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
309 views

Deriving Virasoro algebra question

I'm reading a book Lie groups, Lie algebras, cohomology and some applications in physics by Azcarraga and Izquierdo, and on page 347, when deriving the exact form of the central extension term I came ...
4
votes
1answer
231 views

Question on conserved quantities and Noether's theorem

I have a question about Noether's theorem in the context of QM, which I'll state in the context of the weak interaction but the basic point could be generalized. According to Noether's theorem, given ...
1
vote
1answer
291 views

$t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$

What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & ...
5
votes
2answers
644 views

What is the significance of Lie groups $SO(3)$ and $SU(2)$ to particle physics?

I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics? The application of Lie groups and their ...
1
vote
2answers
429 views

Symmetries & Lie groups in physics

This is not a homework, neither it is any exercise. It is my understanding of $U(1)$ symmetry. I would request if anybody can please correct me on any one of the following understandings: The ...
1
vote
1answer
507 views

How to get Gell-Mann matrices?

How to get Gell-Mann matrices $f_{i}$ (more or less strictly)? What are the requirements for getting them, excluding $||f_{i}|| = 1$, commutational law $[f_{i}, f_{j}] = if_{ijk}f_{k}$ and hermitian ...
8
votes
4answers
309 views

What does “carry a representation” mean (in SUSY algebra)?

I come from a maths background and am struggling with some of the more physical texts on SUSY. In particular they claim that the fermionic generators $Q_A^i$ carry a representation of the Lorentz ...
5
votes
2answers
544 views

Lie Groups and group extensions?

Is $U(1)$ x $SU(2)$ x $SU(3)$ a vector space over a field? I saw an article here http://en.wikipedia.org/wiki/Group_extension that seemed to me that a similar concept to a field extension was ...
0
votes
0answers
63 views

What is ``thermal" about a thermal quotient of EdS and EAds?

This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of ...
7
votes
1answer
280 views

link between real particles, representation of algebra and Young tableau

I know that different representations of this algebra correspond to different spin. One can sort the representation according to the casimir. For any simple Lie algebra, the operator $$ T^2 = ...
1
vote
1answer
346 views

How to theoretically determine the angular momentum of an atom?

To determine if an atom is a boson or a fermion I have to count the fermions that constitute the atom (protons, neutrons and electrons). My question is: How to theoretically (as opposed to ...
4
votes
1answer
157 views

Is the conjecture about $E(11)$ and M-theory (West's conjecture) generally accepted?

I was reading this paper by West, in which it is argued that: Eleven dimensional supergravity can be described by a non-linear realisation based on the group $E\left(11\right)$ From ...
4
votes
1answer
200 views

Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight ...
4
votes
2answers
351 views

Calculating an expression for the trace of generators of two Lie algebra

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie ...
0
votes
1answer
103 views

splitting spin representation when reducible

I'm looking at the spin representation of the orthogonal algebra $so(2m,C)$. This representation is $2^m$ dimensional and an explicit construction with gamma matrices is well known (see for example ...
14
votes
2answers
659 views

Fundamental Representation of $SU(3)$ is a complex representation

Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule: $\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now ...
12
votes
1answer
421 views

How to evaluate this sum of coupling coefficients?

I would like to evaluate the following summation of Clebsch-Gordan and Wigner 6-j symbols in closed form: $$\sum_{l,m} C_{l_2,m_2,l_1,m_1}^{l,m} C_{\lambda_2,\mu_2,\lambda_1,\mu_1}^{l,m} \left\{ ...
4
votes
1answer
653 views

What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
5
votes
1answer
632 views

Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
15
votes
2answers
1k views

Gauge fields — why are they traceless hermitian?

A gauge field is introduced in the theory to preserve local gauge invariance. And this field (matrix) is expanded in terms of the generators, which is possible because the gauge field is traceless ...
5
votes
2answers
228 views

Problem counting spin states

I can't figure out how many different spin states I can create with a four-electron system. I think I can create a spin-zero state, three spin-one states, and five spin-two states. That gives me nine ...
1
vote
1answer
145 views

Lie algebra of lorentz group

I'm stuck in following calcualtion from sredniki's QFT book.(Its actually in the solution manual) How can i get from $$\delta\omega_{\rho\sigma}(g^{\sigma\mu}M^{\rho\nu} - g^{\rho\nu}M^{\mu\sigma}) ...
6
votes
2answers
928 views

Do generators belong to the Lie group or the Lie algebra?

In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've ...
12
votes
2answers
253 views

When are there enough Casimirs?

