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3
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1answer
71 views

On the Lorentz Group representation [on hold]

I am going through the notes on QFT by Srednicki (which is certainly a worth reading on the subject, and can be found online, see http://web.physics.ucsb.edu/~mark/qft.html). When describing ...
9
votes
0answers
74 views

Degenaracy in mass of $8$ and $27$ reps of $SU(3)$ in Coleman's Aspects of Symmetry

In Coleman's Aspect of symmetry he proposes an amusing problem in the first chapter. It asks us to consider a set of eight pseudo-scalar fields transforming in the adjoint representation of $SU(3)$. ...
1
vote
1answer
29 views

How to check if some term in the Lagrangian involving gauge bosons is gauge invariant without explicit computations?

Normally (for fermions and scalars) we can simply use the decomposition of tensor products of gauge group representations to find invariant terms that we can write into the Lagrangian. For example ...
0
votes
1answer
28 views

Do gauge bosons really transform according to the adjoint representation of the gauge group?

Its commonly said that gauge bosons transform according to the adjoint representation of the corresponding gauge group. For example, for $SU(2)$ the gauge bosons live in the adjoint $3$ dimensional ...
9
votes
4answers
344 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 ...
0
votes
1answer
50 views

Rest Mass and Wigner's Classification

I believe (but please correct me if I'm wrong) that I understand the basic philosophy and most of the mathematics involved in Wigner's classification of particles via group representations. But I'm ...
4
votes
1answer
351 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 ...
3
votes
1answer
67 views

Representation of the Standard Model group $SU(3) \times SU(2) \times U(1)$

As the gauge group of the Standard Model is $SU(3) \times SU(2) \times U(1)$, would the associated fermions fields be the product of a triplet, a doublet and a singlet, for all particles, or is that ...
8
votes
1answer
846 views

Decomposing a Tensor Product of $SU(3)$ Representations in Irreps

Can somebody explain in a simple way why, talking about representations $$3\otimes3\otimes3=1\oplus8\oplus8\oplus10~?$$ Here $3$ and $\bar{3}$ are the fundamental and anti-fundamental of $SU(3)$, in ...
1
vote
1answer
94 views

How is the Full Standard Model group representation displayed?

I have often seen, on YouTube lectures and textbooks, the direct product gauge group representation listed below and it is often accompanied with a statement to the effect that "this is how we sum ...
3
votes
1answer
81 views

Why do decompositons like $16 \otimes 16 = 10 \oplus 120 \oplus 126$ tell us which Higgs representations we can use?

EDIT: I found an answer, which I do not understand: In Gürsey - Symmetry breaking patterns in E6 he writes: " Because of Fermi-Dirac statistics of fermions they must occur in the symmetric part of ...
0
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0answers
38 views

Invariant linearly independent scalar potential construction for product groups

Lets say one has a gauge group for example SU(n) or SO(n) and has a scalar field which belongs to a certain representation (m-ranked tensor). If one wants to write down the invariant potential for the ...
7
votes
1answer
176 views

Group representations as vectors and isomorphism between weights and matrix generators

This might be something basic, but it is unclear to me. So I am used to work with representations of groups as matrices. These matrices represent the structure of the Lie algebra by satisfying the ...
2
votes
2answers
112 views

Where does the Lorentz boost for a Dirac spinor come from?

I have read, that if you have a Dirac spinor \begin{equation} \psi = \begin{pmatrix} \phi_R\\ \phi_L \end{pmatrix} \end{equation} that you can apply a Lorentz boost along the $z$-direction with ...
13
votes
3answers
721 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}} = ...
1
vote
1answer
66 views

Do particles have spin because there exist spinor representations for the Lorentz group?

I am reading Peskin and Schroeder's An introduction to field theory. They first describe the spinor representation of the Lorentz group, and then they mention the fact that different particles have ...
1
vote
0answers
37 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 ...
2
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0answers
41 views

Traceless Tensors in $SU(3)$, Georgi's Lie Algebras

I'm doing a self-study through Georgi's Lie Algebra's in Particle Physics and there is a ''note without proof'' in the book that I have not managed to see through myself. In Section 10.3, Georgi ...
3
votes
3answers
62 views

Uniqueness of expression of a Lie group element

Just take the SU(2) group as an example. The three generators are $J_z$, $J_+$, and $J_-$. For an element $ g $, sometimes we want to express it as $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
2
votes
1answer
297 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
62 views

Finding Electronic Energy Levels by Representation Theory

Let $$u=\left( \begin{array}{cccc} c_1&c_2&c_3&c_4 \end{array} \right)^T$$ for $$\psi = c_1\psi_1 + c_2\psi_2 + c_3\psi_3+ c_4\psi_4$$ We assume that $<\psi_i|\psi_j> = \delta_{ij}$ ...
4
votes
2answers
157 views

Traces in different representation

I am actually working with Green-Schwarz anomaly cancellation mechanism in which I have came across a strange formula which relates trace in the adjoint representation (Tr) to trace in fundamental ...
0
votes
0answers
38 views

How to count completely symmetric wavefunctions can be made by 6 (u, d, s, $\bar{u},\bar{d},\bar{s}$) quarks to form Baryons (i.e., 3 particle state)?

