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18
votes
1answer
347 views

Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics: The universal cover: For instance, when representing the rotation group ...
1
vote
0answers
7 views

Writing Breit-Pauli spin-spin-coupling Hamiltonian as a sum of irreducible spin tensor operators

The spin-spin interaction part of the Breit-Pauli Hamiltonian for two electrons is essentially (omitting constants and the contact term) \begin{equation} \frac{3(\mathbf S_1 \cdot \mathbf r)(\mathbf ...
3
votes
4answers
517 views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
0
votes
0answers
32 views

Representations of spinors

I'm trying to understand spinors but I haven't found a good explanation of how to derive them. I'm generally interested. I read that there is a representation of SO(3) as spinors, and also that there ...
2
votes
2answers
46 views

Chiral symmetry breaking and confusion with terminology

I am confused about Chiral symmetry breaking and the terminology we use. First of all, I think the symmetry is started with taking quark masses zero and writing the Lagrangian as; $$ ...
3
votes
1answer
540 views

How to use Clebsch-Gordan coefficients for 3 particles?

I have a Hamiltonian for 3 particles of spin 1 that I boiled down to: \begin{equation} k(\textbf{S}^2+\cdots), \end{equation} where: \begin{equation} \textbf{S}=\textbf{S}_1+\textbf{S}_2+\textbf{S}_3. ...
0
votes
1answer
25 views

Implication of rotational symmetry on scattering matrix/ scattering cross-section [closed]

How does the rotational invariance helps simplifying Non-relativistic quantum scattering problems? Is there any any additional information that can be extracted about the scattering amplitude? It ...
0
votes
1answer
69 views

Susy transformation for gauge multiplet

How can the supersymmetrie transformation $\delta A_\mu = \frac{1}{2} \overline{\epsilon}\gamma_\mu \psi $ be derived from the susy algebra ( or group ). Where $ (A_\mu , \psi)$ are in a gauge ...
4
votes
1answer
111 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
2
votes
1answer
537 views

QCD color factors from quark gluon vertices

The color factors in QCD tell us the relative strength of the coupling of a quark emitting a gluon, a gluon emitting a quark-antiquark pair or a gluon emitting two gluons. To calculate let them we ...
1
vote
0answers
70 views

Books on representation theory [duplicate]

Possible Duplicate: Best books for mathematical background? I'm looking for a textbook on the group/representation theory for a student-physicist. The main questions of interest are ...
1
vote
1answer
53 views

Simple concept question about the dimensionality of a representation in point group

Concept question about the dimensionality of a representation in group theory here: Look at 3.1(c) of problem set, from group theory application to the physics of condensed matter of M.S.Dresselhaus: ...
0
votes
0answers
19 views

Do tensor product tables for irreducible representations apply for non-symmorphic space groups?

I'm reading Dresselhaus's book on group theory for solid-state physics, but I'm having trouble understanding how to get irreducible representations for phonons away from $\mathbf{k} = \mathbf{0}$ for ...
1
vote
1answer
58 views

Decomposition of tensor product space into direct sum [closed]

Consider a tensor product space of two representations $j = \frac{3}{2}$ and $j = 1$. How to show $4 \otimes 3 = 6 \oplus 4 \oplus 2$?
0
votes
0answers
46 views

Is metric $g$ a representation of Lorentz group? What decides it's transformation properties?

I am confused what representation of Lorentz group does a metric transform under? How does it's transformation properties are decided?
3
votes
0answers
26 views

How to build tetra-quark mesons symmetry group?

I was thinking about the issue while reviewing my group theory notes. One can construct mesons with a nonet as an octet and a singlet, $SU(3)\otimes SU(3) = 8\oplus \bar{1}$. In a same way but for ...
2
votes
1answer
137 views

Invariant tensors in a general representation and their physical meaning

I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of $SU(5)$. I can show that the invariant element in $5\otimes\bar{5}$ (or ...
3
votes
1answer
91 views

Why is there no 1/3 spin? [duplicate]

Why do no particles have a 1/3 spin? Why are all particles' spin either a half-integer or integer? How would a particle with such a spin behave, as a fermion, boson, or neither?
0
votes
1answer
42 views

How to normalize matrix representations properly?

In the convention, where the Dynkin index $Tr(T_a T_b)$ of the lowest-dimensional representation is $\frac{1}{2} \delta_{ab}$, how can I normalize a given set of matrices properly? For example, given ...
1
vote
2answers
81 views

The anticommutator of $SU(N)$ generators

For the Hermitian and traceless generators $T^A$ of the fundamental representation of the $SU(N)$ algebra the anticommutator can be written as $$ \{T^A,T^{B}\} = \frac{2N}{d}\delta^{AB}\cdot1_{d} + ...
3
votes
0answers
82 views

Characters of extra representations in the double group of O

Looking at the character table for $\overline{O}$ (double group of $O$) in a book, I noticed that two out of three of the additional irreps (with respect to the five irreps from $O$ itself) are ...
13
votes
3answers
7k views

Adding 3 electron spins

I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
1
vote
1answer
54 views

Poincare group representation and complete set

In Weinberg's book of Qft, chapter 2 of volume 1, he uses the eigenstates of the four-momentum to construct the unitary irreducible representations of the Poincare group. My question is, since ...
19
votes
1answer
3k views

How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?

This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell Show, by explicit calculation, that $(1/2,1/2)$ is the Lorentz Vector. I see that the ...
1
vote
0answers
30 views

Names for various color indices in QCD

In Quantum Chromodynamics with $\mathrm{SU}(3)$ there are at least two types of color indices: Indices $a$, $b$, … that index the eight generators of the group $\mathrm{SU}(3)$. In some sense they ...
4
votes
1answer
121 views

Why are one-particle states called representations of Poincaré group?

