A vector space $\mathfrak{g}$ over some field $F$ and kitted with a bilinear, antisymmetric and Jacobi-identity-fulfilling product ("Lie Bracket" or "commutator"). In physics, most often arises as the Lie algebra (tangent space to the identity) of a Lie group; in gauge theories, basis vectors of the ...

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3
votes
1answer
66 views

$(1/2, 0)$ representation of the Lorentz Group $SO(1,3)$

Let us consider the $(j, j') = \left(\frac{1}{2}, 0\right)$ representation of $SO(1, 3)\cong SU(2) \otimes SU(2)$. $j = \frac{1}{2}$ corresponds to $SU(2)$ generated by $$ \tag{1} N_i^+ = \frac{1}{...
7
votes
2answers
174 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, ...
1
vote
1answer
104 views

How to derive the form of the invariant spinor inner product?

So we have gamma matrices that satisfy the spacetime algebra relations, $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu}$. We know that if we set $\sigma^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ ...
7
votes
1answer
156 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
4
votes
1answer
82 views

Lie Algebra for fermion fields

A key identity (e.g. when deriving BRST symmetry for gauge fields) is that: $$[c,d]_a =f_{abc}c_b d_c$$ where $c$ and $d$ are both Fermion Fields. How do I derive this from the lie algebra ...
3
votes
0answers
45 views

Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

Projective representations of a multiply-connected group is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of mine, it appears ...
1
vote
1answer
71 views

Generators of $SU(2)\times U(1)$

I'm currently reading about spontanous symmetry breaking. In particular about a Lagrangian that is invariant under $SU(2)\times U(1)$, in other words pretty standard QFT stuff. I know that the ...
30
votes
3answers
2k views

Idea of Covering Group

$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence? I heard that this fact is related to the description of bosons and fermions. But how does it ...
6
votes
1answer
330 views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
2
votes
1answer
44 views

How to construct a space which is translation invariant but not rotation invariant

I am just confused by the following idea. Consider a 3-dimensional translation invariant space, we now have 3 translation generators. Then let us start with a point, the full 3-dimensional space ...
0
votes
0answers
26 views

Reference on Lie superalgebras

I wish to study Lie superalgebras, in particular how they are involved with the superconformal symmetry of N=4 SYM in D=4. I have a solid background in Lie algebras from a pure mathematical ...
3
votes
1answer
178 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 ...
5
votes
1answer
318 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
0
votes
0answers
36 views

Glauber formula, Baker

somewhere I read here: $[A,F(B)]=[A,B]F'(B)$ is used to prove Glauber's formula $\exp(A+B).\exp([A,B]/2)$ I have tried and looked everywhere to try and understand this to no avail. The first is in ...
1
vote
1answer
132 views

Question on quotient groups and SLOCC [closed]

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
1
vote
2answers
53 views

What is the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1) \subset su(2)$?

What is meant by the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1)\subset su(2)$? I have read it above eqn. (10) in this paper http://arxiv.org/abs/0812.3572 but have also heard it mentioned in ...
1
vote
1answer
99 views

Virasoro Algebra vs Witt Algebra

I'm reading some notes on CFT, and there's a strange topic that I find quite confusing. We define the Witt algebra to be the generators of conformal transformations on the complex plane. $l_n = -z^{...
2
votes
1answer
138 views

$SU(3)$ Tensor Methods in a Tetraquark

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
3
votes
0answers
23 views

Equivariance Relation - Superconformal Hypermultiplets

I'm concerned with equation 2.24 of http://arxiv.org/abs/1601.00482 The superconformal hypermultiplets in this paper have a conic hyperkahler target manifold and the authors want to gauge some ...
4
votes
1answer
104 views

Integrating elements of a Lie group with respect to parameters of the corresponding Lie algebra

I am working with an operator $\textbf{M}$ that is represented by the Lie group SO(1,3), thus it can be written as, $$ \textbf{M} = \exp{\textbf{L}} $$ where, $$ \textbf{L} = \begin{bmatrix} 0&a&...
1
vote
0answers
27 views

Spontaneous symmetry breaking of scalar multiplet theory

Consider a theory with two multiplets of real scalar fields $\phi_i$ and $\epsilon_i$, where $i$ runs from $1$ to $N$. The Lagrangian is given by: $$\mathcal L = \frac{1}{2} (\partial_{\mu} \phi_i) (\...
7
votes
2answers
128 views

Lie groups with same algebra

I had a problem when considering symmetry breaking in an SO(4) gauge theory: $\mathcal{L} = \left| D_\mu\phi \right|^2$ where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some ...
-2
votes
1answer
44 views

Doesn't modelling using Lie Groups assume spacetime is continuous?

Lie groups are used to some behaviors of quantum mechanics, as well as forming a basis for Kaluza-Klein, Yang-Mills, and String theory. But Lie groups are defined as involving a differentiable ...
0
votes
0answers
28 views

Construct any Hamiltonian that is the linear combination of existing constructable Hamiltonians

In the paper Quantum Computation over Continuous Variables, it states that since $$e^{iAt}e^{iBt}e^{-iAt}e^{-iBt} = e^{-[A,B] t^2} + O(t^3)$$ when $t\rightarrow 0$, if one can apply a set of ...
0
votes
1answer
27 views

Decomposition of the adjoint representation of a spontaneously broken compact group

Let be $G$ a compact group, symmetry of the theory I am working with. $G$ is broken into one subgroup $H$. I define the generators of G as $T_A = \{T_a,T_\hat{a}\}$, where the first are the unbroken ...
3
votes
1answer
73 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) ...
4
votes
0answers
65 views

Why are generators defined oppositely in Weinberg's vs. Maggiore's QFT books?

