# Tagged Questions

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. ...

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 4-...
51 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 ...
147 views

A transformation $\Lambda$ is a Lorentz transformation if it satisfies $\Lambda^T g \Lambda = g$, for the flat metric $g = \left( \begin{array}{cccc} 1 &&& \\ & -1 &&& \\ &... 1answer 121 views ### Representation of U(1) on fock space I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ... 0answers 194 views ### Monstrous Moonshine outside of String Theory My question concerns applications of monstrous moonshine, which is the connection between the$j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ... 0answers 73 views ### Supermultiplet dimensions from Young Tableaus In John Terning's book, on pages 14 and 15, there are lists of$\mathcal{N} = 2$and$\mathcal{N} = 4$supermultiplets, labeled in terms of the dimensions of the corresponding R-symmetry$d_R$and ... 3answers 149 views ###$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$? It seems it should be. Is ... 0answers 37 views ### symmetry group of multi-electron atom Neglecting spin effects, the energy levels of multi-electron atoms are characterized by states of definite total orbital ($L^2$) and spin angular momentum ($S^2$). From this it seems that the ... 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 ... 2answers 183 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 ... 1answer 126 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$U$... 0answers 84 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 ... 0answers 82 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$R$... 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 $$g = e^{i a J_+} e^{i b J_z} e^{i c J_-} .$$ ... 1answer 265 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 ... 2answers 168 views ###$su(1,1) \cong su(2)$? The three generators of$su(2)$satisfy the commutation relations $$[J_0 , J_\pm] = J_\pm , \quad [J_+, J_- ] = +2J_0 .$$ The three generators of$su(1,1)$satisfy the commutation relations $$[... 2answers 102 views ### How to construct generators and Lie Algebra for Lorentz group? I'm trying to figure out Lorentz group in 2+1. First of all, I'd like to think the special orthgonal group as a combination of rotation and boost in space. Then I construct it as below. First rotation ... 0answers 72 views ### Which representation do we start with in Grand Unified Theories? The conventional approach in GUTs is to put all left-chiral fields F_L of the standard model into one representation of the GUT group. For example, the 16 rep for SO(10) GUT:$$ 16_L \rightarrow ... 2answers 141 views ### Why gauge fields are traceless Hermitian? So I've had a read of this, and I'm still not convinced as to why gauge fields are traceless and Hermitian. I follow the article fine, it's just the section that says "don't worry about this ... 1answer 282 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 ... 1answer 105 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 ... 2answers 89 views ### Correct vector space of eigenkets of angular momentum When we say an particle is in the state: $$|l,m\rangle,$$ what is the underlying state space, as a vector space? Is it a tensor product vector space, of dimension: \begin{... 1answer 122 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 $$D_{\mu}\Phi(x) = (\partial_{\mu}+A_{\mu})\... 2answers 159 views ### Normalising Generators of a Lie Algebra Ok, so I'm asking this in physics because I'm currently working through part of Srednicki's text on QFT, even though it's really a maths question. In Srednicki's chapter on non-Abelian gauge theory, ... 1answer 594 views ### Why do we need complex representations in Grand Unified Theories? EDIT4: I think I was now able to track down where this dogma originally came from. Howard Georgi wrote in TOWARDS A GRAND UNIFIED THEORY OF FLAVOR There is a deeper reason to require ... 1answer 97 views ### General construction of equations of motion for free particles I've got a question regarding the different Symmetrie-Lie-Groups of Newtonian Mechanics and special realtivity. Is there a canonical way to obtain the equations of motion for a free particle only by ... 0answers 69 views ### The Lie algebra of the Lorentz group is su(2) \oplus su(2). Is there a similar relation for the algebra of the Poincare group? It can be shown easily, by introducing new generators from the usual ones that we can think of the Lie algebra of the Lorentz group as being built up by two copies of the SU(2) Lie algebra:$$ \... 1answer 124 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 ... 3answers 295 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 "N"... 0answers 48 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 here?... 1answer 54 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 "problem"... 1answer 163 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 ... 1answer 312 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 ... 3answers 628 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 $$\lvert\Psi\rangle=\cos\frac{\theta}{2} \... 2answers 84 views ### About Lorentz Group In definition of orthogonal matrices we say that the a matrix A is orthogonal if A^TA = I, while for Lorentz Group it is written as \Lambda^Tg\Lambda = g . And we say that Lorentz transformation ... 1answer 70 views ### Coset construction of Tricritical Ising CFT In http://iopscience.iop.org/1742-5468/2008/03/P03010 the authors state that the Tricritical Ising Model (TIM) CFT can be obtained from a Wess Zumino Witten construction based in the coset \frac{(... 1answer 133 views ### Clarification: Why the gauge symmetry of pure Yang-Mills is PU(n) and not SU(n)? [closed] I am quoting the following from the Wikipedia article on the projective unitary group: In the pure Yang–Mills SU(n) gauge theory, which is a gauge theory with only gluons and no fundamental ... 0answers 32 views ### Singular points of an orbit space I am wondering what, precisely, the singular point of an orbit space is. Specifically, I am looking at quantum statistics and the orbit space M^N/S_N, where M^N is the classical configuration ... 1answer 59 views ### Parity transformation is proper orthochronous? In 3+1 dimensional spacetime the parity transformation is$$P^\mu_{\;\,\nu}=\begin{pmatrix}+1&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}.$$This is ... 1answer 94 views ### Group Theory VS Quantum Mechanics [closed] We all know that a quantum state or an observable, for example |\phi> is a vector in Hilbert space. What is the equivalent of a quantum state (or simply a state) in group theory? 1answer 100 views ### Decomposition of group representation using tensor method I am dealing with the decomposition of the representation 5\otimes5 of SU(5):$$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)==\... 1answer 173 views ### About$SU(2)_L \times U(1)_L = U(2)_L In the many textbook of standard model, i encounter the relation \begin{align} SU(2)_L \times U(1)_L ~=~ U(2)_L. \end{align} HereL$means the left-handness. (It is a physical meaning(representation)... 1answer 62 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: $$\psi(x) \rightarrow \psi'(x) =e^{i \... 1answer 354 views ### SU(3) irreducible representations with tensor method I am dealing with the tensor product representation of SU(3) and I have some problems in understanding some decomposition. 1) Let's find the irreducible representation of 3\otimes\bar{3} we have ... 0answers 104 views ### Uses of the accidental isomorphism SO(5)\sim Sp(2)? Some of the accidental isomorphisms of low dimensional Lie algebras have very important applications in physics. The theory of angular momentum makes use of the fact that SO(3)\sim SU(2). ... 1answer 156 views ### SU(2) generators and creation annihilation operators The algebraic method to find the irreducible representation of the SU(2) group makes use of the operators:$$J_z\\J_+=\frac{1}{\sqrt{2}}(J_x+iJ_y)\\J_-=\frac{1}{\sqrt{2}}(J_x-iJ_y)$$In the book ... 1answer 81 views ### Eigenvalues of the total angular momentum operator in$d>3\$? [closed]

How to calculate them? What are their degeneracy?
63 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 ...