I've always heard of such statements as $SU$ group describing our world even before I seriously learned physics.   When learning about spin, someone spoke of $SU(2)$ but did not seriously explain. When learning isospin and hypercharge for particles made of $u, d, s,$ someone spoke of $SU(3)$ but did not give detailed explanation. When talking about the discovery of J/ψ, teacher (a experimentalist) say, well now with the charm quark, we need $SU(4)$.   I am crazy, $SU$ is just special unitary matrix group, what are they talking about when relating these matrix with spin or particles? I hope someone could give me a detailed explanation about how these group is related to real world substances or recommend a book.
123556
I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.
28383
I'm a chemist so bear with me:  I understand the  $\Delta^{+}$ and $\Delta^{0}$ to be in some sense spin (and isospin) quartet states of the proton and neutron. These can decay straight to a proton or neutron through emission of a pion, which is spin-0.  I've been puzzling about the mechanism by which this decay conserves spin. I note via Wikipedia that  into either a lepton pair or a photon pair, for which it's easy to show spin is conserved, however pion production seems to induce a loss of $J=1$.  This seems to suggest momenta of:  $\Delta^{+} (J=\frac{3}{2}) \rightarrow{} p^{+} (J=\frac{1}{2}) + \pi^{0} (J=0)$  Am I missing a boson? Is my understanding of spin flawed? I assume that if this decay exists, similar things turn up everywhere else.
3495
I often hear about subatomic particles having a property called "spin" but also that it doesn't actually relate to spinning about an axis like you would think. Which particles have spin? What does spin mean if not an actual spinning motion?
117543
I know that many kind of groups can be represented by matrices; for example: rotation groups can be represented by matrices.  Especially all elements of rotation groups can be represented by orthogonal matrices with determinant positive one. Also Permutation group can be represented by matrices.  But I need to know:    are there any kind of group which can not be represented by matrices?           Or is there any kind of group which does not have a representation?                 Can you show me sample?
72794
I am looking for references (books or web links) which have "simple" examples on the use of group theory in physics or science in general.   I have looked at many books on the subject unfortunately they usually require extensive technical coverage of the basics, i.e. the 1st 100 pages or something, to be able to start discussing applications.  I believe that there is an easy way to explain anything (it's just hard to find it).
114153
Why is there no analog for $\Theta_\text{QCD}$ for the weak interaction? Is this topological term generated? If not, why not? Is this related to the fact that $SU(2)_L$ is broken?
120471
Ok so I read the following article the other day:  and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit and Foote's book on abstract algebra and skimmed through it. It didn't say anything about Lie groups however. $E_8$ is coming to be rather famous so maybe other people are interested in this question too. Lets suppose I wanted to learn about lie groups. What books should I read to be ready to learn about Lie groups and what is a good book that talks about Lie groups. I'm guessing its a combination of group theory (representation theory in specific) and also differential geometry. Is this correct? Thank you very much for your time.
62877
I have seen a number of articles on Zitterbewegung claiming searches for it such as this one: . Others such as the so-called ZBW interpretation by Hestenes seemingly propose to explain electron spin as a consequence of ZBW.  According to Itzykson and Zuber p.62, Zitterbewegung is an artefact of considering a single-particle theory. It has been pointed out in this question's replies that it is not a physical phenomenon: .   How does the upgrade to many-particle theory solve the issue?
93992
I am reading a text on Lie groups.  There is a whole chapter devoted to the group of invertible real or complex matrices of degree $n$, which are called the general linear groups (complex and real).   Further in the text, there is a reference to a general linear group of a vector space, in general. What is meant exactly by that? And why do we use the same terminology?
