Su(2) Definition and 121 Threads

Is it purely coincidental that the internal symmetry related flavor quantum numbers(like isospin and weak isospin) and the spacetime symmetry related spin quantum number have SU(2) as underlying group? They refer to seemingly unrelated things but it is remarkable how ubiquitous SU(2) is.

A paper I'm reading says "Our starting point is the SU(N) generalization of the quantum Heisenberg model: H=-J\sum_{\langle i,j \rangle}H_{ij}=\frac{J}{N}\sum_{\langle i,j \rangle}\sum_{\alpha , \beta =1}^N J_{\beta}^{\alpha}(i)J_{\alpha}^{\beta}(j) The J_{\beta}^{\alpha} are the generators of...

In Ryder's Quantum Field Theory it is shown that the Lie Algebra associated with the Lorentz group may be written as \begin{eqnarray} \begin{aligned}\left[ A_x , A_y \right] = iA_z \text{ and cyclic perms,} \\ \left[ B_x , B_y \right] = iB_z \text{ and cyclic perms,} \\ \left[ A_i ,B_j...

In Howard Georgi's Lie Algebras in Particle Physics (and other texts I'm sure), it is determined that the Pauli matrices, \sigma_1 \sigma_2 and \sigma_3, in 2 dimensions form an irreducible representation of the SU(2) algebra. This is a bit confusion to me. The SU(2) algebra is given by...

Well I am trying to understand the adjoint representation of the su(2) algebra. We know that the algebra is given: [X_{i}, X_{j}]= ε_{ij}^{k} X_{k} (maybe I forgot an i but I am not sure). The adjoint representation is then ( in the matrix representation) defined by the ε_{ijk} structure...

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, From perusing books on QFT, I've gathered that the photon is written as a 4-vector in field theory and transforms under the standard Lorentz group operators, while an electron for instance is a 2-component spinor and transforms under a special representation of the Lorentz group as part of...

Hi, I think I need a sanity check, because I've been working on this for a while and I can't see what I'm doing wrong! According to several authors, including Sakurai (Modern QM eq 3.3.21), a general way to write an operator from SU(2) is...

Could anyone explain why these are invariant under U(1) X SU(2)? H^{dagger}H (H^{dagger}H)^{2} What is the condition for invariance under U(1) and similarly, under SU(2)? Unfortunately, I am not familiar with tensor contraction or tensor products...

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...

Homework Statement Find the 3D representation of what I think are the commutators [T_a,T_b] for the SU(2) groupHomework Equations I think the generators(X_i) in SU(2) group are the 3 Pauli matrices, which are 2X2 matrices... I assume I need to find the matrices for these generators as 3x3...

Homework Statement Compute the volume of the group SU(2) Homework Equations Possibly related: in a previous part of the problem I showed that any element g = cos(\theta) + i \hat{n} \cdot \vec{\sigma}sin(\theta) The Attempt at a Solution How do I compute the infinitesimal...

Hi, I'm an undergraduate taking the basic quantum classes and on my own, I'm trying to wrap my mind around how symmetry and group theory applies in Q.M. and theoretical physics in general; it's coming along slowly but surely! Can someone please explain why the ammonia molecule is said to...

I am confused with SU(2). How do you prove that the generators are pauli matrices (1/2 sigma)? I would appreciate any link or reference with details of all algebraic steps, taking one from the conditions of unitarity and det=1 to the well know exponential form. In general, how do you find...

I have a trivial mathematical problem with SU(2) parametrization. In www.mat.univie.ac.at/~westra/so3su2.pdf , section 3, there is a sentence starting with "We first assume b = 0 and find then(...)". My question is: doesn't assuming that b = 0 reduce generality of our parametrization? If not, why?

SU(2) a double cover for Lorentz group? I'm presently reading the new book, "Symmetry and the Standard Model", by Matthew Robinson. On page 120, he writes, "the Lorentz group (SO(1,3), pg 117) is actually made up of two copies of SU(2). We want to reiterate that this is only true in 1+3...

Homework Statement I nedd some help to write a left-invariant and right invariant metric on SU(2) Homework Equations The Attempt at a Solution

Hello! I have two questions regarding QM. First, I tried to rotate J_z into J_y for the j=1 representation by the transformation rule for matrices.. I took my rotationvector to be (1,0,0) and rotated about 3/4 pi radians which I thought would give me S_y. Instead I got S_z again! Wolfram...

I ran across the following passage in the Wikipedia article on mass-energy equivalence: This level of physics is way over my head, but I'm wondering: "What happens to the quarks that comprise the protons and neutrons?" Are they conserved in the neutrinos and antielectrons? Chris

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...

Hi all, do you know where i can find the 3x3 matrix representation of SU(2)? Which means basically rotation matrices for particles of spin 1. Thanks!

Hi, Could someone explain to me how to split SU(2) into its axial and vector subgroups, what does this mean? (The context I'm trying to understand this in is the U(2)_L x U(2)_R global flavour sym of chiral Lagrangian) A related question: I know that the three axial generators of SU(2)_L x...

Homework Statement Derive the pure SU(2) YM theory on \mathbb{R}^4 from the action. Let A_{\mu} (x) be a solution to these equations. Show: \tilde{A}_{\mu} (cx) is also a solution (with the same action).Background The Euclidean YM action \mathbb{S} = - \int_{\mathbb{R^4}} Tr (F \wedge...

