Su(2) Definition and 121 Threads

All of the formulations of the Lie algebra of ##\frak{so}(3)## (or ##\frak{su}(2)##) utilizing raising/lowering operators that I have seen in the literature involve complexification to ##\frak{su}(2) + i \frak{su}(2) \cong \frak{sl}(2,\mathbb{C})##. I have found explicit derivations in a...

Hi, I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements \begin{pmatrix} e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\ ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}...

Hi, ##SU(2)## group as topological space is homeomorphic to the 3-sphere ##\mathbb S^3##. Since ##SU(2)## matrices are unitary there is a natural bijection between them and points on ##\mathbb S^3##. In order to define an homeomorphism a topology is needed on both spaces involved. ##\mathbb...

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...

For two quantum oscillators, I have raising and lowering operators and , and the number operator . I need to check if operators below follow commutation relations. Now as far as I know, SU(2) algebra commutation relation is [T_1, T_2] = i ε^ijk T_3. So, should I just get T_1 and T_2 in...

Could anyone help with some of the later parts of the derivation for Dirac spinors, please? I understand that an arbitrary vector ##\vec v## $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$ can be defined as an equivalent matrix V with the components $$ \begin{bmatrix} z & x - iy \\ x + iy...

hi, i have studied Standard Model for particle physics - at present it is described by three groups - i have studied - these groups but could not establish what particular feature suggest of these group to be used to describe SM. Thanks

1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...

Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: $$\Phi^\dagger...

Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3##...

While writing down the basis for SU(2), physicists often choose traceless hermitian matrices as such, often the Pauli matrices. Why is this? In particular why traceless, and why hermitian?

Hello guys, I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here...

Hello! I want to make sure I understand the relation between this and rotation (mainly between SU(2) and SO(3), but also in general). Also, I am a physics major, so I apologize if my statements are not very rigorous, but I want to make sure I understand the basic underlying concepts. So SU(2) is...

Hello! I am reading some representation theory and I am a bit confused about some stuff. I read that SU(2) is the double covering of SO(3), so to each matrix in SO(3) corresponds one in SU(2). I am not sure I understand this. So if we have a 3D representation of SU(2), the 3D object it acts on...

Homework Statement [/B] I am looking at this document. http://www.math.columbia.edu/~woit/notes3.pdf Homework Equations [/B] ad(x)y = [x,y] Ad(X) = gXg-1The Attempt at a Solution [/B] I understand how ad(S1) and X is found but I don't understand what g and g-1 to use to find Ad(X). Also...

Greg Bernhardt submitted a new PF Insights post A Journey to The Manifold SU(2) - Part II Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post A Journey to The Manifold - Part I Continue reading the Original PF Insights Post.

I'm reading a paper on physics where it's said it can be shown that every irreducible representation of ##SU(2)## is equivalent to the one which uses the Ladder Operators. I am a noob when it comes to this subject, but I'd like to know whether or not the proof is easy to carry out.

Hello! I am reading some Lie Algebra and at a point the author says that for a vector with 3 cartesian components ##V_i## i =1,2,3 the commutation relations with the generators of rotation are: ##[J_i,V_j]=i\epsilon_{ijk}V_k##. Can someone explain this to me? I am confused as ##V_j## is a number...

From "Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson", it states that 'SU(2) matrix has one of the real parameters fixed,leaving three real parameters'. I don't really get this part and hope someone can clear my doubt...

Homework Statement Hi, I'm trying to self-study quantum mechanics, with a special interest for the group-theoretical aspect of it. I found in the internet some lecture notes from Professor Woit that I fouund interesting, so I decided to use them as my guide. Unfortunately I'm now stuck at a...

I know that for SU(2), weak interaction, in the standard model the right handed leptons are singlets, (and right handed neutrinos don't exist). For right handed quarks are they singlets or doublets in the standard model. So is it (u d)R or is it just u(R) and d(R)

Homework Statement Using the irreducible representation of ##su(2)##, with ##j=\frac{5}{2}##, calculate ##J_z##, ##exp(itJ_z)## and ##J_x##. Homework EquationsThe Attempt at a Solution There seem to be loads of irreducible representations of ##su(2)## online, but no reference at all to a...

Hi all, I read in Cheng and Li's book "Gauge theory of elementary particle physics" Ch 11, specifically : Eq. (11.46) that the hypercharge commutes with the SU(2) generators, i.e., ##[Q-T_3,T_i]=0##, I'd like to understand what that mean and how this could be proved ?

The matrix representation ##U## for the group ##SU(2)## is given by ##U = \begin{bmatrix} \alpha & -\beta^{*} \\ \beta & \alpha^{*} \\ \end{bmatrix}## where ##\alpha## and ##\beta## are complex numbers and ##|\alpha|^{2}+|\beta|^{2}=1##. This can be derived using the unitary of...

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...

In QCD, quark is in fundamental representation of SU(3) and thus it has to have 3 charges (what we came to call "colors"). Gauge bosons are in adjoint representation and there are 8 of them. The choice how to assign color charges to them is not unique, one popular choice is based on Gell-Mann...

Let the generators of the SU(2) algebra be ##\tau_{1}##, ##\tau_{2}## and ##\tau_{3}##. Consider an ##N## dimensional representation, which means that the ##\tau_{i}## are ##N \times N## matrices which act on some ##N##-dimensional vector space. Consider the ladder operators...