57578
My impression for how Poincare and Lorentz representations are linked in $3+1$ dimensions is:   Assuming positive mass for simplicity, irreducible representations of the Poincare group are indexed by their mass $M$ and spin $s$. Irreducible representations of the Lorentz group are indexed by two numbers $(s_1, s_2)$ and when restricted to the rotation subgroup contains spins from $|s_1 - s_2|$ up to $s_1 + s_2$ in integer steps. Particles transform in representations of the Poincare group, while fields transform in representations of the Lorentz group. They are related by defining $$|\mathbf{p}, \lambda \rangle = a^\dagger(\mathbf{p}, \lambda) |0 \rangle$$ where $\lambda$ is the helicity, and the field is defined in terms of the creation and annihilation operators in the usual way. The only requirement is that the Lorentz representation used for the field contains the spin $s$ used in the Poincare irrep. A relativistic wave equation is imposed to eliminate the unwanted degrees of freedom.   For example, a massive spin $1$ particle may be embedded in a vector field, which is in the $(1/2, 1/2)$ irrep of the Lorentz group. This vector field has one extra degree of freedom that is eliminated by the equation $\partial_\mu A^\mu = 0$.  Assuming this is right, I'm wondering why I've never heard of "alternative" embeddings. For instance, why can't one embed a spin $1$ particle inside the $(1, 0)$ irrep, which doesn't have any extra degrees of freedom at all, or inside the $(1, 1)$ irrep? For massive particles, I've only ever seen spin $0$ in $(0, 0)$, spin $1/2$ in $(1/2, 0) + (0, 1/2)$, and spin $1$ in $(1/2, 1/2)$. Are other embeddings very clunky, or not allowed, or are they actually used somewhere?
100064
What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix}  0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \end{pmatrix},\,\frac{1}{\sqrt{2}}\begin{pmatrix}  0 &amp; -i &amp; 0 \\ i &amp; 0  &amp; -i \\ 0 &amp; i &amp; 0 \end{pmatrix},\,\begin{pmatrix}  1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; -1  \end{pmatrix} \right\} \end{equation}  I'm a bit puzzled with this, these are not the generators of the triplet representation of $SU(2)$ (the triplet in $SU(2)$ is the adjoint rep which is real and constructed with the structure constants), nevertheless they are used as if they were in much literature. What are really these generators? They seem like the extension to 3 dimensions of the Pauli matrices. The 3 dimensional fundamental rep (this does not make sense to me)? some kind of non-irreducible representation?
83590
I am looking at the tensor product of $n$ spin halfs (fundamental of $SU(2)$): \begin{equation} \left(\frac{1}{2}\right)^n = \frac{1}{2} \times \frac{1}{2} \times ... \times \frac{1}{2} = \frac{n}{2} + ... \end{equation} For example, for $n = 3$ \begin{equation} \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{2} \times \left(1 + 0\right) = \left(\frac{3}{2}\right)_1 + \left(\frac{1}{2}\right)_1 + \left( \frac{1}{2} \right)_0 \end{equation} where lower indexes on the right hand side specify from which tensor product representation came from. My question is how do I read off if a certain term on the right is a symmetric, antisymmetric or a mixed symmetry representation of $SU(2)$? I came up with two ways of approaching the problem.  First is Young Tableaux. As far as I understand rows of Young Tableaux correspond to symmetric indexes while columns to antisymmetric ones. So, for example, a diagram with just one row (like one that corresponds to the highest spin $\frac{3}{2}$ from the example - three boxes in a row) describes a totally symmetric representation. A diagram that has one column (maximum 2 boxes since we are dealing with $SU(2)$) describes totally antisymmetric representation. Everything that has both rows and columns is a mixed symmetry representation. Following this logic it looks like only the highest representation in the product can be totally symmetric, singlet is antisymmetric and everything in between has mixed symmetry. So, for the example \begin{equation} \left(\frac{3}{2}\right)_1: \text{symmetric rep}\\ \left(\frac{1}{2}\right)_{0,1}: \text{mixed symmetry reps}\\ \end{equation} Is this is a correct logic?  Another way to think about this is the following. Let's build representation the way we do in QM starting from the highest one, when all the spins are up. Then all components of the highest multiplet can be generated by applying lowering operators to the highest state. So, all the relative signs in the expressions are &quot;+&quot; and thus the highest representation is always symmetric as 1. also suggests. Then all the other representations must be constructed by writing down expressions orthogonal to components of the highest multiplet, so they will necessary get some &quot;-&quot; signs and as a result must be either antisymmetric or have mixed symmetry, but can not be totally symmetric.   Is this correct logic? Is it true that only the highest representation in the product can be symmetric and all the others have mixed symmetry or antisymmetric?
