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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3
votes
1answer
50 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = ...
3
votes
2answers
39 views

Normalising Generators of a Lie Algebra

Ok, so I'm asking this in physics because I'm currently working through part of Srednicki's text on QFT, even though it's really a maths question. In Srednicki's chapter on non-Abelian gauge theory, ...
2
votes
0answers
22 views

Recipe to compute dimension and decompose product of $SO(N)$ group representations [migrated]

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
16
votes
1answer
337 views
+50

Why do we need complex representations in Grand Unified Theories?

EDIT4: I think I was now able to track down where this dogma originally came from. Howard Georgi wrote in TOWARDS A GRAND UNIFIED THEORY OF FLAVOR There is a deeper reason to require ...
4
votes
1answer
71 views

General construction of equations of motion for free particles

I've got a question regarding the different Symmetrie-Lie-Groups of Newtonian Mechanics and special realtivity. Is there a canonical way to obtain the equations of motion for a free particle only by ...
0
votes
0answers
20 views

How can we determine the Hypercharges in a GUT like SO(10)?

I understand how the assignment works for a symmetry breaking like $$SO(10) \rightarrow SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_X$$ The Hypercharge can then easily computed by $$ ...
0
votes
0answers
20 views

Casimir operators of the Poincare Group

Say, $P^2$ and $W^2$ are the Casimir operators of the Poincare Group. Should the commutator of these Casimir operators be zero, because then they would be independent of each other and form a unique ...
-1
votes
0answers
15 views

Spinor Representations in terms of the eignvalues of the Casimir Operators of the Lorentz Group [duplicate]

The Casimir operators of the Lorentz Group are $S^2$ and $T^2$. As such, particles can be identified by the following representations of the Lorentz Group: (0,0) = scalar/singlet representation: Why ...
0
votes
0answers
40 views

The Lie algebra of the Lorentz group is $su(2) \oplus su(2)$. Is there a similar relation for the algebra of the Poincare group?

It can be shown easily, by introducing new generators from the usual ones that we can think of the Lie algebra of the Lorentz group as being built up by two copies of the $SU(2)$ Lie algebra: $$ ...
4
votes
1answer
63 views

Where does in GUT symmetry breaking $U(1)$ come from?

In GUTs one starts with some larger group, like $SU(5)$, which is then broken into smaller groups, for example $$SU(5) ~\longrightarrow~ SU(3) \times SU(2) \times U(1)$$ This can be seen, for ...
9
votes
3answers
144 views

Why is the “real” gauge group of the standard model $SU(3) \times SU(2) \times U(1) /N$?

In this paper John Baez says the real gauge group of the standard model is $SU(3) \times SU(2) \times U(1) /N$. Can someone explain the logic behind this line of thought? Firstly, does this group ...
1
vote
0answers
41 views

How to get from $E_8 \rightarrow E_7 \rightarrow E_6 \rightarrow …$

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
0
votes
1answer
28 views

What is the current state of axion research theoretically?

Is the problem (the strong CP problem) that is solved by axions still considered a really big problem, like the missing mass terms that could be solved by the Higgs mechanism? Or is it more a ...
1
vote
0answers
45 views

Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$? [migrated]

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
2
votes
1answer
29 views

Dimension and Basis properties of $SU(N)$

$SU(N)$ is the group of special unitary matrices of dimension $N$, i.e., the set of all unitary ($U^{\dagger}U=I$) $N\times N$ matrices with $\det(U)=1$. For $N=2$, these matrices are spanned by the ...
0
votes
0answers
27 views

How to write $su(3)$ Lie algebra as a sum of two subspaces? [migrated]

Let $K,F\subset\mathfrak{su}(3)$ be subspaces, such that $K \oplus F =\mathfrak{su}(3)$, and $K$ has a $\mathfrak{su}(2)$ structure. How can we show that $[K,K] = K$ (i.e., commutator of any two ...
4
votes
1answer
145 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 ...
6
votes
2answers
70 views

Why is $\theta \over 2$ used for a Bloch sphere instead of $\theta$?

I'm a beginner in studying quantum info, and I'm a little confused about the representation of a qubit with a Bloch Sphere. Wikipedia says that we can use $$\lvert\Psi\rangle=\cos\frac{\theta}{2} ...
0
votes
0answers
41 views

Landau theory of phase transation

In his article http://www.ujp.bitp.kiev.ua/files/journals/53/si/53SI08p.pdf, Landau defines probability distribution $\rho$ which is related to symmetry of crystal. If crystal has certain symmetry ...
0
votes
2answers
53 views

About Lorentz Group

In definition of orthogonal matrices we say that the a matrix $A$ is orthogonal if $A^TA = I$, while for Lorentz Group it is written as $\Lambda^Tg\Lambda = g $. And we say that Lorentz transformation ...
2
votes
1answer
35 views

Coset construction of Tricritical Ising CFT

In http://iopscience.iop.org/1742-5468/2008/03/P03010 the authors state that the Tricritical Ising Model (TIM) CFT can be obtained from a Wess Zumino Witten construction based in the coset ...
2
votes
1answer
78 views

Clarification: Why the gauge symmetry of pure Yang-Mills is $PU(n)$ and not $SU(n)$? [closed]

I am quoting the following from the Wikipedia article on the projective unitary group: In the pure Yang–Mills $SU(n)$ gauge theory, which is a gauge theory with only gluons and no fundamental ...
1
vote
0answers
25 views

Singular points of an orbit space

I am wondering what, precisely, the singular point of an orbit space is. Specifically, I am looking at quantum statistics and the orbit space $M^N/S_N,$ where $M^N$ is the classical configuration ...
2
votes
1answer
44 views

Parity transformation is proper orthochronous?

In 3+1 dimensional spacetime the parity transformation is $$P^\mu_{\;\,\nu}=\begin{pmatrix}+1&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}.$$ This is ...
-2
votes
1answer
66 views

Group Theory VS Quantum Mechanics [closed]

We all know that a quantum state or an observable, for example $|\phi>$ is a vector in Hilbert space. What is the equivalent of a quantum state (or simply a state) in group theory?
0
votes
1answer
66 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: $$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$ ...
3
votes
1answer
87 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 ...
0
votes
1answer
45 views

Does invariance under infinite small transformation imply invariance to the finite one?

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it. The chiral transform is defined as: $$\psi(x) \rightarrow \psi'(x) =e^{i ...
2
votes
1answer
88 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
0answers
42 views

Uses of the accidental isomorphism $SO(5)\sim Sp(2)$?

Some of the accidental isomorphisms of low dimensional Lie algebras have very important applications in physics. The theory of angular momentum makes use of the fact that $SO(3)\sim SU(2)$. ...
2
votes
1answer
53 views

$SU(2)$ generators and creation annihilation operators

The algebraic method to find the irreducible representation of the $SU(2)$ group makes use of the operators: $$J_z\\J_+=\frac{1}{\sqrt{2}}(J_x+iJ_y)\\J_-=\frac{1}{\sqrt{2}}(J_x-iJ_y)$$ In the book ...
0
votes
0answers
22 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 ...
2
votes
0answers
29 views

Group theory of quark model [closed]

I am trying to understand the group theoretical aspects of quark model. In chapter 11 - Hypercharge and Strangeness- in the book titled 'Lie Algebras in Particle Physics' by H. Georgi, I am not able ...
0
votes
0answers
27 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 ...
5
votes
1answer
108 views

Complete derivation of generator of rotations

I have been look all across the internet and every book I could find trying to get a full derivation of the generator of rotations and more specifically angular momentum as a generator of rotations. I ...
2
votes
2answers
97 views

Half-integer spin and infinitesimal rotations

On p. 692 of 'Quantum Mechanics' by Cohen-Tannoudji, he states that: Every finite rotation can be decomposed into an infinite number of infinitesimal rotations, since the angle of rotation can ...
1
vote
1answer
73 views

Problem with determining number of goldstone bosons

Consider a theory $$\mathcal{L}=(\partial_\mu\Phi^\dagger)(\partial^\mu\Phi)-\mu^2(\Phi^\dagger\Phi)-\lambda(\Phi^\dagger\Phi)^2$$ where $\Phi=\begin{pmatrix}\phi_1+i\phi_2\\ ...
2
votes
0answers
51 views

Finding Casimir operators for the Poincare group $ISO(1,2)$

I was asked to write the generators for translations and Lorentz-transforms in 1+2 dimensions and then to find the Casimir operators. For the generators I can take the same ones as in 1+3 case ...
0
votes
0answers
18 views

The Wigner 3j-Symbol and Kronecker delta

If you look up the definition of the Wigner 3j-Symbol (e.g. on Wolfram) you'll find $m_1+m_2=M$ must be satisfied. Does that mean that, for an arbitrary Wigner 3j-Symbol I could replace: $ ...
0
votes
1answer
59 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. ...
0
votes
2answers
57 views

How would I relate $\Lambda=e^{-i\omega_{\mu\nu}J^{\mu\nu}/2}$ to the Lorentz boost matrix?

$\omega_{\mu\nu}$ contains infinitesimal parameters and $J^{\mu\nu}$ contains generators of boost and rotation. Any 4-vector transforms as $p^\mu=\Lambda^\mu_\nu p^\nu$. Starting from given ...
2
votes
0answers
52 views

Relation between representations/classifications

Generally a quantum system can be characterized in the following way: its states form a representation space for every symmetry group of that system. The representation has to be unitary (or ...
5
votes
2answers
187 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
2
votes
0answers
55 views

What is the difference between the groups $PSU(N)$ and $SU(N)$? [closed]

What is the difference between the groups $PSU(N)$ and $SU(N)$? For example how is $PSU(2,2|4)$ different than $SU(2,2|4)$?
1
vote
1answer
69 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
57 views

Why is the Poincaré group non-abelian?

Based on what I've learned, I gather the Poincaré group is the group of isometries of Minkowski spacetime and it is a non-abelian Lie group. Why is it non-abelian? Or perhaps rather, does the fact ...
1
vote
0answers
47 views

Why use class multiplication to describe topological entangling and merging?

I'm studying some references about topological defects in ordered media like Soft matter physics: An introduction by Kleman and the Review modern physics paper The topological theory of defects in ...
1
vote
2answers
219 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
84 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 ...