I've been confused about the sign conventions used in Weinberg's QFT book for a long time. Here's my question: The generators $J^{\mu\nu}$ are defined in this book as $$U(1+\omega)=1+\frac{i}{2}\...
2
votes
1answer
157 views

Casimir Invariants of the Galilean group

I had studied a couple of things about Galilean and Poincare group. But in the Galilean group, there is not enough clarity on how to calculate generators for boosts ($B_i$), which if I do it seems I ...
2
votes
3answers
568 views

Pauli Matrices proof

The original question was to prove $$e^{i\sigma_z\phi} \sigma_y e^{-i\sigma_z\phi}=e^{2i\sigma_z\phi} \sigma_y.$$ Since left hand side is equal to $\sigma_y$, I thought in order for right hand ...
1
vote
0answers
128 views

Matrix Representations of Galilean group

The general group element (in the vector representation) $$ \left [{ \begin{array} {c} \bar x^1 \\ \bar x^2 \\ \bar x^3 \\ \bar t \\ 1 \\ \end{array} } \right] = \left[ {\begin{array}{...
1
vote
1answer
48 views

ADHM construction and Momentum Map

while I was reading about ADHM construction I had some troubles with precise geometrical identification of the various quantities. My doubts is well manifest in these two Wikipedia pages 1) ADHM ...
3
votes
1answer
174 views

Commutator of SU(2) Casimir operators in su(3)

I have a $\mathfrak{su}(3)$ Lie algebra spanned by 8 generators, $$\left\{J_1,J_2,J_3,J_4,J_5,J_6,J_7,J_8 \right\}$$ Now, I can choose infinitely many $\mathfrak{su}(2)$ sub-algebras composed of ...
0
votes
1answer
91 views

Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
5
votes
2answers
243 views

What is the exact relation between $\mathrm{SU(3)}$ flavour symmetry and the Gell-Mann–Nishijima relation

I'm trying to understand how the Gell-Mann–Nishijima relation has been derived: \begin{equation} Q = I_3 + \frac{Y}{2} \end{equation} where $Q$ is the electric charge of the quarks, $I_3$ is the ...
11
votes
3answers
284 views

Group representations as vectors and isomorphism between weights and matrix generators

This might be something basic, but it is unclear to me. So I am used to work with representations of groups as matrices. These matrices represent the structure of the Lie algebra by satisfying the ...
1
vote
1answer
36 views

Structure constant of the commutators of generators in broken symmetry

When I read a paper related to spontaneously global symmetry breaking, I cannot understand a statement: If we use the notation $T^i$ for the unbroken group generators in $H$ and $X^a$ the broken ...
0
votes
2answers
79 views

Commutator relationships and the exponential

I am currently trying to prove that the two following commutator relationships are equivalent (for an operator $\hat{A}(s)$ that depends on a continuous parameter $s$), so if one holds the other one ...
3
votes
1answer
188 views

The derivation of the irreducible representations of the Lorentz group

I took the way of classification of Lorentz group representations from Sexl, Urbantke, Relativity, groups and particles (Germ. ed. 1975). But I don't understand it as I outline in the following: In ...
2
votes
2answers
266 views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
3
votes
1answer
170 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 "...
0
votes
2answers
64 views

Good reference on the parametrization of $SU(3)$ and $SU(N)$

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}\cos(\theta) & -e^{-i\beta}\sin(\theta) \\ e^{i\beta}\sin(\...
3
votes
1answer
234 views

What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
20
votes
1answer
574 views

Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics: The universal cover: For instance, when representing the rotation group $\mathrm{SO}(...
26
votes
2answers
743 views

Finding the vacuum which breaks a symmetry

I will start with an example. Consider a symmetry breaking pattern like $SU(4)\rightarrow Sp(4)$. We know that in $SU(4)$ there is the Standard Model (SM) symmetry $SU(2)_L\times U(1)_Y$ but depending ...
2
votes
0answers
70 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 [\...
3
votes
1answer
845 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. ...
1
vote
0answers
77 views

Books on representation theory [duplicate]

Possible Duplicate: Best books for mathematical background? I'm looking for a textbook on the group/representation theory for a student-physicist. The main questions of interest are ...
0
votes
0answers
23 views

Why should an generator acts on an operator with the Lie bracket?

When we deal with ordinary symmetries which form a Lie group, we have an corresponding Lie algebra with a structure of Lie bracket $[,]$. A infinitesimal transformation can act on a state or an ...
1
vote
1answer
89 views

Decomposition of tensor product space into direct sum [closed]

Consider a tensor product space of two representations $j = \frac{3}{2}$ and $j = 1$. How to show $4 \otimes 3 = 6 \oplus 4 \oplus 2$?
3
votes
1answer
187 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 ...