# 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

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.

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1. ### 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...

20. ### 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. ### 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. ### 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. ### 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...

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

How can we find a parametrization for SU(3) in terms of angles?
48. ### Defining Integration over SU(3)

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