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2
votes
1answer
69 views

$SU(3)$ irreducible representations with tensor method

I am dealing with the tensor product representation of $SU(3)$ and I have some problems in understanding some decomposition. 1) Let's find the irreducible representation of $3\otimes\bar{3}$ we have ...
0
votes
1answer
48 views

Question about angular momentum operator

To show that the eigenvalue to $L^2$ is proportional to $\hbar^2$ is shown from $L_z=xP_y-yP_x$ $p_y=-i\hbar\frac{\partial}{\partial y}$ $p_x=-i\hbar\frac{\partial}{\partial x}$ ...
0
votes
0answers
14 views

Explanation for orientation entanglement

I have to write a summary for "orientation-entanglement": the state of an object/subsystem depends in general not only (locally) on its configuration in space, but also (nonlocally) on its topological ...
0
votes
0answers
17 views

normal degeneracy and the “span” of an irreducible representation

In Tinkham's "Group Theory and Quantum Mechanics", Tinkham defines normal degeneracy so that the span of the action of the Hamiltonian's symmetry group on any energy eigenstate yields all possible ...
2
votes
1answer
83 views

Embedding of particles into fields

For the classification of particles (Wigner 1939), we look for unitary representations of the Poincaré/Lorentz group. There are are only infinite-dimensional (non-trivial) unitary representations! To ...
0
votes
1answer
61 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 ...
2
votes
2answers
75 views

Explicit Spinor Representations in $SO(3)$ and $SO(4)$

The beautiful physical interpretation of the decomposition of the elasticity stress tensor into it's irreducible representations (mentioned in context below) has inspired the following question on ...
0
votes
1answer
53 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. ...
1
vote
1answer
56 views

If isospin is conserved under strong interactions why it is represented by SU(2)?

As far as I know from my readings SU(2) is a representation group of isospin symmetry which shows deep symmetry of the strong force which conserves flavor. Isospin symmetry is broken under weak ...
1
vote
1answer
68 views

Tensor operators and transformation of $O^s_{\ell}|j,m,\alpha\rangle$

In H. Georgi's Lie Algebras in Particle Physics one defines a tensor operator transforming under the spin-$s$ representation of $SU(2)$ as the set of operators $O^s_{\ell}$ (for $\ell=-s...s$) such ...
1
vote
2answers
110 views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
0
votes
3answers
71 views

$SO(3)$, $SU(2)$ and symmetries in quantum mechanics [duplicate]

A rotation in the vector space $\mathbb{R}^3$ is represented by the known 3x3-matrices. But at this point I'm really confused how to get from there to Quantum Mechanics. The group of ...
2
votes
1answer
57 views

Does $GL(N,\mathbb{R})$ own spinor representation? Which group is its covering group? (Kaku's QFT textbook)

In Kaku's QFT textbook page 54, there is a saying: $GL(N)$ does not have any finite-dimensional spinorial representation. This implicates that $GL(N)$ owns infinite-dimensional spinorial ...
4
votes
0answers
78 views

Group theory and quantum optics

This is a question about application of group theory to physics. The starting point is the group $SU(n)$. I have a representation $R$ of $SU(n)$ that takes values on the unitary group on an infinite ...
0
votes
0answers
17 views

How to know whether a representation is in $d$ or $\bar{d}$ using Young Tableaux?

For the sake of example, suppose we work with $SU(3)$ and we find that some product of reps decompose into a sum which contains a box consisting of four boxes: something like $$\tag{1}\Box\Box ...
1
vote
1answer
45 views

What is the point of defining the lie algebra of the proper Lorentz group in a “covariant” way?

In Muller-Kirsten's book Introduction to Supersymmetry, the author first defines the proper Lorentz group's lie algebra basis in the standard manner - antisymmetric matrices consisting of $0$s and ...
3
votes
2answers
141 views

Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of complex $2\times 2$ matrices?

Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as $$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 ...
3
votes
2answers
88 views

Identifying irreps of $SU(2)$

How does one verify that, the representations of $SU(2)$ corresponding to $j=1/2$ or $j=1$ is irreducible? I think showing the irreducibility (taking the representative matrices into a block-diagonal ...
0
votes
1answer
40 views

Why are Majorana fields usually used to introduce gravity in the Rarita-Schwinger Lagrangian?

When first introducing the gravitational interaction for a spin-3/2 Rarita-Schwinger field, Majorana fields are usually used (see for example here at chapter 4, or in Ramond, (6.4.112) ). Why is ...
2
votes
1answer
83 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 ...
8
votes
1answer
89 views

Group notation $\otimes$ and $\oplus$ used for representations of quarks and mesons

I've been trying to figure out this statement from the PDG quark model summary (PDF). Following $\mathrm{SU}(3)$, the nine possible $q\bar{q}′$ combinations containing the light $u$, $d$, and $s$ ...
2
votes
0answers
46 views

Decomposing a representation under a subgroup [closed]

I am trying to understand what is the method for decomposing representations of a group under one of its subgroups. I already had a look in Slansky, but I could not extract a concrete set of ...
4
votes
1answer
101 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
2
votes
2answers
88 views

Group representation acting on operators (QFT)

I have found in many texts the following statement: Let $T_g$ be a representation of a group (of transformations, e.g. rotations, translations, Lorentz transformations ) acting on a given Hilbert ...
2
votes
1answer
56 views

How does the Lorentz group act on a 4-vector in the spinor-helicity formalism $p_{\alpha\dot{\alpha}}$?

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I ...
1
vote
0answers
28 views

What is the largest subgroup of the Galilean group and the Lorentz group?

What is the largest subgroup of the Galilean group and the Lorentz group? In the book Structure of Dynamical Systems - A Symplectic View of Physics by J.-M. Souriau, the author mentions (p. 168), ...
2
votes
1answer
73 views

How to diagonalise the Lagrangian mass term with SU(4) symmetry and self-dual tensors

I should write the mass term of the Lagrangian with global SO(4) symmetry in tensor representation with anti-symmetric tensors and then diagonalise this term with defining a new set of tensors ...
2
votes
1answer
71 views

Unitary representations of $SO(3)$ and $so(3)$

According to my skript: Quantum mechanic states $ψ ∈ \mathcal H$ changes under a rotation $R ∈ \text{SO(3)}, \vec{x} \rightarrow R\vec{x}$ according to $ψ \rightarrow U(R)ψ$, whereas $U(R)$ is a ...
3
votes
3answers
406 views

In physics, what is the importance of distinguishing between a matrix and a group? [closed]

On the topic of Pauli matrices, I have noticed that some authors tend to use the term matrix and group interchangeably. I am asking because I do not see see any profound difference referring to the ...
1
vote
0answers
50 views

Proving a relation for representations of gauge groups [closed]

I have asked this question in Math Stackexchange as well. However, given that it is closely related to gauge theories studied by physicists who will probably be more familiar with the language and ...
0
votes
0answers
59 views

Lorentz transformations of spinors in $SL(2,\mathbb{C})$

I was wondering what the matrix representations of all the coordinate rotations and Lorentz boosts of the $SL(2,\mathbb{C})$ were along with a general method of solving for them. I've been able to do ...
0
votes
0answers
27 views

Real representations of chiral fields

Why we can´t have real representations of chiral fields, i.e. why does a multiplet of chiral field (Weyl spinors) under a real representaiton of a Lie Group transforms as a "vector". It is easy to see ...
1
vote
0answers
26 views

What is the physical meaning of U- and V-spin?

