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2
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1answer
57 views

On the Lorentz Group representation

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
72 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
27 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 ...
0
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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 ...
3
votes
1answer
66 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 ...
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 ...
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 ...
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
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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
votes
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 ...
1
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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 $<\psi_i|\psi_j> = \delta_{ij}$ ...
0
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0answers
29 views

Axial vector and vector representions of C$_{4v}$ group [migrated]

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way $$ \vec{V} ...
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
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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) ...
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
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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 ...
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 ...
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: $$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$ ...
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 ...
1
vote
1answer
89 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 ...
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 $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
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
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. ...
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 ...
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 ...
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- ...
0
votes
0answers
41 views

Why are the azimuthal quantum number $\ell$ and magnetic quantum number $m$ integers? [duplicate]

Why are the azimuthal quantum number $\ell$ and magnetic quantum number $m$ integers?
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
1answer
182 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
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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 ...
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". ...
3
votes
1answer
94 views

About $SU(2)_L \times U(1)_L = U(2)_L $

In the many textbook of standard model, i encounter the relation \begin{align} SU(2)_L \times U(1)_L ~=~ U(2)_L. \end{align} Here $L$ means the left-handness. (It is a physical ...
2
votes
1answer
121 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
78 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
32 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
32 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
99 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 ...
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
116 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
73 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
72 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 ...
0
votes
1answer
73 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 ...
0
votes
2answers
309 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
89 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
63 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
101 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 ...