Understanding the Standard Model

[FeDeX_LaTeX]

Hello;

I apologise for any typing errors, as I am typing from a handheld device. However, there is something that I do not understand about the standard model.

I have read that this equation is a 'lagrangian' but I do not know what this means or how to work with one. According to what I have read, this equation describes how different particles interact with each other. My question is "how?"... How does this equation tell us this? Thanks.

 

[Vanadium 50]

It's like a movie - you can't always come in the middle and expect to understand what's going on. A few months ago you were asking about Ohm's Law. To explain what a Lagrangian is in quantum field theory takes about six years of additional study.

 

[FeDeX_LaTeX]

So there is no brief explanation?

 

[Vanadium 50]

Not for someone who is still earning Ohm's Law. Even classically a Lagrangian is not a simple thing - it requires calculus of variations to understand what it's doing. Then you add QM, and then field theory.

 

[kaksmet]

Hello;
I have read that this equation is a 'lagrangian' but I do not know what this means or how to work with one. According to what I have read, this equation describes how different particles interact with each other. My question is "how?"... How does this equation tell us this? Thanks.

I am not, yet, an expert on the subject but I will still give it a go so that someone else can point out my misstake. It is definitely correct that it takes time to be able to understand how to work with a lagrangian, but also refuse to believe that there isn't some way to try to explain the picture in a simplified way.

The lagrangian in classical mechanics is L= K - V, where K is the kinetic energy and V is the potential energy. Using what is called the euler lagrange equation, which basically comes from the thought that the path taken will be an extremum of the action.
In the standardmodel, the lagrangian is a combination of different fields, which can be associated to the particles. If we then for example have a term of aABC where ABC are three different fields this will give us an interaction between these three fields (or particles) and the a is a parameter that describes the strength of the coupling between these fields.

Makes any sense?

 

[Naty1]

You can can get an introductory idea from reading/skimming here:

http://en.wikipedia.org/wiki/Lagrangian

Note the last section "SEE ALSO" for further references...

 

[Pythagorean]

I did about half a semester of coursework on the classical lagrangian in my undergrad and, despite being able to do the math, I still don't really understand it intuitively.

 

[petergreat]

I did about half a semester of coursework on the classical lagrangian in my undergrad and, despite being able to do the math, I still don't really understand it intuitively.

It's amazing (and bewildering) that each of the pillars of theoretical physics, including classical mechanics, electromagnetism, quantum mechanics, quantum field theory, and general relativity, can be encoded into the mysterious mathematical entity known as the Lagrangian. What's so wrong with "action" that Nature desperately wants to minimize it?

P.S. I realize that for classical theories, even including general relativity, the reason is trivial: any 2nd order differential equation can be derived from a variational principle. However, the fact that the action principle also incorporates quantum theory (via the path integral) is a genuine surprise, because its equivalence to canonical quantization isn't entirely obvious.

 

[element4]

It's amazing (and bewildering) that each of the pillars of theoretical physics, including classical mechanics, electromagnetism, quantum mechanics, quantum field theory, and general relativity, can be encoded into the mysterious mathematical entity known as the Lagrangian. What's so wrong with "action" that Nature desperately wants to minimize it?

Well, this might be amazing and it might not. Depends on one thing.

Given an differential equation, does there always exist a functional s.t. when minimized one gets the original differential equation? Anybody that know the precise mathematical conditions for this?

If this is always possible, then Lagrangians are just a convenient way of describing the physical problem and not necessarily something "deep". But I guess that there might be some serious conditions in order for such a functional to exist.

 

[element4]

It's amazing (and bewildering) that each of the pillars of theoretical physics, including classical mechanics, electromagnetism, quantum mechanics, quantum field theory, and general relativity, can be encoded into the mysterious mathematical entity known as the Lagrangian. What's so wrong with "action" that Nature desperately wants to minimize it?

P.S. I realize that for classical theories, even including general relativity, the reason is trivial: any 2nd order differential equation can be derived from a variational principle. However, the fact that the action principle also incorporates quantum theory (via the path integral) is a genuine surprise, because its equivalence to canonical quantization isn't entirely obvious.

You are too fast for me! You edited you own post and included the answer to my question, before I had posted my question! :)
Do you have a reference about "any 2nd order differential equation can be derived from a variational principle"?
With the quantum part, I agree. Not obvious at all.

 

[petergreat]

Do you have a reference about "any 2nd order differential equation can be derived from a variational principle"?

I must admit I got this opinion via diffusion rather than rigorous math... After some Googling I found this was the "Inverse Problem for Lagrangian Mechanics":
http://en.wikipedia.org/wiki/Inverse_problem_for_Lagrangian_mechanics"

 

[inempty]

This is my understanding below. I'm trying to explain the points in a relatively simple way. As I'm not very familiar with written English, please omit grammatical and wording mistakes in it.

Elementary Particles are considered fields at present. For a primary understanding, you can just regard a lagrangian as some quantity which describes natures of fields and their interactions. And as a standard procedure, instead of writing down equations physicists often draw diagrams named Feynman diagrams to calculate what they want. Feynman diagrams can tell us what will happen when particles interact with each other. For example, a Feynman diagram can be like this: we can see an electron emits a photon and the photon is received by another electron, etc. There are a couple of rules to calculate quantities with these diagrams.

And what is lagrangian? Classical mechanics can be summarized by a single principle:the least action principle, which means the real path is chosen from arbitrary virtual paths which share the same initial and final time and status in the way that the real path's "action" is the least, where action is the product of lagrangian and difference between the initial and the final time. In other words, given a dozen of paths, the one has the least product of time and lagrangian is the real path. How can we get the lagrangian? There isn't any general method and physicists find them by analogy and other physical principles, such as the principle of symmestry.

After gaining a proper lagrangian we actually know the all natures of fields involved. Then with a procedure called quantization and others physicists can derive the rules to calculate Feynman diagrams.

I hope explanations above can help you and you must learn more fundamental knowledge on classical mechanics and QM

 

[hamster143]

In other words, given a dozen of paths, the one has the least product of time and lagrangian is the real path.

On the other hand, in quantum mechanics, there's an infinite number of different paths, and the lagrangian let's you calculate the "amplitude" (probability, of sorts) of any particular path. Amplitudes of different paths can add or subtract, which means that some final outcomes are more likely and some are less likely. The "classical" path is the one that is most likely, and it is provably the one that extremizes the action.

 

[inempty]

Right, the classical action is just the phase of complex amplitude of a path.Then we can quantize the classical field into a quantum field with this simple rule.

 

[Naty1]

What's so wrong with "action" that Nature desperately wants to minimize it?

Quantum mechanic accidents can lead to the regularities in the universe...such 'accidents' produce emergent regularities...it's more likely a minimum action will result from random processes rather than one requiring a far more complicated process...