I've got a question about the identification of SU(2) with O(3) in Ryder's QFT book (2nd edition) pages 34 - 35. The other posts on this topic I could find don't seem to address this question, so here goes. He derives the matrix in eqn 2.47: $$H= \left[\begin{array}{cc} -\xi_{1}\xi_{2} &...

Hello! I just started reading about SU(2) (the book is Lie Algebras in Particle Physics by Howard Georgi) and I am confused about something - I attached a screenshot of those parts. So, for what I understood by now, the SU(2) are 2x2 matrices whose generators are Pauli matrices and they act on a...

Homework Statement Good day, From my reading, SU(4) have 15 parameter and SU(2) has 3 paramater that range differently with certain parameter(rotation angle). And all the parameter is linearly independent to each other. My question are: 1. What the characteristic of each of the parameter? 2...

Homework Statement How do I use the commutation relations of su(2) and so(3) to construct a Lie-algebra isomorphism between these two algebras? Homework Equations The commutation relations are [ta, tb] = i epsilonabc tc, the ts being the basis elements of the algebras. They basically have the...

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).

The Lie group SU(2) is the set of unitary 2x2 matrices with determinant 1. These matrices can be written a b -b* a* Thus, as a manifold, we can think of a coordinate chart consisting of the four real numbers making up the two complex numbers a and b. It is a...

Consider pion states composed of ##q \bar q## pairs where ##q \in \left\{u,d \right\}## transforms under an ##SU(2)## isospin flavour symmetry. These bound states transform in the tensor product ##R_1 \otimes R_2## of two representations ##(R_1, R_2)## of ##SU(2)##. Take ##R_2## as the...

Within my project thesis I stumbled over the term SU(2)_V, SU(2)_A transformations. Although I know U(1)_V, U(1)_A transformations from the left and right handed quarks( U(1)_V transformations transform left and right handed quarks the same way, while U(1)_A transformations transform them with a...

I have a left-handed ##SU(2)## lepton doublet: ## \ell_L = \begin{pmatrix} \psi_{\nu,L} \\ \psi_{e,L} \end{pmatrix}. ## I want to know its transformation properties under conjugation and similar 'basic' transformations: ##\ell^{\dagger}_L, \bar{\ell}_L, \ell^c_L, \bar{\ell}^c_L## and the general...

In section 3.5 of his textbook Quantum Field Theory, Ryder discusses an example of a non-Abelian gauge theory. He considers a 3D internal space and rotations in this space. At first he shows that the fields in this internal space transform like ##\delta \vec \phi=-\vec \Lambda \times \vec \phi...

Hello everyone, I have a question that will probably turn out to be trivial. I have the following matrix: $$ U=\text{diag}(e^{2i\alpha},e^{-i\alpha},e^{-i\alpha}). $$ This seems to me as an SU(2) matrix in the adjoint representation since it's unitary and has determinant 1. Am I right? If so...

Homework Statement Construct the decompositions ##\mathbf 2 \otimes \mathbf 2 = \mathbf 3 \oplus \mathbf 1##, where ##\mathbf N## is the representation of su(2) with ##\mathbf N## states and thus spin j=1/2 (N-1). Homework Equations Substates within a state labelled by j can take on values -j...

I am curious as to the meaning of, and name given to the phase ##\xi(t)## which may be added as a prefix to the time evolution operator ##\hat{U}(t)##. This phase acts to shift the energy of the dynamical phase ##<{\psi(t)}|\hat{H}(t)|\psi(t)>##, since it appears in the Hamiltonian along the...

Hi, a fairly quick question. I'm reading Bruce Schumm's book "Deep Down Things" and he says that in SU(2) you have to rotate by 720 degrees to return to your starting point. This is clearly the same definition as a spinor. My question is, then, does rotation in SU(2) automatically imply the...

When we start by postulating local SU(2) gauge invariance for our weak isospin doublets \begin{align} \psi &= \begin{pmatrix} \nu_{e} \\ e^{-} \end{pmatrix}_{L} \end{align} etc., we have to introduce massless gauge fields to preserve the Lagrangian's invariance. For SU(2) this demands 3...

Is it possible to prove that weak isospin associated with SU(2) is conserved using Noether's theorem?

Hi. I know that the \sigma matrices are the generators of the rotations in su(2) space. They satisfy [\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k It is conventional therefore to take J_i=\frac{1}{2}\sigma_i such that [J_i,J_j]=i\epsilon_{ijk}\sigma_k . Isn't there a problem by taking these...

I've been having fun with my new Lie Algebra text and it occurred to me that working out a couple of basic examples of my own would be a good idea. I got rather large surprise. The example I'm working with is SU(2) and I'm going through some basic properties it has. For all its uses in...

Please, I'm stuck. How to get the rhs from the lhs? X^{\alpha}_{\ \ \alpha^{\ \prime}}X^{\beta}_{\ \ \beta^{\ \prime}}\epsilon^{\alpha^{\ \prime}\beta^{\ \prime}}=det X\epsilon^{\alpha\beta}

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...

SU(2) matrices act isometrically on the Riemann sphere with the chordal metric. At the same time the group of automorphisms of the Riemann sphere is isomorphic to the group SL(2, C) of isometries of H 3(hyperbolic space) i.e. every orientation-preserving isometry of H 3 gives rise to a Möbius...

I am somewhat confused with the connection between the two groups. In the text I'm reading (An Introduction to Tensors and Group theory for physicists N. Jeevanjee), there is a chapter quite early on (in the group theory part) which outlines a homomorphism from SU(2) to SO(3), however I find...