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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4
votes
1answer
160 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) ...
1
vote
0answers
76 views

Bondi-Metzner-Sachs (BMS) related Question(s)

I started studying the BMS group in connection with the set of papers by A. Strominger et al., also related with the supposed solution of the "Black Hole Information Paradox" by S. W. Hawking ...
6
votes
1answer
80 views

Is there a systematic way to obtain all conserved quantities of a system?

I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's ...
0
votes
0answers
23 views

How to compute remnant discrete charges?

When a $U(1)$ gauge symmetry is broken by Higgs field with $U(1)$ charge $g$, we have a remnant $\mathbb{Z}_q$ symmetry. How can I compute the corresponding charges under this remnant group for ...
1
vote
1answer
70 views

Center group and Symmetry breaking

First, I will state the definition of center, $i.e$, the center of group $C_G$ is defined a sthe part of $G$ which commutes with all generators. I want to know the procedure of breaking gauge ...
2
votes
1answer
81 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: ...
0
votes
0answers
29 views

Dilations in momentum space

I don't quite understand what's going on here. Let's suppose I have a dilation in real space. The generator is $D=x^j \partial_j$, so an infinitesimal dilation is $\delta x^i = Dx^i = x^j \partial_j ...
2
votes
0answers
57 views

What does complexification mean for our particles in physics

As gauge group let's consider the popular $SO(10)$ group. The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional $$ \pi: \mathfrak{so}(10) ...
2
votes
1answer
164 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
12
votes
1answer
197 views

Highest symmetric non-maximally symmetric spacetime

What is the highest number of symmetries (Killing vectors) that a (4-dimensional) spacetime can have without being maximally symmetric? From what I can see, it seems to be 7 (which includes the ...
-1
votes
1answer
98 views

Derivation of the Lorentz algebra explicity [closed]

I need the complete proof for commutation relation of the Lorentz group generators. The proof of Lorentz algebra using this commutation relation.
3
votes
1answer
190 views

Permutation symmetry - a continuous symmetry?

From quantum mechanics it is known that permutation between identical particles does not change the Hamiltonian. Assuming that the quantum system consists of a very high number of particles such that ...
1
vote
1answer
77 views

Regarding different representations of the Lorentz Group & its defining properties

Take $\Lambda$ to be a Lorentz matrix, it satisfying $\Lambda^T \eta \Lambda=\eta$. By writing $\Lambda=\exp[-\frac{i}{2}\omega_{\mu\nu}\mathcal J^{\mu\nu}]$, we find that the generators satisfy $$ ...
0
votes
0answers
31 views

trace of spherical components of an angular momentum multiplet

My basic mathematical question is whether or not there exist selection rules regarding the traces of the spherical tensor components of some operator acting on some subspace of definite total angular ...
1
vote
0answers
53 views

Given a VEV how can I compute which generators remain unbroken using tensor methods?

This is a follow up to this question. A generator $T_a$ of a given gauge group $G$ remains unbroken after some Higgs field $\Phi$ gets a vev if $$ T_a \langle\Phi\rangle =0 $$ I'm trying to ...
0
votes
0answers
52 views

Why is the tensor product $n \otimes n = 1$ for $SO(n)$ not the usual scalar product?

For concreteness let's consider $SO(4)$. The quantum numbers for the four states in the fundamental representations are (schematically) $$ (1, 1) ,(-1, 1) ,(1, -1) ,(-1, -1 )$$ thus $$ 4= ...
2
votes
0answers
77 views

What do people mean with a vev=diag( -2,-2,-2,-2,-2,-2,3,3,3,3)?

For example, in this paper on page 21 the authors write the vev that breaks $SO(10)$ to $SU(4)\times SU(2) \times SU(2)$ $$ <54>= 1/5 \cdot diag( -2,-2,-2,-2,-2,-2,3,3,3,3) \omega_s$$ where ...
3
votes
2answers
321 views

Rotation matrix of Euler's equations of rotation relative to inertial reference frame

I was playing with simulation of Euler's equations of rotation in MATLAB, $$ I_1\dot{\omega}_1 + (I_3 - I_2)\omega_2\omega_3 = M_1, $$ $$ I_2\dot{\omega}_2 + (I_1 - I_3)\omega_3\omega_1 = M_2, $$ ...
0
votes
0answers
62 views

Explicit Matrix Representation for 54 Higgs rerpesentation for $SO(10)\rightarrow$ Pati-Salam Breaking

Which Higgs field or linear combination of Higgs fields gets a vev if we want to break $$ SO(10) \rightarrow SU(4) \times SU(2) \times SU(2) ? $$ Formulated a bit differently, I'm looking for an ...
1
vote
1answer
46 views

Are the mass matrices the same if Higgs corresponding to different Cartan generators get a vev?

I'm trying to understand what happens when a Higgs field in the adjoint representation of a given gauge group gets a vacuum expecation value (vev). Normally, the fermions do not couple to adjoint ...
3
votes
1answer
69 views

Anticommutative Sets of SU(N) Generators? Anticommutative Analogue to Cartan subalgebra?

I am currently studying SU(N) generators in order to find bases that may suit a problem at hand. I am especially interested in getting as large anticommuting sets within a basis as possible. In SU(2) ...
1
vote
2answers
49 views

Notation in crystallography

I'm trying to comprehend the proof that for a crystal with translational symmetry only 1,2,3,4 or 6 rotational axes exist. The proof I'm trying to follow however uses a weird notation I haven't seen ...
3
votes
1answer
82 views

How to find the remaining subgroup after some linear combination of Higgs fields gets a VEV?