(Excluding antisymmetric part i.e., colour part) Peskin said 6 (particles + Anti-Particles) "guys" make (6(i.e., total "guys") x7x8)/(3x2)= 56 total symmetric state. How? There are 10 (decuplate) ...
12
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3answers
5k 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 ...
5
votes
3answers
424 views

Explaining a quote by Weinberg about the signifcance of symmetry groups in physics

When skimming through a book, I found this quote: The universe is an enormous direct product of representations of symmetry groups. —Steven Weinberg I am a mathematician (so I know only basic ...
0
votes
1answer
58 views

Getting to spins of arbitrary direction

Let me rephrase this question: Let us assume we know that symmetry transformations always look like this: $$U(s)=e^{iKs} $$ with a hermitian Operator K. This tells us that for very small $s$: ...
1
vote
1answer
60 views

Representations of Lorentz algebra

It is well known that the Lorentz algebra can be written as two $SU(2)$ algebras. By defining $$N_i=\frac{1}{2}(J_i+iK_i), \qquad N^{\dagger}_i=\frac{1}{2}(J_i-iK_i)$$ we have ...
1
vote
0answers
19 views

symmetry group of multi-electron atom

Neglecting spin effects, the energy levels of multi-electron atoms are characterized by states of definite total orbital ($L^2$) and spin angular momentum ($S^2$). From this it seems that the ...
1
vote
0answers
34 views

Decomposition of a tensor under transformations

To illustrate my question I'll take an example from theory of relativity: An arbitrary 4-tensor $A^{ik}$ changes under a general coordinate transformation: $$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$ ...
4
votes
1answer
122 views

Why $SU(3)$ has eight generators?

The generators of $SU(3)$ group are Gell-Mann matrices and one can construct these generators from Pauli spin matrices, basically expanding in 3d and rotating about each axis. Take $\sigma_3$, assume ...
1
vote
1answer
90 views

Spin commutation relations

For orbital angular momentum defined as $L= r \times p $ we can prove, in quantum mechanics, the commutation relations. Also, we could prove these relationships through the study of rotations ...
1
vote
2answers
90 views

What are the unitary operators for various transformation?

Transformations, at least in lagrangian-symmetries context, are usualy described as uintary operators. I dont understand what are these operators exactly. For example, let's look at the Lorentz ...
1
vote
2answers
66 views

Calculations with angular momentum

Is the following correct, when adding 3 angular momenta/spins: \begin{align} 1\otimes 1\otimes \frac{1}{2}&=\left(1\otimes 1\right)\otimes \frac{1}{2} \\ &=\left(2\oplus 1\oplus ...
3
votes
1answer
104 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. ...
3
votes
1answer
59 views

3D isotropic oscillator and angular momentum algebra

In our QM class, the prof said: "We are ready to begin constructing the individual states of the 3D isotropic harmonic oscillator system. The key property is that the states must organize ...
2
votes
1answer
82 views

Spinors and Möbius strips

I asked this question on Math.SE as I thought the perspective of representation theory might be enlightening. But since the question was provoked by a description of Spinors describing the spin of ...
14
votes
1answer
690 views

Wick rotation and spinors

I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; ...
5
votes
2answers
63 views

SU(3) singlets and triplets

I was reading my prof's notes and came across a passage that I didn't understand: Consider a chiral SU(3) symmetry, under which the left-handed parts of the spin-1/2 fields of a fermion-number- ...
15
votes
1answer
2k 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 (I'm reading this for fun- it isn't a homework problem.) Show, by explicit calculation, that ...
0
votes
2answers
267 views

Azimuthal quantum number $\ell$ and magnetic quantum number $m$ are from angular momentum?

Azimuthal quantum number $\ell$, and magnetic quantum number $m$ are defined when we do derivation for $L^2f=\ell(\ell+1){\hbar}^2f$ and $L_zf=\ell{\hbar}f$. This is my own conclusion after studying ...
0
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0answers
41 views
4
votes
1answer
83 views

Representations of Lie group symmetries on Hilbert space

I have some troubles understanding Hilbert representations for (eg) the standard free quantum particle On the one hand, we can represent Heisenberg algebra [Xi,Pj]= i delta ij on the space of square ...
4
votes
2answers
142 views

Where does the $50^*$ in $SU(5): 10\otimes10= 5^*\oplus45^*\oplus 50^*$ in A. Zee QFT?

See A. Zee, QFT in a nutshell, Appendix B, eq. (24) (p. 469 in first edition with a typo $55^*\to50^*$, cf. Zee errata; p. 530 in second edition.) Where does the $50^*$ in $SU(5)$: $$10\otimes10= ...
3
votes
2answers
159 views

Unitary groups and infinitesimal transformations - Schwingers way of deriving Lie groups

In Schwinger's source theory book, he suggests if $G_a$ are the hermitian generators of the Unitary group, then we have an infinitesimal transformation is given by : $$ G = \sum_{a=1}^n ...
4
votes
1answer
183 views

Lorentz Algebra Representation and QFT

I just have a trouble making a full analogy between Lorentz Algebra Representation in Quantum Field Theory (QFT) and SU(2) representation in Quantum Mechanics (QM). To make my point, I will write few ...
1
vote
0answers
49 views

Notation - d.o.f.'s for Grassmann delta functions in a SUSY field theory amplitude

I was reading the following paper http://arxiv.org/pdf/1306.2962v1.pdf as I stumbled upon an issue concerning counting and assigning the Grassmann degrees of freedom that appear in grassmann delta ...
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7answers
1k views

Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
5
votes
3answers
544 views

The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices

The Pauli spin matrices $$ \sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}), \qquad\qquad \sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
0
votes
1answer
75 views

Decomposition of group representation using tensor method

I am dealing with the decomposition of the representation $5\otimes5$ of $SU(5)$: $$5\otimes5=15\oplus10 $$ demonstration: $$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$ ...
0
votes
1answer
79 views

How do states in Hilbert Space act like irreducible representations?

I am reading Georgi's book on group theory and I came across this sentence..." Hilbert space of any parity invariant system can be decomposed into states that behave like irreducible representations". ...