$SU(3)$ group has three $SU(2)$ subgroups. The first one (with generators $\lambda_{1}, \lambda_{2}, \lambda_{3}$ of corresponding algebra) is called I-spin, the second one (with generators ...
3
votes
0answers
40 views

Spin of an operator in supersymmetric theories

How exactly is the spin of an operator in the context of a supersymmetric theory defined? For example, in page 25 of [1], $\mathcal{N} = 2$ supersymmetry is defined to have operators $J, G^{+}, G^{-}, ...
1
vote
1answer
57 views

Scalar products in the spinor helicity formalism

In A. Zee's book Quantum Field Theory in a Nutshell (2nd edition), Chapter N.2, page 486, the momentum $p$ is written as a $2\times 2$ matrix: $$ p_{\alpha\dot{\alpha}} = p_{\mu} ...
2
votes
0answers
23 views

Different ways of derivation of Gell-Mann-Okubo mass formula

Recently my teacher have told me that there are many ways of derivation of Gell-Mann-Okubo mass formula by using group representation theory (by using dynamical group etc). Where can I read about ...
8
votes
1answer
311 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 ...
2
votes
3answers
73 views

What is “a vector of $SO(n)$”?

I'm watching (or trying to watch) this lecture from NPTEL on classical field theory. I've understood everything in the series up till this point, including the first half of the lecture on elementary ...
8
votes
2answers
236 views

Does the lagrangian contain all the information about the representations of the fields in QFT?

Given the Lagrangian density of a theory, are the representations on which the various fields transform uniquely determined? For example, given the Lagrangian for a real scalar field $$ \mathscr{L} = ...
1
vote
0answers
33 views

Difference between the 1/2 representation of $SU(2)$ and the (1/2,1) representation of $SU(2)\times SU(2)$? [closed]

What's the difference between the $j = 1/2$ representation of $SU(2)$ and the $(j,j') = ( 1/2 , 1 )$ representation of $SU(2)\times SU(2)$?
2
votes
1answer
70 views

Does the $\bf{1+3}$ representation of $SU(2)$ also represent $SU(2)\times SU(2)$?

I'm a bit confused about this following issue concerning representations of $SU(2)$. Denote by 1 the 1-dimensional representation of the group $SU(2)$ (=the spin 0). Similarly, denote by 2 and 3 the ...
0
votes
1answer
83 views

Spin operator eigenstate in Fock space

I am creating an operator group from representation of spin 1 operators $$J_{x} = \frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{array} ...
2
votes
0answers
79 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
65 views

Energy as charge with respect to time translations in QM

Consider a non relativistic quantum mechanical system with Hamiltonian $\mathcal{H}$, and denote the states by $\psi \equiv \psi(t) \equiv | \psi(t) \rangle$. From the Schrödinger equation we know ...
2
votes
1answer
58 views

Solution space of a differential equation with 3D rotational symmetry

We know that the space of solutions will be invariant under 3D rotations, but why can we say that the space of solutions will constitute a representation of the rotation group $SO(3)$? We know that a ...
0
votes
1answer
79 views

Derivation of the irreducible representations of SO(3)

Is there a way to derive the representations of $SO(3)$ without the usual method with the ladder operators which also gives the ones of $SU(2)$? The usual way to do these calculations is to start ...
1
vote
0answers
43 views

How to break a irreducible representation into its subgroups

In Grand Unified Theories (though I'm sure this a general group theory result) people write the irreducible representations of a group (i.e., the gauge bosons) using a sum of irreducible ...
1
vote
1answer
64 views

What do position and momentum representations represent in QM?

In QFT we classify field operators according to how they transform under a given symmetry, i.e. in their being a basis for some representation of the symmetry group of the Hamiltonian/Lagrangian. This ...
1
vote
1answer
43 views

Explicit Symmetry Breaking: Where do the additional d.o.f. come from?

Massless vector bosons have only two independent degrees of freedom, while massive ones have three. In spontaneous symmetry breaking, the massless vector belonging to the broken group becomes massive ...
1
vote
2answers
64 views

SU(2) kinetic term as a trace

Is there a easy way to rewrite the SU(2) kinetic term as a trace? As in $$\mathcal{L} = -\frac{1}{4}\vec{F}_{\mu\nu}\vec{F}^{\mu\nu}\\[1cm] = -\frac{1}{2}\mathrm{tr}\Bigg[\bigg(\vec{F}_{\mu\nu}\cdot ...