Hi, this is probably very simple. 1) What is the product of two singlets? 2) What is the product of two singlets and a doublet? It looks like (2) breaks SU(2) symmetry and (1) doesn't, but I don't really understand why =/. Any help would be appreciated, Thanks!

Hello, I'm reading Ryder's Quantum Field Theory book, and I'm reading the preliminary part where he discusses a little bit of Groups before he introduces the Dirac equation. So, as an example, he is talking about the identification between the SU(2) group and the O(3) group. Before he gets there...

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.

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.

one standard basis of su(2) are the 2x2 matrices (i 0;0 -i), (0 i; i 0), (0 1;-1 0) whereas the standard basis of sl(2) are (1 ; 0 -1), (0 1; 0 0), (0 0;-1 0) Why then is su(2) called a real algebra, but not sl(2)? thanks

Increasingly QG is being done with a q-deformed symmetry group: replacing SU(2) by the quantum group SUq(2). What do you see as the intuitive basis for this? One intuitive justification is that in a universe with a minimum measurable size and an a maximum distance to horizon there is an...

Hi all, (Also - if anybody could tell me how to get the latex to work on this page that'd be very handy!) While not technically homework this is a problem I've found I'm stuck on during my revision. Any help would be greatly appreciated. Homework Statement "By demanding that the covariant...

what is meant by the custodial SU(2) approximate symmetry as a 5, a 3? it is always mentioned in the Higgs triplet, charged higgs context.

I have been struggling with this for a long time. I gave up to review GR and came back to Ryder. I started in Chapter 2 and the material was easier and more intuitive on this second pass. But the same topic, "SU(2) and the rotation group" has trapped me yet again. I am bogged down specifically...

can the neutrino mass eigenstate couple to the group of SU(2) doublet?if we intentionally not impose any flavor symmetry on it. \left(\begin{array}{c}\nu_{1}\\e\end{array}\right)

In the context of LQG spin networks are derived based on Ashtekar's formulation implementing local SU(2) gauge symmetry in tangent space. Here SU(2) and 3+1 dim. spacetime are deeply related. Forgetting about this derivation and starting with spin networks, talking about dimensions does no...

The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))? Is a representation R of some group G a group too? If so, is it true that G is representation of R?

all elements of su(2) can be written as \exp(iH) with H being a traceless hermitian matrix thus H can be written as the sum of \sigma_x,\sigma_y,\sigma_z H=\theta (n_x \sigma_x + n_y \sigma_y+ n_z \sigma_z). Here (n_x,n_y,n_z) is a unit vector in R^3. we can take \theta in the...

why group SO(3) is not any good reference on the relation of SU(2) and SO(3)?

I have not seen why SU(2) and SO(3) groups are isomorphic?

Hi all, If I define Tij = a+i aj, then C2 = T11T11 + T12T21 + T21T12 + T22T22 is a second order casimir operator. For SU(2), it's \frac{N}{2} (\frac{N}{2} + 1) But as I calculate it directly, C2 = a+1 a1a+1 a1 + a+1 a2a+2 a1 + a+2 a1a+1 a1 + a+2 a2a+2 a2 = a+1 a1a+1 a1 + a+1...

I am pretty confused about how to construct states to make symmetric / anti-symmetric combination so I would like to ask some questions. For example, for SU(2), states of three spin-half particles can be decomposed as 2 x 2 x 2 = 4 + 2 + 2, 3 irreducible combination with dim 4, 2, 2. -if...

I want a good reference (not difficult) for su(2) for Its representaion Its lie algebra It's applications in angular momentum

Can I check with someone - is the following pauli matrix in SU(2): 0 -i i 0 Matrices in SU(2) take this form, I think: a b -b* a* (where * represents complex conjugation) It seems to me that the matrix at the top isn't in SU(2) - if b=-i, (-b*) should be -i...

Homework Statement How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)? The Attempt at a Solution I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had...

Why is it that the Pauli spin matrices ( the operators of quantum spin in x,y,z) are the generators of a representation of SU(2)? I understand that we use the 2X2 representation as it is the simplest, but why is it that spin obeys this SU(2) symmetry and how is it that we come up with the Pauli...

Homework Statement Finding the matrix A such that, exp(sA) is in SU(2) Homework Equations My attempt is in trying to solve \left(e^{sA}\right)^{t} B \left(e^{sA}\right) = B for A, where A is some 2x2 (complex?) matrix. and B is the matrix representing the group of SU(2) matrices. Trouble...

Homework Statement I'm working through a bit of group theory (specifically SU(2) commutation relations). I have a question a bout symmetries in the SU(2) group. It's something I'm trying to work through in my lecture notes for particle physics, but it's a bit of a mathsy question so I thought...

I know that there are many reasons why SU(2) can't be the electroweak gauge group, but I want to have some clarifications about the following one, that disergads neutral currents: in this case the currents are (considering only the lepton sector of the first generation)...

If you have doublet Q=(u,d) , and want to give the u-quark mass, you have to connect it to the Higgs VEV H=(\nu,0) doublet through the adjoint opertion: H^{\dagger i}Q_i Connecting H and Q through the Levi-Civita symbol e_{ij} : e^{ji} H_{ i}Q_j results in d-quark mass, not u-quark...

Hi, 1: I just want to ask that what does SU(3) x SU(2) x U(1) means? and when a lagrangian is invariant under SU(3) x SU(2) x U(1) what does that mean? Does it mean that lagrangian L is invariant under SU(3) and then SU(2) and then U(1)? so if one wants to check SU(3) invariance of L...