I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators ...
8
votes
2answers
2k views

Definition of Casimir operator and its properties

I'm not sure which is the exact definition of a Casimir operator. In some texts it is defined as the product of generators of the form: $$X^2=\sum X_iX^i$$ But in other parts it is defined as an ...
6
votes
1answer
447 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
5
votes
2answers
588 views

In quantum mechanics(QM), can we define a high-dimensional “spin” angular momentum other than the ordinary 3D one?

Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define angular momentum in other than three dimensions? , now I get ...
12
votes
4answers
2k views

Trace and adjoint representation of $SU(N)$

In the adjoint representation of $SU(N)$, the generators $t^a_G$ are chosen as $$ (t^a_G)_{bc}=-if^{abc} $$ The following identity can be found in Taizo Muta's book "Foundations of Quantum ...
3
votes
2answers
1k views

Quantization of orbital angular momentum

Probably a very simple question, but I can't find the answer on the Internet. I know nearly to nothing about quantum mechanics, but in statistical physics I'm confronted with the idea that the orbital ...
3
votes
1answer
159 views

Mathematically, how do we deduce that angular momentum is bounded?

So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$? I.e. that m is bounded by j. We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its ...
1
vote
1answer
62 views

Please provide the simplest example you can think of, of generators of time evolution and generalized coordinates

I was reading the Wikipedia article about Noether's theorem and this thing popped out: Then the resultant perturbation can be written as a linear sum of the individual types of perturbations ...
2
votes
1answer
395 views

Dirac trace theorem

I am unable to prove exactly one trace identity that appears in the appendix of Peskin and Schroeder's QFT book. Can someone help me? The theorem [Appendix A.4 eqn (A.28)] says that the order of ...
2
votes
2answers
271 views

Representations of Lie algebras in physics

Why is an invariant vector subspace sometimes called a representation? For example in Lie algebras, say su(3), the subspace characterized by the highest weight (1,0) is an irreducible representation ...
3
votes
3answers
448 views

Quantum mechanical angular momentum and spin formalism/notation

I am currently stuck on the following notation: $\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$ No matter what I tried, I couldn't derive the identity. I am sure that ...
6
votes
1answer
254 views

Equivalent Representations of Clifford Algebra

I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94. We've considered the standard chiral representation of the Clifford Algebra, ...
9
votes
2answers
503 views

How to model a symmetry using Lie Groups?

I have been reading lately about Lie groups, and although all books keep listing the groups, and talk about Lie algebras and all that, one thing I still don't know how is it made, and I guess it's the ...
1
vote
2answers
391 views

high spin atoms SU(2) representation

I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry. Why not $SU(N)$?
2
votes
1answer
229 views

Taylor series for unitary operator in Weinberg

On page 54 of Weinberg's QFT I, he says that an element $T(\theta)$ of a connected Lie group can be represented by a unitary operator $U(T(\theta))$ acting on the physical Hilbert space. Near the ...
4
votes
1answer
524 views

Holstein-Primakoff and Dyson-Maleev representation

In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
9
votes
3answers
205 views

Embedding of $F(4)$ in $OSp(8|4)$?

Is the superconformal algebra in five dimensions, $F(4)$, a subalgebra of the (maximal) six-dimensional superconformal algebra $OSp(8|4)$?
7
votes
2answers
2k views

Killing vector fields

I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.
9
votes
1answer
411 views

Why does a transformation to a rotating reference frame NOT break temporal scale invariance?

Naively, I thought that transforming a scale invariant equation (such as the Navier-Stokes equations for example) to a rotating reference frame (for example the rotating earth) would break the ...
3
votes
0answers
127 views

Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
2
votes
1answer
448 views

What is the Lie algebra of the Galilean group and what is the structure of it?

I read Freeman Dyson's article Missed Opportunities, in which he talked about the mathematical attractiveness of the Lorenz group compared to the Galilean group. I am reading Florian Scheck's book on ...
1
vote
1answer
392 views

Commutators with a density matrix

The equation describing the evolution of our system is as follows: $ \dot{\rho} = u_1(t)(a^\dagger a \rho - 2a\rho a^\dagger +\rho a^\dagger a) + u_2(t)(a a^\dagger \rho - 2a^\dagger\rho a +\rho a ...
5
votes
2answers
480 views

Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
1
vote
1answer
672 views

Wigner-Eckart projection theorem

I'm following the proof of Wigner-Eckart projection theorem which states that: $$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
8
votes
2answers
686 views

Lie bracket for Lie algebra of $SO(n,m)$

How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by $$[J_{ab},J_{cd}] ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$ ...
5
votes
1answer
533 views

Wigner-Eckart theorem of SU(3)

I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
3
votes
1answer
404 views

How do I find the tensor components of all weights of a representation of $SU(3)$, e.g. the six dimensional representation $(2,0)$?

How do I find the corresponding tensor component $v^{ij}$ of the six dimensional representation of $SU(3)$ with Dynkin label $(2,0)$?