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22 views

Double groups in Crystallography

I'm currently studying double point groups and their applications in condensed matter physics. Let me start by giving you the definition of the double group that is used in my textbook: Let $G$ be a ...
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1answer
101 views

How unique are the quantum numbers we commonly use?

We use the eigenvalues of the Cartan generators (=diagonal generators) of a given gauge group as quantum numbers in physics. Are these numbers somehow fixed and if not, what transformations are ...
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2answers
87 views

What does it mean by saying the generators of translations transform as vectors under the Lorentz Group?

The commutator of generators of Lorentz transformation and translation is as follow: $$[M^{\mu\nu},P^\sigma]=i(P^\mu\eta^{\nu\sigma}-P^\nu\eta^{\mu\sigma} ).$$ Then from this we usually say that the ...
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1answer
89 views

On the Lorentz Group representation [closed]

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 ...
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83 views

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

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 $SU(3)$. ...
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1answer
32 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 ...
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1answer
33 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 ...
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2answers
70 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 ...
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1answer
75 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 ...
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1answer
96 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 ...
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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 ...
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1answer
71 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 ...
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0answers
41 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 ...
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0answers
43 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 ...
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0answers
63 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 $<\psi_i|\psi_j> = \delta_{ij}$ ...
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3answers
432 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 ...
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0answers
40 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) ...
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1answer
60 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 $s$: ...
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1answer
63 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 ...
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0answers
20 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 ...
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2answers
113 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 ...
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0answers
37 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: $$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$ ...
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1answer
97 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 ...
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1answer
122 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 ...
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3answers
68 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 $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
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1answer
130 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 ...
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2answers
68 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
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1answer
125 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
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1answer
88 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
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1answer
61 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 ...
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2answers
70 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- ...
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0answers
43 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?
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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
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1answer
193 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 ...
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0answers
53 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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1answer
77 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: $$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$ ...
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1answer
82 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". ...
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1answer
101 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
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1answer
128 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
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1answer
78 views

Question about angular momentum operator

To show that the eigenvalue to $L^2$ is proportional to $\hbar^2$ is shown from $L_z=xP_y-yP_x$ $p_y=-i\hbar\frac{\partial}{\partial y}$ $p_x=-i\hbar\frac{\partial}{\partial x}$ ...
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0answers
35 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 ...
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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
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1answer
107 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 ...
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2answers
93 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 ...
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2answers
121 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 ...
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1answer
101 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. ...
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1answer
74 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 ...
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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 ...
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2answers
338 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 $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
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3answers
90 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 ...