In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.
The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on
C
n
{\displaystyle \mathbb {C} ^{n}}
. It is itself a subgroup of the general linear group,
SU
(
n
)
⊂
U
(
n
)
⊂
GL
(
n
,
C
)
{\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n,\mathbb {C} )}
.
The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.
Spin 1/2 particles are two states system in C^2 and so it is natural for the rotations to be described by SU(2), for three states systems like spin - 1 particle, Why do we still use SU(2) and not SU(3) to describe the rotations? Is it possible to derive them without resorting to the eigenvalue...
The question may be ambiguous but it's really simple. One says that the baryon octet is the D(1,1) representation of SU(3), but then uses the same one for mesons. D(1,1) means one quark and one antiquark, which corresponds perfectly to mesons. But how can it explain baryons?
My information and...
I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states.
How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...
Summary: if we use up, down and staring quarks and their own antiparticle we can create the Eightfold way and understand mesons by the hyper charge and isospin projections.
I don't understand how the conjugate representation of SU(3) allows us to create a vector space of dimension 3, while...
Hello there,
Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way?
In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the...
If we consider the generators of SU(3) in the standard model. Is there a direct correspondence between them and a physical quantity, esp. if we only consider ##T_+=\frac{1}{2}(\lambda_1+i\lambda_2)\; , \;U_+=\frac{1}{2}(\lambda_6+i\lambda_7)\; , \;V_+=\frac{1}{2}(\lambda_4+i\lambda_5)##, do they...
Homework Statement
Determine the Lie bracket for 2 elements of SU(3).
Homework Equations
[X,Y] = JXY - JYX where J are the Jacobean matrices
The Attempt at a Solution
I exponentiated λ1 and λ2 to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector...
I am trying to work out the weights of the adjoint representation of SU(3) by calculating the 2 Cartan
generators as follows:
I obtain the structure constants from λa and λ8 using:
[λa,λb] = ifabcλc
I get:
f312 = 1
f321 = -1
f345 = 1/2
f354 = -1/2
f367 = -1/2
f376 = 1/2
f845 = √3/2
f854 =...
Hi everyone,
this is something i know because i saw it many times, but i have never fully understand it. Suppose i have a quark field (singlet under SU(2) let's say) ##q## and i would like to build an invariant term to write in the Lagrangian. The obvious choice is to write a mass-term...
The lagrangian of a non interacting quark is made to be invariant under local SU(3) transformations by introduction of a new field, the gauge field, giving rise to the gluon. This gives us a locally gauge invariant lagrangian for the quark field and together with the construction of a locally...
Hello! I am reading something related to algebra in particle physics and I want to make sure I got it. So, they say the u, d and s quarks can represent the basis of the SU(3) representation when the diagonalizable matrices are Y=B+S and ##I_3##. But, if I want to look only in the ##I_3## space...
i used to get pauli matrices by the following steps
it uses the symmetry of a complex plane sphere i guess so..?
however i can't get the 8 gell mann matrices
please help !
method*: (x y) * (a b / c d ) = (x' y')
use |x|^2 + |y|^2 = |x'|^2 + |y'|^2
and |x| = x * x(complex conjugate)
this way...
I read that hadrons are in colour singlet state and that gluons are not and that the colour singlet gluon is forbidden for the reason of making strong force a long range force otherwise (and that SU(3) has 8 generators and thus 8 gluons) but my question is: are mesons in a colour singlet state...
Not sure if this is the correct forum but here goes.
I am trying to prove [Ta,Tb] = ifabcTc
Where (Ta)bc = -ifabc and fabcare the structure constants for SU(3).
I picked f123 and generated the three 8 x 8 matrices .. T1, T2 and T3.
The matrices components are all 0 except for,
(T1)23 = -i...
Hi,
I trying to understand. If there is non-trivial SU(3) group, is it always possible to find SU(2) as part of SU(3)?
And same question about SU(2) and U(1).