The one-particle states in the Hilbert space of a quantized relativistic field theory are said to form representations of the Poincaré group. Why is that? I mean, popular texts in QFT do not ...
3
votes
0answers
67 views

Representations of SO(3) and the classification of relativistic massive particles as in Weinberg's “The Quantum Theory of Fields”

I'm reading about the classification of relativistic massive particles in Weinberg's "The Quantum Theory of Fields", and I found something that doesn't convince me. In Chapter 2, paragraph 5, having ...
10
votes
2answers
427 views

Vector spaces for the irreducible representations of the Lorentz Group

EDIT: The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain ...
5
votes
1answer
210 views

Is a $SU(2)$ supergauge theory really a $SU(2)$ gauge theory?

Consider $SU(2)$ supergauge theory with $A$, a doublet of two chiral superfields in the fundamental representation. $$A= \begin{pmatrix} \Phi_1\\ \Phi_2 \end{pmatrix}$$ where $\Phi_1$ and $\Phi_2$ ...
1
vote
0answers
48 views

$SU(3)$ Tensor Methods in a Tetraquark

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
2
votes
2answers
108 views

Does every Hilbert Space carry a representation of Poincare group?

We know all infinite dimensional Hilbert Spaces are unitarily equivalent. It should follow therefore that if I have an unitary representation of say Lorentz or Poincare group on one infinite ...
1
vote
2answers
63 views

Complex conjugation of Weyl Spinors

Let $\chi$ be a left-handed Weyl spinor transforming as $$\delta\chi=\frac{1}{2}\omega_{\mu\nu}\sigma^{\mu\nu}\chi.$$ In my lecture notes it is explicitly stated that complex conjugation interchanges ...
1
vote
0answers
4k views

How is multiplicity given by 2S+1?

Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}$, $l_1 = 1$ and $s_2 = \frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ ...
3
votes
1answer
89 views

Representation of the Lorentz group

Is there any representation of the Lorentz group where $$U^{-1} f(x) U = f(\lambda^{-1}x)$$ other than the (0,0) representation? If not then is it possible for a field (with a well defined ...
1
vote
0answers
39 views

Is the coordinate transformation of an object the same of the action of a group on this same object?

I am having troubles in understanding frame transformations in physics from the mathematical point of view. What I understand for a coordinate transformation is just a function to one chart to another ...
3
votes
0answers
93 views

What defines the spin of a certain field? (formally)

Update: see the restatement of the question below! I've seen this question over and over through the archive of questions, but so far the closer to an answer was this. But I still don't understand. ...
1
vote
2answers
330 views

Operator vs. Matrix in quantum formalism

We use in Dirac formalism of QM the tool of operators and kets in spatial and spin spaces to obtain eigenvalues and eigenkets. But the operation here is simply that of a matrix multiplication. Now ...
1
vote
2answers
238 views

Is the fundamental representation of $SU(3)$ irreducible?

I want to check if the fundamental representation of $SU(3)$ is irreducible. The algebra is $$\mathbb{su}(3) = \{ m \in Mat(3,\mathbb{C} )\ |\ m = -m^+,\ Tr[m] = 0 \}$$ and I've found the generators. ...
3
votes
1answer
95 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) ...
13
votes
3answers
933 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...
5
votes
1answer
637 views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
1
vote
1answer
121 views

Trivial representation in Clebsch-Gordan decomposition

My professor defined the Clebsch-Gordan series as the direct sum decomposition of the tensor product of two representations of the Lie group SU(2): $$ D_{j_1} \otimes D_{j_2} = D_{j_1+j_2} \oplus ...
1
vote
1answer
83 views

Unitarily Inequivalent Representations

The definition of unitarily equivalent representations I am using is the one given here: https://en.wikipedia.org/wiki/Haag%27s_theorem. Now in this text ...
1
vote
1answer
62 views

Regarding different representations of the Lorentz Group & its defining properties

Take $\Lambda$ to be a Lorentz matrix, it satisfying $\Lambda^T \eta \Lambda=\eta$. By writing $\Lambda=\exp[-\frac{i}{2}\omega_{\mu\nu}\mathcal J^{\mu\nu}]$, we find that the generators satisfy $$ ...
4
votes
1answer
156 views

Anomaly cancellation in the standard model (calculating the symmetrized trace of generators)

The Problem We can show that the condition for the Standard Model to be anomaly-free is that the symmetrized trace over the generators of the gauge group vanishes: \begin{align} \text{tr} ...
1
vote
0answers
45 views

Mass, Spin, Internal Energy and 1-Particle States in Galilean Quantum Mechanics

I have been reading an article discussing the unitary representation of Galilean group and non-relativistic quantum mechanics. The link to the article is given below. http://arxiv.org/abs/1107.2442 ...
4
votes
0answers
113 views

The classification of particles or fields in general spacetime- Is it still meaningful to say spin-0, 1/2 ,1 field in general spacetime? [closed]

In 3+1 dim Minkovski spacetime, the classification of particle or field, that is spin-0, 1/2 , 1..., depends on the representation of the universal covering group of $SO(1,3)$, that is $SL(2,C)$. When ...
0
votes
0answers
27 views

trace of spherical components of an angular momentum multiplet

My basic mathematical question is whether or not there exist selection rules regarding the traces of the spherical tensor components of some operator acting on some subspace of definite total angular ...
9
votes
5answers
481 views

Seeking a quality plain-language description of the Wigner-Eckart theorem

I'm a third year physics undergrad with a very cursory knowledge of quantum mechanics and the formalism involved. For instance, I understand roughly how tensors work and what it means for a tensor to ...
2
votes
1answer
89 views

Representation of U(1) on fock space

I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ...