ones?...:frown: :frown:
 
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  • #2
What you have is a toy example of a single integral. A quantum field theory requires iterations of such substitutions across an infinite tower of integrals of progressively higher dimension as you move up in perturbative order.

In renormalizable theories the iterative substitution works. After performing the substitutions you worked out at the lower orders you're left with an unproblematic substitution at the current order. In non-renormalizable theories you are not. There will be an order of perturbation theory with an integral that even after the lower order substitutions will be left with ##M > N## divergences, with ##N## the number of parameters available to perform substitution with.
 

What is renormalization and why is it important?

Renormalization is a mathematical technique used in quantum field theory to remove infinities (known as divergences) from equations and make them physically meaningful. It allows for accurate predictions and calculations in particle physics and other fields of physics.

What are divergences and why do they occur?

Divergences occur in quantum field theory when equations produce infinite or undefined values. They occur due to the nature of quantum mechanics and the fact that particles can exist in multiple states simultaneously.

How does renormalization work?

Renormalization involves redefining certain parameters in equations to account for the effects of virtual particles and interactions between particles. This allows for the removal of infinities and the calculation of physically meaningful values.

What are the different types of divergences and how are they handled?

The two main types of divergences are ultraviolet (UV) and infrared (IR) divergences. UV divergences occur at very small distances and are removed through renormalization. IR divergences occur at very large distances and are typically ignored or accounted for through additional calculations.

What are the limitations of renormalization?

Renormalization is not a perfect or complete solution and has its limitations. It can only remove certain types of divergences and may introduce new uncertainties in calculations. It also does not fully explain the underlying physical processes and is still an area of active research in theoretical physics.

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