Suppose that in the tensor component ##T^a_b ## the upper index is the ## \bf{3}## component and the lower index is the ##\bf{\bar{3}} ## component. To be concrete, consider the decomposition
u^iv_j= \left( u^iv_j-\frac{1}{3}\delta^i_j u^kv_k \right) +\frac{1}{3}\delta^i_j u^kv_k
which...
'In the SU(3) quark model there are two singlet vector states $$|\omega_8 \rangle = \frac{1}{\sqrt{6}} \left(|u \bar u \rangle + |d \bar d \rangle - 2 |s \bar s \rangle \right) $$ belonging to the octet and the pure singlet state $$|\omega_1 \rangle = \frac{1}{\sqrt{3}} \left(|u \bar u \rangle +...
Dear All
I just have a question. We say that the SU(2) doublet have the same value of isospin but the particles of this multiplet differs by I3. Now what quantum number the particles of SU(3) multiplet share.
Thank you
Hi.
General question: Is there a fixed way to find all invariant tensor for a generic representation?
Example problem: Suppose you search for all indipendent quartic interactions of a scalar octet field ## \phi^{a} ## in the adjoint representation of SU(3). They will be terms like
##...
Hi all,
The trace of two SU(3) generators can be calculated by:
## T_{ij} T_{ji} = \frac{1}{2} ##, now how to calculate the trace of SU(3) generators:
## T_{il} T_{lk} T_{kj} T_{ji} ## ?
Homework Statement
I'm trying to figure out this question:
"Show that the 10-dimensional representation R3,0 of A2 corresponds to a reducible representation of the LC[SU(2)] subalgebra corresponding to any root. Find the irreducible components of this representation. Does the answer depend on...
'Using the following normalization in the su(3) algebra ##[\lambda_i, \lambda_j] = 2if_{ijk}\lambda_k##, we see that ##g_{ij} = 4f_{ikl}f_{jkl} = 12 \delta_{ij}## and, by expanding the anticommutator in invariant tensors, we have further that $$\left\{\lambda_i, \lambda_j\right\} =...
I am trying to work out with Young graphs the tensor product of:
\bar{3} \otimes \bar{3}
The problem is that I end up with:
\bar{3} \otimes \bar{3} = 15 \oplus 6 \oplus 3 \oplus 3
Is that correct? It doesn't seem correct at all (dimensionally speaking I should have taken something like...
I am still learning about all the Groups related to the Dirac Equation for spin 1/2 particles. Apparently, the reason that the Hilbert Space for spin 1/2 particles is 2-dimensional is because when you try to map SU(2) to SO(3), the mapping is 2-to-1, i.e. SU(2) is a double cover for SO(3)...
I have been reading Georgi "Lie Algebras in Particle Physics" and on page 183 he mentions how that the SU(3) defining representation decomposes into an SU(2) doublet with hyperchage (1/3) and singlet with hypercharge (-2/3). I am confused on how he knows this. I apologize if this is not the...
Hello,
I am trying to understand the concept of symmetries, SU(2), SU(3), unitary group, orthogonal group SO(1)...so on.
I don't know from where to start and what would be the first group to study and then move on step by step into the other.
Also, I need to have a basic (theoretical)...
I understand that when the quark theory was being developed that SU(3) was used to explain the mesons that were ultimately found to be composed of the up, down, and strange quarks. I also get that the SU(3) is grouped as an octet and a singlet, with the eta prime meson being the singlet. But I'm...
I have a question regarding symmetry groups. I've often heard that the Standard Model is a SU(3) x SU(2) x U(1) theory. From what I understand these groups contain the symmetries under which the Lagrangian function is invariant. If so, what does every one of the 3 groups above contain (what...
Hi All,
I'm working through the theory of the strong interaction and I roughly follow it. However I have some questions about the meaning of the terms.
The book I use gives the gauge transformation as: \psi \rightarrow e^{i \lambda . a(x)} \psi
First question ... What are the a(x)...
