What is the purpose of Faddeev-Popov ghosts in the standard model lagrangian?

[um0123]

Does anyone have a high resolution picture of the standard model lagrangian? i want to put it on a t shirt and give it to my math teacher who worked at Bell Laboratories for 10 years and knows just about everything about physics and math.

P.S. i only think its called the standard model lagrangian, what I am talking about is the really long equation that takes up like a long scroll.

P.P.S. A picture of the equation itself and not the scroll with the original equation is just as good.

 

[humanino]

From wiki
http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html
http://blogs.discovermagazine.com/cosmicvariance/files/uploads/smshort.pdf

 

[um0123]

http://nuclear.ucdavis.edu/~tgutierr/files/sml2.pdf

that can't be a REAL exercise question. can it!?

btw, thanks for the response, but is there a way to get that pdf into an image?

 

[humanino]

http://nuclear.ucdavis.edu/~tgutierr/files/sml2.pdf

that can't be a REAL exercise question. can it!?

The website author calls it "fun yet soul-crushing exam question". Whether it may be real depends on the cruelty of the teacher convoluted with the talent of the students.

btw, thanks for the response, but is there a way to get that pdf into an image?

Now this is not really a physics question anymore
I attached the pdf converted to png, hopefully that suits your puproses

 

[um0123]

The website author calls it "fun yet soul-crushing exam question". Whether it may be real depends on the cruelty of the teacher convoluted with the talent of the students.
Now this is not really a physics question anymore
I attached the pdf converted to png, hopefully that suits your puproses

Haha, what teacher would do such a thing!

and thanks for the image conversion, i know it wasnt a physics question or even anyones responsability but my own, so i think you for the effort.

 

[um0123]

oh i forgot to ask, why is when i type standard model lagrangian on google it comes up with a different equation. One that isn't a completely connected equation. Like this:

Is that related to it? it must be, google wouldn't lie to me.

 

[hamster143]

#1 only contains matter-related terms in compact notation, it's missing all Higgs-related terms (all terms with H from the original link) and all gauge field self-coupling and propagation terms.

#2 only contains electroweak terms (GWS subscript stands for Glashow-Weinberg-Salam).

Humanino's link, on the other hand, contains, in addition to usual SM, some stuff which I'm having trouble identifying right away (what's G, X, and Y, for example?)

 

[humanino]

G, X, and Y

X and Y are ghosts, and IIRC G is the glue field (g is the graviton, so depending with whom the f's are, they represent different structure constants).

 

[hamster143]

I can accept X and Y to be ghosts, though they shouldn't really be part of the lagrangian. G is not the glue field, two-index g is: notice that the fourth term couples gluon with two quarks. G couples to itself (fifth term) and there's a vertex between two G's and a gluon (sixth term), and it has no other interactions.

I'm guessing that $$\phi$$'s are also ghosts? That's a very ghostly lagrangian ...

 

[humanino]

G is not the glue field

You are right, my bad, I should have paid attention, there is no graviton. Thank you for the correction.

That's a very ghostly lagrangian ...

Indeed, all bosons have their ghosts.
Y for the photon, 3 X's for the weak vectors, 8 G's for the gluons, and $\phi$'s for the charged and neutral Higgs' ghosts.

 

[um0123]

ghosts? scary! Do i need to have advanced mathematical knowldege to know what those are? or can it be explained to me?

So there are different types of the standard model lagrangian to calculate different things, and the one that humanio posted is the one that includes everything (even the higgs which hasnt been discovered).

Do each have names, like is the first one i posted call the Matter Lagrangian? and would the second be the Electro-weak Standard model?

 

[humanino]

ghosts? scary! Do i need to have advanced mathematical knowldege to know what those are? or can it be explained to me?

They are not as romantic as their Faddeev-Popov name suggests. Mostly they enable us to conveniently include a jacobian in a path integral, where the jacobian is necessary because we have a delta function to enforce gauge invariance.
http://en.wikipedia.org/wiki/Faddeev-Popov_ghost