Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. ...

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30 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 ...
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49 views

Use Cartan subalgebra in spinor representation to find weights of vector representation

For $SO(2n)$ we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the Clifford algebra. Up to some prefactor the elements $ S_{\mu \nu} = \alpha ...
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22 views

Generators of cubic group: Euler angles

I am trying to build numerically all members of the cubic group in the representation of Wigner D matrices. Angular-momentum quantum number may be small, e.g. 2, so I use the formula stated e.g. at ...
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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 ...
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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 ...
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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: ...
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0answers
58 views

Conjugate elements of $SO(3)$ group [migrated]

Composition of two rotations in 3d space yields another rotation $$R_1 R_2 = R_3, $$ and I can understand this by help of some figures in my book. So, the rotations in 3d space forms group. Then ...
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1answer
43 views

Symmetry and Group theory book

I would like to start learning about symmetries in physics and how they affect physical quantities. As far as I know, the mathematical language that describes symmetries is the Group Theory. So, I ...
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1answer
50 views

What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?

What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?
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1answer
36 views

Mesons and Young Tableaux

I need some help conecting Young Tableaux with actual particles. I think I have some feel for using Young Tableaux for instance: a baryon in SU(3) where the states are u,d,s can be represented by ...
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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 ...
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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 ...
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37 views

Is it proper definition of the free motion? The orbit of free motion is a free group [closed]

That's what I wrote in my notes but I don't understand this definition, I've studied group theory but free groups were not included. Can someone explain this definition, please?
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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?
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39 views

What is $\mathrm{U(1)}$ vector and axial?

In hadron physics we talked about $\mathrm{U(1)_V}$ (vector) and $\mathrm{U(1)_A}$ (axial) as well as $\mathrm{SU(3)_L}$ (left) and $\mathrm{SU(3)_R}$ (right). There are certain relations between them ...
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0answers
28 views

Weyl Transformations and Group actions [migrated]

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
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0answers
65 views

Mathematical definition of reversible processes

If I label an initial thermodynamic state as $\psi$ and the final thermodynamic state as $\xi$ then can I say that under a reversible process the two states are related to each other by a continuous ...
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1answer
46 views

Lorentz group in SUSY

Why do we carry Lorentz group to be included also in supersymmetry? That is after we extend our symmetry to supersymmetry, we carry with us the Lorentz group. Why not other group instead?
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0answers
36 views

Is it possible to define a symmetry group for the Einstein metric?

I was just wondering if there exists a group of transformations that act on the metric such that the EFE are invariant. At first I thought it would be the group of 2nd roots of unity. That is, the set ...
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1answer
53 views

Is spin angular momentum conserved?

According to the Noether theorem, we only have the conserved quantity $$J+S,$$ where $J$ is the orbital angular momentum and $S$ is the spin angular momentum. But I am always impressed that the spin ...
2
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1answer
136 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 ...
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0answers
19 views

How to introduce symmetry of particles to a layperson? [closed]

I want to introduce the concept of special unitary symmetry and how it is important in particle physics to a layperson. Without being technical, is there a way to explain the fundamental concept?
3
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2answers
74 views

AdS/CFT Group Theory

I have a two part question about AdS/CFT: Is the only necessary ingredient that the isometry group of AdS matches the conformal group in one dimension less or are there other prerequisites to build ...
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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 ...
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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 ...
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1answer
98 views

Enhancing the QED $U(1)$ gauge symmetry

QED is a gauge theory based on $U(1)$ gauge symmetry, which gives rise to photon as the gauge boson mediating the interaction. Mathematically, I think it is perfectly allowed to implement a ...
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26 views

Correct Yukawa Term with a SU(2) Higgs Triplet?

Given $SU(2)$ doublet fermions $\Psi^1$ and $\Psi^2$ and a $SU(2)$ triplet Higgs $H$, how does the correct Yukawa term look like in tensor notation? Schematically, we have $$ 2 \otimes 2 \otimes 3 ...
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27 views

Charge Conjugation for $SU(N)$?

