What is renormalization exactly?

[ayush solanki]

i am really confused about it.i know this much that it is used to cancel out the infinities to combine the theory of relativity and quantum physics.i just want to know how.and also what is super symmetry?please help.

 

[atyy]

There are two main ideas in renormalization.

1) The traditional procedure which cancels out the infinities and is nonsensical but it works.

2) The Wilsonian viewpoint in which our theory has a cutoff, representing the energy above which unknown degrees of freedom like undiscovered particles become important. Below that cutoff, we guess that we know the degrees of freedom and the symmetries, and we write down all terms consistent with this guess. Then we ask what happens if we do low energy experiments with finite resolution. The answer we get reproduces the answer from the traditional procedure, with small corrections. This is the way renormalization is conceived of nowadays. However, we still calculate with the traditional procedure, since it works.

A presentation of the Wilsonian viewpoint is given by Srednicki in http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf, Chapter 29, Effective Field Theory.

 

[ayush solanki]

thank you very much

 

[radium]

A good way to get the conceptual idea is to look at the original paper by Wilson and Kogut. Cardy's book is also very nice. Basically renormalization is done when a theory gives nonsensical results, like for the Casimir force, because our theory is flawed, we have included irrelevant information. In the Casimir force case we have to realize that the high energy modes don't matter, they would just tunnel right through. So we must add a cutoff. It doesn't matter what this is. Sometimes there is a natural cutoff, for condensed matter the natural one is a lattice spacing. When you use the cutoff to make a regulator, as long as the regulator dies sufficiently fast, you get the same answer. Casimir showed this in his original paper. When calculating loops in QED, we realize the coupling constant and the loop itself are not observable, we only carry about things that are like scattering. Observables must be finite. The coupling constants depend on distance

The business with things like counterterms comes from renormalization conditions you impose (there are different conventions) which modify the vertex on a loop.

The main idea in terms of the Ising model block spin picture is that all spins within a correlation length behave like on block. At the phase transition, the correlation length diverges. So as you go to this you can create renormalization equations which will flow to a fixed point, a point invariant under the renormalization transformation. This is done since this point must be scale invariant.

 

[A. Neumaier]

Renormalization is primarily a means of reparameterizing a system of differential equations.

In the context you place it (infinities) it is a means for choosing the parameterization such that a singular limit can be taken that gives physically relevant results.

In particular, renormalization is therefore used to give sense to quantum systems defined in terms of singular operators (in quantum mechanics) or fields (in quantum field theory). This is nontrivial since singular objects cannot be handled in the standard way without producing meaningless results. For an explanation how it works in simple cases where one can fully understand the meaning of the infinities and how to avoid them see my renormalization tutorial.

 

[bhobba]

If you want to see something really simple (for some maybe too simple) check out a paper I wrote on it:
https://www.physicsforums.com/threads/renormalisation-made-easy-comments.813025/

Thanks
Bill

 

[ayush solanki]

Thank u guys a lot nd also if u could explain me the gauge symmetry it wud be great.it is a part of renormalisation,is'nt it?

 

[ayush solanki]

Ur

Renormalization is primarily a means of reparameterizing a system of differential equations.

In the context you place it (infinities) it is a means for choosing the parameterization such that a singular limit can be taken that gives physically relevant results.

In particular, renormalization is therefore used to give sense to quantum systems defined in terms of singular operators (in quantum mechanics) or fields (in quantum field theory). This is nontrivial since singular objects cannot be handled in the standard way without producing meaningless results. For an explanation how it works in simple cases where one can fully understand the meaning of the infinitis and how to avoid them see my renormalization tutorial.

Your paper really helped thank you.

 

[A. Neumaier]

Gauge symmetry is not really related to renormalization. It is instead about the additional difficulties in quantizing constraints.

 

[ayush solanki]

Gauge symmetry is not really related to renormalization. It is instead about the additional difficulties in quantizing constraints.

Ok

 

[radium]

Gauge symmetry is actually not a real symmetry. A symmetry relates different states which are related by that symmetry. For examples, in band theory, symmetry in the little group at certain k points in the Brillouin zone correspond to degeneracies in the eigenvalue spectrum. The Ising model breaks Z2 symmetry in the ordered phase. The magnetic field picks a preferred direction for the system to order. However, in zero field, these two orientations are degenerate. They correspond to the two minima in the potential instead of the one minimum in the paramagnetic phase. However, you cannot tunnel from one into the other without breaking symmetry so these two states are topologically the same. Nonetheless, in zero field they are degenerate and can be related by a pi rotation.

Gauge symmetry does not related two degenerate states, it instead labels two seemingly different states as the same. For example in electromagnetism, you have a U(1) gauge. This corresponds to adding a phase to the wavefuntion. This represents charge conservation. There is a kinetic term in the Lagrangian corresponding to the charge. In the BCS it seems that you have broken U(1) gauge symmetry, however you have only broken it locally, not globally. In a BCS state, the current carriers are cooper pairs made of two electrons. If you modify the coupling to be 2e, you restore the gauge symmetry. Charge is conserved modulo two.

In my research involves the Ising lattice gauge theory. We construct states these resonating spin singlet pairs on the lattice in mean field theory. However, in doing this, we are over counting the number of states. Two get rid of this redundancy, there is an emergent gauge structure.