70320
Setup  Introducing this spinorbital notation:  \begin{align} \Psi_1=\chi_{(r1)}\alpha_{(\omega1)}  = 1 \\ \Psi_1=\chi_{(r1)}\beta_{(\omega1)}  = \bar{1} \end{align}  and the Slater's determinant, for a system of two electrons, is:  \begin{align} |\Psi\rangle = \frac{1}{\sqrt{2}} \Big|  \begin{array}{cc} 1 &amp; \bar{1}  \\ 2 &amp; \bar{2}  \\ \end{array}  \Big| = \frac{|1\bar{2}\rangle-|\bar{1}2\rangle}{\sqrt2} \end{align}  Question  Based on the example, what would be the Slater's determinant representation for an excited state like $\Psi_a^r$ (i.e. with parallel spins like: $|\bar{1}\bar{2}\rangle$) and $\Psi_{ab}^{rs}$ (feel free to use any number) ?
64229
From my reading, $SU(4)$ has 15 parameters and $SU(2)$ has 3 parameters that range differently with certain parameters (rotation angle). And all the parameters are linearly independent to each other.  My question are:   What the characteristic of each of the parameters? If I choose two of them, what is the reason behind the theory?  Can anyone explain to me or suggest any book/paper to me to understand the parameters themselves?
35511
If I understand correctly, spin is an intrinsic property of particles, which follows the algebra of angular momentum, but has nothing to do with an "orbital angular momentum" in that the particle is not like a small sphere that rotates on itself, to which we could attribute an angular momentum as usual in classical mechanics, cf.  and  Phys.SE posts.   My question is: why does spin combine with orbital angular momentum to give total angular momentum? To me this is surprising, as spin, in fact, has nothing to do with a classical angular momentum. Any insight on that?
80150
In the  of particle physics, there are three generations of quarks (up/down, strange/charm, and top/bottom), along with three generations of leptons (electron, muon, and tau).  All of these particles have been observed experimentally, and we don't seem to have seen anything new along these lines.  A priori, this doesn't eliminate the possibility of a fourth generation, but the physicists I've spoken to do not think additional generations are likely.     Question: What sort of theoretical or experimental reasons do we have for this limitation?   One reason I heard from my officemate is that we haven't seen new neutrinos.  Neutrinos seem to be light enough that if another generation's neutrino is too heavy to be detected, then the corresponding quarks would be massive enough that new physics might interfere with their existence.  This suggests the question: is there a general rule relating neutrino masses to quark masses, or would an exceptionally heavy neutrino just look bizarre but otherwise be okay with our current state of knowledge?  Another reason I've heard involves the Yukawa coupling between quarks and the Higgs field.  Apparently, if quark masses get much beyond the top quark mass, the coupling gets strong enough that QCD fails to accurately describe the resulting theory.  My wild guess is that this really means perturbative expansions in Feynman diagrams don't even pretend to converge, but that it may not necessarily eliminate alternative techniques like lattice QCD (about which I know nothing).  Additional reasons would be greatly appreciated, and any words or references (the more mathy the better) that would help to illuminate the previous paragraphs would be nice.
37015
It seems to me that we could change all the current spin values of particles by multiplying them by two. Then we could describe Bosons as even spin particles and Fermions as odd spin particles. Is there some consequence or reason why we can't simply change 1/2 as the smallest unit of spin into 1?  It just seems prettier this way. In fact, there's almost an interesting relation to even and odd functions in that if you switch any two bosons around the wave function stays the same while if you switch any two fermions around the wave function becomes negative.