What is Su(3): Definition and 67 Discussions

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



{\displaystyle \mathbb {C} ^{n}}
. It is itself a subgroup of the general linear group,









{\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.

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  1. spin_100

    A SU(2) and SU(3) representations to describe spin states

    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...
  2. James1238765

    I Use of Gell-Mann matrices as the SU(3) basis for gluon states?

    The 8 gluon fields of SU(3) can be represented (generated) by the 8 Gel-Mann matrices: $$ \lambda_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \lambda_2 = \begin{bmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , \lambda_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0...
  3. joneall

    A Why does D(1,1) representation of SU(3) give baryon octet?

    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...
  4. RicardoMP

    A Decomposing SU(4) into SU(3) x U(1)

    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...
  5. Jelly-bean

    I Why do we need two representations of SU(3)

    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...
  6. P

    A Highest weight of representations of Lie Algebras

    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...
  7. fresh_42

    I SU(3) Generators & Physical Quantity Corresp.

    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...
  8. N

    Lie Bracket for Group Elements of SU(3)

    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...
  9. N

    SU(3) Cartan Generators in Adjoint Representation

    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 =...
  10. Luca_Mantani

    A Invariant combination of SU(3) states

    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...
  11. C

    I SU(3) Gauge Group: QCD & SM Invariance Explained

    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...
  12. S

    I Can Quarks Form a Basis for SU(2) Using Only the I3 Space?

    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...
  13. K

    B Matrices of su(3) and sphere symmetry

    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...
  14. SWFvanRijk

    A Are Mesons in Colour Singlet State?

    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...
  15. N

    I Bases for SU(3) Adjoint representation

    What are the bases for the adjoint representation for SU(3)?
  16. N

    I Adjoint representation of SU(3)

    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...
  17. A

    A Is SU(3) always contains SU(2) groups?

    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).
  18. P

    A How Do SU(3) Tensors Decompose into Irreducible Components?

    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...
  19. C

    I SU(3) quark model and singlet states

    '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 +...
  20. S

    A Exploring the Quantum Numbers of SU(3) Multiplets

    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
  21. Lamia

    SU(3) octet scalar quartic interactions

    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 ##...
  22. Safinaz

    How do I calculate the trace of SU(3) generators in the adjoint representation?

    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} ## ?
  23. D

    Representations of SU(3) Algebra

    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...
  24. C

    How Is the Anticommutator Derived in SU(3) Algebra?

    '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\} =...
  25. ChrisVer

    Decomposing an SU(3) product in irreps

    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...
  26. LarryS

    Spin 1 Particle Representations of SO(3) and SU(2)

    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)...
  27. K

    SU(3) defining representation (3) decomposition under SU(2) x U(1) subgroup.

    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...
  28. shounakbhatta

    SU(2), SU(3) and other symmetries

    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)...
  29. G

    Eta prime meson as SU(3) singlet

    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...
  30. Q

    Symmetry Groups of the Standard Model: SU(3) x SU(2) x U(1)

    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...
  31. N

    Meaning of terms in SU(3) gauge transformation

    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)...
  32. L

    Tensor techniques in $3\otimes\bar 3$ representation of su(3)

    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...
  33. D

    Calculate the Clebsch-Gordon coefficient for the SP(3,R) by SU(3)

    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.
  34. P

    SU(3) and dark U(1) coupling to dark matter

    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.
  35. lonewolf219

    What is common theme between U(1) SU(2) and SU(3)

    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...
  36. Safinaz

    What are the SU(3) structure constants and how can they be calculated?

    Hi, How can I calculate such terms d^Aae d^ebD f^AaE f^EbD d^Aae d^EbD f^AaE f^ebD Cheers, Safinaz
  37. Einj

    Problem with SU(3) generators's trace

    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...
  38. naima

    Length of roots in su(2) su(3) and other Lie algebras.

    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...
  39. F

    U(1), SU(2), SU(3) are symmetry of what?

    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.
  40. N

    How many ''charges'' are there in SU(2) and SU(3) symmetry?

    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.
  41. R

    Decomposition of SU(3) and particles

    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...
  42. N

    Does Ward Identity in QCD has origin of U(1) or SU(3) symmetry?

    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)...
  43. Z

    How to prove that SU(3) is compact

    How to prove that SU(3) is compact?I have no idea how to do this . And What is the significance of The compactness of SU(3) on the quark model?
  44. N

    How do 6 quarks manifest hiden SU(2) symmetry(together SU(3) symmetry)?

    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...
  45. A

    Meaning of X as in SU(3) X SU(2) X U(1)

    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.
  46. S

    What is the center of SU(3) group

    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...
  47. Y

    Finding a Parametrization for SU(3) in Terms of Angles

    How can we find a parametrization for SU(3) in terms of angles?
  48. J

    Defining Integration over SU(3)

    How is integration over the group SU(3) defined?
  49. J

    Spontaneous Symmetry Breaking of SU(3)

    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 =...
  50. L

    Can anyone explain this term is antisymmetric in SU(3)

    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...