This is a follow-up question to this question. How can I compute which generators remain unbroken when a linear combination of Higgs fields $a \Phi_1+ b\Phi_2$ get a vev? If I compute the unbroken ...
2
votes
1answer
114 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 ...
2
votes
1answer
62 views

What is the definition of the duality group $E_{7(7)}$?

What is the definition of the duality group $E_{7(7)}$ that appears in ${\cal N}=8$ Supergravity and what are the basics properties? Moreover what is the relation with the Lie Algebra $E_7$? ...
0
votes
0answers
57 views

How can I compute the orbit of a Higgs field?

In many papers that deal with symmetry breaking a concept called orbit is introduced: It is worth noting that if the potential is a minimum $\phi_0$ at a value of the field, then from (3.13) it is ...
2
votes
1answer
60 views

QCD Color Structure relation [closed]

i want to proof the following relation : \begin{equation} t^a t^b \otimes t^a t^b = \frac{2}{N_C} \delta^{ab} \mathbb{1} \otimes \mathbb{1} - \frac{1}{N_C} t^a \otimes t^a \end{equation} Right now I ...
9
votes
2answers
521 views

Why are band maxima / minima often (always?) at high-symmetry points?

(inspired by this question.) In every semiconductor that I can think of, the valence band maximum and conduction band minimum are at a high-symmetry point in the Brillouin Zone (BZ). Often the BZ ...
3
votes
1answer
79 views

Symmetry breaking to a special subalgebra?

This is a follow-up to my question here. For regular subalgebras of some group's Lie algebra the root system of the subalgebra is a subset of the root system of the original's group algebra. In ...
0
votes
0answers
55 views

Resource for (String) Symmetry Breaking in Terms of Roots and Weights?

I'm currently searching, for quite a while now, for a paper/book that discusses symmetry breaking in terms of roots and weights. Any suggestions would be much appreciated!
0
votes
1answer
60 views

Role of SU(2) group in isospin and in the weak interaction

I know that the SU(2) group describes internal symmetries such as isospin and the weak interaction. But isospin and weak interactions are quite different, so more precise what is the role of SU(2) in ...
1
vote
1answer
120 views

How to find the remaining subgroup after some Higgs field gets a VEV?

Say we have a group $G$ and a set of Higgs fields in a representation $R$ of $G$. One of the Higgs fields in $R$ gets a VEV, how can I determine the remaining subgroup after this symmetry breaking? ...
2
votes
1answer
69 views

Why are bare mass terms for W-bosons forbidden, but coupling terms to Higgs doublets allowed?

The $W$ bosons live in the adjoint rep of $SU(2)$, which is three dimensional. The standard model Higgs lives in a $SU(2)$ doublet, i.e. the two dimensional rep. The $W$ bosons get their mass ...
1
vote
2answers
140 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 ...
5
votes
1answer
143 views

Why does Wikipedia equate hidden symmetry with broken symmetry for the standard model?

I have recently started studying the basic ideas of symmetry and group representation in order to understand the basic principles behind the standard model. I do follow the difference between a global ...
1
vote
1answer
60 views

Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
1
vote
0answers
29 views

Conserved charge for boosts? [duplicate]

In (3+1) dimension Poincare group has three types of Symmetries : a) Four space-time translations b) Three spatial rotations and c) Three boosts Among them, (a) implies "conservation of ...
1
vote
0answers
50 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 ...
4
votes
2answers
144 views

Does reversal of one spatial direction count as a discrete Lorentz transformation?

A transformation $\Lambda$ is a Lorentz transformation if it satisfies $\Lambda^T g \Lambda = g$, for the flat metric $g = \left( \begin{array}{cccc} 1 &&& \\ & -1 &&& \\ ...
2
votes
1answer
116 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 ...
6
votes
0answers
190 views

Monstrous Moonshine outside of String Theory

My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ...
1
vote
0answers
73 views

Supermultiplet dimensions from Young Tableaus

In John Terning's book, on pages 14 and 15, there are lists of $\mathcal{N} = 2$ and $\mathcal{N} = 4$ supermultiplets, labeled in terms of the dimensions of the corresponding R-symmetry $d_R$ and ...
3
votes
3answers
147 views

$SO(3)$ vs 3-Torus ${(S_1)}^3$

From rigid body rotations point of view, why are $SO(3)$ and 3-Torus not the same. Every rigid rotation is rotation about three axes. So how come $SO(3)$ is not ${(S_1)}^3$? It seems it should be. Is ...
1
vote
0answers
36 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

How to find the generators of a deformed boost?

I'm reading the paper arXiv:gr-qc/0012051 on doubly special relativity. In page 7, the author wants to find the generators of a deformed boost that preserves $$E^2 = p^2 + m^2 - l_p p^2 E$$ ($l_p$ is ...
3
votes
2answers
175 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 ...
2
votes
1answer
123 views

Classical spin viewed as $SU(2)$

In which sense is the configuration variable of a classical spin $SU(2)$? I can view a classical spin as a unit vector in $\mathbb{S}^2$ (2-dim. sphere), but it seems it is really given by a matrix ...
2
votes
0answers
82 views

Are mass terms forbidden in the Lagrangian because of parity violation or because fermions live in a complex representation?

Normally one argues that we can't write down Lorentz AND gauge invariant mass terms, because of parity violation, i.e. l-chiral and r-chiral fields transform differently. This means that mass terms ...
3
votes
0answers
80 views

Is Witten's claim that gauge group representations get exchanged with its dual under parity correct?

I'm currently reading Physics and Geometry by Witten, which I really liked up to the point where he claimed that we exchange representations $R$ and $\tilde R$ under parity transformations, where $R$ ...
3
votes
3answers
73 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 $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...