The standard model, a couple of q's

[karnten07]

I have a couple of questions about the standard model:

1.) I read somewhere that the standard model describes 3 closely related Yang-Mills systems, but i thought yang-mills systems were non-abelian whereas U(1) is abelian. So is the statement false or have i misunderstood?

2.)Why do i see subscripts for the gauge symettries of SU(3)xSU(2)xU(1) and they are soemtimes different depending where i read from. Can anyone guide me on this please?

Many thanks

karnten07

 

[blechman]

I have a couple of questions about the standard model:

1.) I read somewhere that the standard model describes 3 closely related Yang-Mills systems, but i thought yang-mills systems were non-abelian whereas U(1) is abelian. So is the statement false or have i misunderstood?

this is just semantics. Some people use "Yang-Mills" to refer to nonabelian gauge theory, while others use "Yang-Mills" and "gauge theory" interchangably. It doesn't really make a difference what you call it! ;-)

2.)Why do i see subscripts for the gauge symettries of SU(3)xSU(2)xU(1) and they are soemtimes different depending where i read from. Can anyone guide me on this please?

Many thanks

karnten07

SU(3) or SU(3)_C refers to QCD - the gauge theory of the strong interaction; the C stands for "color", referring to the three color charges of QCD.

SU(2) or SU(2)_W or SU(2)_L all refer to part of the gauge theory of the weak nuclear force. The W stands for "Weak" while the "L" stands for "left-handed", since left-handed fields are charged under this group while the right handed fields are not.

U(1) or U(1)_Y refers to the other part of the electroweak force, called "hypercharge" (denoted "Y" purely for historical reasons). The SU(2) x U(1) break to make U(1)_EM, the electromagnetic force.

Hope that helps!

 

[karnten07]

this is just semantics. Some people use "Yang-Mills" to refer to nonabelian gauge theory, while others use "Yang-Mills" and "gauge theory" interchangably. It doesn't really make a difference what you call it! ;-)



SU(3) or SU(3)_C refers to QCD - the gauge theory of the strong interaction; the C stands for "color", referring to the three color charges of QCD.

SU(2) or SU(2)_W or SU(2)_L all refer to part of the gauge theory of the weak nuclear force. The W stands for "Weak" while the "L" stands for "left-handed", since left-handed fields are charged under this group while the right handed fields are not.

U(1) or U(1)_Y refers to the other part of the electroweak force, called "hypercharge" (denoted "Y" purely for historical reasons). The SU(2) x U(1) break to make U(1)_EM, the electromagnetic force.

Hope that helps!

That is most helpful, thankyou. Just so I'm clear, are the subscripts used to show that we are doing physics and not just talking about the mathematical groups or are they just there to remind us of what each is representative of?

Also, i read that the SM involves these 3 systems, the higgs field and several matter fields. What are the matter fields?

 

[blechman]

That is most helpful, thankyou. Just so I'm clear, are the subscripts used to show that we are doing physics and not just talking about the mathematical groups or are they just there to remind us of what each is representative of?

yes... and yes. why do you have an "or"?!

Also, i read that the SM involves these 3 systems, the higgs field and several matter fields. What are the matter fields?

"matter fields" are the fermions - quarks and leptons.

 

[Vanadium 50]

this is just semantics. Some people use "Yang-Mills" to refer to nonabelian gauge theory, while others use "Yang-Mills" and "gauge theory" interchangably. It doesn't really make a difference what you call it! ;-)

Indeed. Particularly since the original theory of Yang and Mills was incorrect. They tried to use this to explain the interaction between hadrons and failed. What we call Yang-Mills today are generalizations of this original idea - which I have to say was brilliant, even if it was originally applied to the wrong thing. But where you draw the line at generalizations is, at some level, arbitrary.

I would probably be inclined to take a historical approach, and call everything that isn't a U(1) a Yang-Mills theory. But that's my choice, and others probably differ.

 

[karnten07]

yes... and yes. why do you have an "or"?!



"matter fields" are the fermions - quarks and leptons.

Ok that's very helpful, is it a matter of choice to include these subscripts or is it a must to include them for things like papers or dissertations?

 

[blechman]

Indeed. Particularly since the original theory of Yang and Mills was incorrect. They tried to use this to explain the interaction between hadrons and failed. What we call Yang-Mills today are generalizations of this original idea - which I have to say was brilliant, even if it was originally applied to the wrong thing. But where you draw the line at generalizations is, at some level, arbitrary.

The same can be said about Kaluza-Klein modes, which now refer to any extra modes you get from a field when compactifying on an extra dimension, whereas THE KK modes of Kaluza's and Klein's 1920-something paper refer to a pure gravitational theory with a single extra dimension compactified on a circle. But the idea was generic enough to keep the name for any generalization.

I would probably be inclined to take a historical approach, and call everything that isn't a U(1) a Yang-Mills theory. But that's my choice, and others probably differ.

since we're collecting opinions, with all due respect to Yang and Mills, of course, I just refer to all of these as "Gauge Theories" and don't worry about whether or not there are abelian factors. But whatever...

Ok that's very helpful, is it a matter of choice to include these subscripts or is it a must to include them for things like papers or dissertations?

Are you writing a thesis? I guess you include the subscripts if you want to emphasize the point. But usually people can figure it out. If you see the words

"Standard Model"

followed by the symbols

SU(3) x SU(2) x U(1)

I think it's pretty clear what the paper is talking about!