For $SU(2)$ the charge conjugation operator $C$ reads explicitly $$ C \Psi = i \sigma_2 \Psi^\star ,$$ where $\sigma_2$ is a Pauli matrix. What is the generalized charge conjugation for $SU(N)$?
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1answer
45 views

determining electrostatic field using only symmetries

As an exercise, I'm trying to (rigorously) determine as much as possible about the electrostatic field due to a infinite line of charge (along the z-axis) without using Maxwell's equations or any of ...
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0answers
39 views

Spacetime as a coset of a symmetry group

In the introduction to his nice PNAS paper on symmetry, David Gross said Einstein’s great advance in 1905 was to put symmetry first, to regard the symmetry principle as the primary feature of ...
3
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1answer
89 views

Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

I am quite confused with the groups Diff$(M)$ and $GL(4,\mathbb{R})$ in the context of general relativity. I understand that the symmetries of GR are the transformations that leave the equations ...
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 ...
4
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3answers
69 views

In most physical cases, the elements of a group can be represented by unitary matrices. Why no time-reversal?

In Dresselhaus's group theory page 19, a theorem writes: Every representation (of a Hamitonian's group) with matrices having non-vanishing determinants can be brought into unitary form by an ...
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1answer
68 views

Direct Sum representation of multiple particles in Quantum Mechanics

Suppose that I have three non-interacting spin-1/2 particles such that I can represent the combined system in a basis of \begin{align} D^{(1/2)}_1 \otimes D^{(1/2)}_2 \otimes D^{(1/2)}_3 & ...
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1answer
50 views

spin representations and polynomials

I'm reading Group Theory and General Relativity by Moshe Carmeli and his discussion of spin representations of SU(2) and the isomorphism to the space of homogenous polynomials is confusing me. I'll ...
1
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1answer
40 views

Matrices belonging to orthochronous Lorentz group

My Professor says that all members of the orthochronous Lorentz group may be written as $e^\Gamma$, where $$ \Gamma^{\mu}_{\nu}=\Lambda^{\mu \rho} \eta_{\rho \nu}$$ Here $\Lambda$ is an antisymmetric ...
5
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0answers
55 views

$(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$ [duplicate]

I am confused about the notation. What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ? ...
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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} + ...
2
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1answer
50 views

Space group setting of a crystal structure

I was reading a PDF of a crystal phase in order to draw its structure, when I noticed that it was, apparently, ambiguously described. The PDF lists two descriptions of the monoclinic structure: 1) ...
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 $\Phi_2$ ...
0
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0answers
10 views

Can i extend the non-linear realization of the chiral group to $U$ with complex pions?

In chiral perturbation theory we build a Lagrangian invariant under $SU(2)_L\times{}SU(2)_R$ which acts on the matrix $U$ that accommodates the pion degrees of freedom in the following way ...
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0answers
53 views

Irreducible representations of $SU(2)$, Tensor-operators under rotations

First of all: this is my first question so please give feedback to the way I'm formulating the question! The question is about an exercise I have to solve, but I simply get nowhere. It is given the ...
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0answers
42 views

The universal covering group of a symmetry group [duplicate]

In Weinberg QFT Vol.1, it says one can enlarge the symmetry group $H$ to the universal covering group $C$ such that one obtains a trivial cocycle or $C$ is simply connected whereas $H$ is not. I get ...
1
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0answers
76 views

What does Addition of Angular Momenta tell us about Group Theory?

I've come across this a lot, but I've never understood it. I do know basic Group Theory including Lie Groups. In Introduction to Quantum Mechanics, Griffiths ends the chapter on spin with the remark ...
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0answers
24 views

What is therelation between nonlinear sigma model, complex projective group?

The O(N) nonlinear sigma model has topological solitons only when N=3 in the planar geometry. There exists a generalization of the O(3) sigma model so that the new model possess topological solitons ...
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2answers
329 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 ...
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0answers
26 views

Textbooks in group theory for budding theoretical particle physicists [duplicate]

I am looking for textbooks in group theory. I have just finished my undergraduate studies in Physics and am looking to specialise in theoretical high-energy physics. Therefore, textbooks in group ...
1
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1answer
84 views

Matrix represenation of total angular momentum operator

I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively. For the total angular ...
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0answers
37 views

Does Bose-Statistics mean that we are only allowed to write symmetric tensor products of Higgs fields in the Lagrangian?

Given Higgs fields, transforming as the representation $R$ of a group $G$. Does the physical fact that scalars=boson commute mean that we are only allowed to write symmetric tensor products of the ...
2
votes
3answers
92 views

Showing a mapping between $SU(2)$ and $SO(3)$

I know this has been done on this site in a different manner but I'm wondering if it's possible to show the 2:1 Lie group homomorphism between $SU(2)$ and $SO(3)$ using exponentials of the generators ...