Group and Quantum Field Theory

[Calabi]

Good afternoon : I now what I've written here : https://www.physicsforums.com/showthread.php?t=763322 in the first message. I've made the Clebsh Cordon theorem with the components. Which can be represented by the Young tableau.

There also the $$SU(3)$$ and the $$su(3)$$ representation of dimension 3. The base represent the quarks. I know nothing on the quarks. Just that each quarks is represent by a vector of a base. The anti quarks are in the conjugued representation.

What is the link with what I said here and here https://www.physicsforums.com/showthread.php?t=763322 in the first message and the quantum field theory please?

I want to know now who to make Physiks with this.

How this represnetation of $$SO(3)$$, $$SU(2)$$ and $$SU(3)$$ and their Lie algebra intervene in the Quantum Field Theory please?

This space matrix are group.

What is a gauge please?

Thank you in advance and have a nice afternoon.

 

[PhilDSP]

Hi Calabi,

There is a very good book (in my opinion) covering this that was originally written in French. However, it's graded at the graduate level so you might need some introductory material.

"Spinors in Physics" by Jean Hladik.

 

[Calabi]

Good morning and many thanks. I'm going to see for the book.
I can offer it to me but I've not a library near to me to buy it. And I don't trust on the Internet.

I'll have to order it in a Bibliothèque near to me.

However what do you think of what I said please?

I need somme help. I'm lost.

Thank you in advance and have a nice morning.

 

[PhilDSP]

I think you need to ask one very focused question at a time. There are really too many there above to address.

One basic observation is that SU(2) involves a 2 x 2 matrix of complex numbers. So it can be applied to 3 physical or real spatial dimensions plus potentially time or another one dimensional basis. SU(3) involves a 3 x 3 matrix of complex numbers that can be applied to a richer set of dimensions, far greater than the 3 spatial dimensions.

 

[Calabi]

I don't really now how to start.

For exemple : I've said that the quarks could be represented by the vectors of a base in a 3 dimension space representation of $$SU(3)$$.

We have already said to me that $$SU(3)$$ is the gauge group of Strong Interaction which link the quarks.

I'm sure it's not a coincidence.

What is a gauge grup in that case please? What does it mean please?

Thank you in advance and have a nice afternoon.

 

[Avodyne]

If you do not know what a "gauge group" is, you need to start by reading a QFT text. You can't expect people here to write out standard textbook material for you.

 

[Calabi]

If you do not know what a "gauge group" is, you need to start by reading a QFT text. You can't expect people here to write out standard textbook material for you.

I'm already reading a course on this subject thanks. I just want to have an idea of what it is. I never ask people to wright a course here.

Just a way to follow to help.
And as I say I'm a few lost. I want to now how to make Physics withe those Lie Group Representation.

Have a nice afternoon.

 

[microsansfil]

For exemple : I've said that the quarks could be represented by the vectors of a base in a 3 dimension space representation of $$SU(3)$$.

It is not my understand of Groups and Representations.

Here a course on "Quantum Theory, Groups and Representations". Cover the basics of quantum mechanics, up to and including basic material on relativistic quantum field theory, from a point of view emphasizing the role of unitary representations of Lie groups in the foundations of the subject.

Patrick

 

[ChrisVer]

gauge groups are in general connected to symmetries under those transformations (gauge transformations).
One can recall from a quantum mechanic course, that the wavefunctions that are connected by an overall phase are equivalent, which means that either you use $\psi (x)$ or $e^{ia} \psi (x)$ with $a \in R$, you get the same information from the wavefunction. This is a global U(1) "rotation" symmetry. The photon field $A_\mu$ is also arbitrary up to a gauge transformation (known from electrodynamics) where you know that you can choose gauges on which to work (eg, Lorentz gauge, Coulomb gauge, and a whole set of gauge families).
That's a general idea someone has from known (before QFT) physics. The general idea is then given by the fact that you want to make symmetric objects under gauge groups, and that's how one can start building the Lagrangians needed for the field theory.
In general, in most introductions to QFT, one tries to create a fermionic theory (Dirac Lagrangian) obey a local U(1) transformation, local stands for the fact that the transformation parameter $a$ I used above can depend on the spacetime position $a \rightarrow a(x)$. This eventually lead to the reproduction of "electromagnetic" theory, with $A_\mu$ the photon field playing the role of the connection (knowing some GR can help you see some analogies with SR to GR and the introduction of the connection there, with $\eta _{\mu \nu} \rightarrow g_{\mu \nu} (x)$).
Gauge transformation for example is the reason the photon has one less physical degree of freedom than working in any other gauge (choosing a particular equivalent gauge, you can kill one component). If you don't set a specific gauge, you are going to reproduce more (unphysical) degrees of freedom for the photons (sometimes called Faddeev Popov Ghosts). These objects will only contribute in internal loops and not as final assymptotic states [that's why they are unphysical].

For SU(3) now, I suppose you ask for the flavor representations (of up, down and strange quarks) you have the representation $3$ and $\bar{3}$, from combinations of these representations you can have the rest irreducible representations. The $3$ is a three-dimensional representation, the basis vectors are then the quarks u,d,s...
For example $3 \otimes 3 = 6 \oplus \bar{3}$ and $3 \otimes \bar{3} = 8 \oplus 1$. The last means that the quarks (uds) and their antiquarks, can form an 8 dimensional represention (the octatet of mesons) and a singlet (that's the $\eta'$ meson).
And then you can have the
$3 \otimes 3 \otimes 3 = 3 \otimes (6 \oplus \bar{3} ) = (3 \otimes 6) \oplus (3 \otimes \bar{3})= 10 \oplus \bar{8} \oplus 8 \oplus 1$.
The $3 \otimes 3 \otimes 3$ which means the combination of 3 quarks of different flavors, can be decomposed to the irreducible representations of dimension 10, 8 (and its conjugate) and a singlet representation. These corresponds to the hadron 10plet,8plets and singlets.
In general the $SU(3)$ flavor is an approximate symmetry, which is broken because of the differences in the quark masses.
The Young tableaux can help you see the combinations applicable for these sets of n-plets.

The more appropriate symmetry, the one that is used in QCD, is the $SU(3)_{color}$ where instead of having quarks (u,d,s), you have colors (R,G,B). This symmetry is also broken at low energies (because of non-perturbative QCD effects, which are dealt in general by lattice QCD). In the Lagrangian language you are looking for objects that don't change under the symmetry transformations. These objects are combinations of $3$ (colors) and $\bar{3}$ (anticolors) that contain the singlet (singlets don't change under symmetry transformations, since they can be considered "scalar" quantities). From the above you've seen that $3 \otimes \bar{3}$ contains the singlet, as well as $3 \otimes 3 \otimes 3$. So if you take a field which belongs in $3$ you can either combine it with 2 other same objects or with 1 existing in $\bar{3}$. That way you can have lagrangians that don't change under SU(3) color, and this gauge symmetry is a symmetry of your problem.