Hi everyone!
I would like to ask you a very basic question on the decomposition 3\otimes\bar 3=1\oplus 8 of su(3) representation.
Suppose I have a tensor that transforms under the 8 representation (the adjoint rep), of the form O^{y}_{x}
where upper index belongs to the $\bar 3$ rep and the...
Dear Everyone,
I have a problem to be solved now. how to calculate the symplectic group SP(3,R) non-compact induced by the SU(3) group? Any reference provided will be appreciated.
Is anyone familiar with a
$$ SU\left( 3 \right) \otimes U \left( 1 \right){}_d$$ or $$SU \left( 3 \right) \otimes SU\left( 2 \right) \otimes U \left( 1 \right)_d $$ model? Kind of what I'm currently interesting in working with, but I don't have access to anything other than the arxiv.cheers.
Hello! If all the elements of a Unitary group can be found using Euler's formula, does that mean that each unitary group represents some kind of cyclic transformation, since we are talking about a circle? I think I read that U(1) is a phase transformation, and SU(2) is a spin transformation...
Hi everyone. I'm not sure this is the correct section for this topic and if not my apologiez.
I'm studying SU(3) and my professor wrote down the following equality:
$$Tr\left(\left[ T^a_8,T^b_8\right] T^c_8\right)=i\frac{3}{2}f^{abc}$$
where Ts are generators of the adjoint...
Hi all
I found these equalities from Gordon Brown (1963).
He uses the killing form to measure the length of the roots in a semi simple algebra.
First and second equalities are quite obvious and come from the definition.
Could you help me for the last one which prove that we have a...
The Standard Model symmetries are U(1), SU(2), and SU(3). But I'm not sure whether these are symmetries of the Action intgral or if they are symmetries of the background spacetime.
Please teach me this:
How many conserved observations(''charges'') are there in SU(2) and SU(3) symmetries?I know that U(1) has only one charge that is electric charge.
Thank you very much for your kind helping.
As we know the algebra of SU(3) consist of two Cartan generators and 6 raising and lowering operators. We define the eigenstates of the Cartan operators as u,d,s, correspoding to the three lightest quarks.
Now when we study the 3\otimes 3 tensor product we can show that the Hilbert space of...
Please teach me this:
Can we deduce Ward Identity in QCD from U(1) symmetry of QED?Because QCD is a theory of quarks and quarks have electric charge.So we need not deduce the Ward Identity from SU(3) symmetry,but we can be able to demontrate the Ward Identity( considering gluons)with U(1)...
Please teach me this:
It seem to me that lepton manifests broken symmetry SU(2) with couple electron and neutrino(electron is a state with mass,neutrino is a state with nearly zero mass).Similarly for 2 other families of lepton,we have a state with mass and a state with nearly zero mass.But I...
I am wondering what the meaning of X is in formulations such as SU(3) X SU(2) X U(1). The symbol is used a lot but I've never seen it explained. I'm assuming it's not any kind of multiplication but ... Clarification would be appreciated.
Dear Every One,
In literatures on QCD confinement, I usually see the words ``center of group''.
It is defined to be the subgroup of some parent group and consists of elements which
commutes with all elements from the parent group. But what is the center of SU(3)
group? I need...
Homework Statement
The generators of SU(3) are the Gell Mann matrices, \lambda_a. Consider symmetry breaking of an SU(3) theory generated by a triplet of complex scalar fields \Phi = \left(\phi_1, \phi_2, \phi_3\right). Assuming the corresponding potential has a minimum at \Phi_0 =...
Hi,
I'm reading the SU(N) chapter in Jones' Group theory book. In SU(3) we have these 3 component spinors which transform as \psi^{'}_{a}=U_{a}^{..b}\psi_{b} and we have upper spinors defined by \psi^{a}=\epsilon^{abc}\phi_{[bc]}
Now if consider building up higher-dimensional reps, by taking...