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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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 ...
26
votes
2answers
738 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 ...
20
votes
1answer
471 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 ...
20
votes
1answer
206 views

Any use for $F_4$ in hep-th?

In high energy physics, the use of the classical Lie groups are common place, and in the Grand Unification the use of $E_{6,7,8}$ is also common place. In string theory $G_2$ is sometimes utilized, ...
17
votes
2answers
914 views

Is the G2 Lie algebra useful for anything?

Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. ...
17
votes
1answer
582 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 ...
16
votes
2answers
2k 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 ...
16
votes
1answer
2k views

Mathematically, what is color charge?

A similar question was asked here, but the answer didn't address the following, at least not in a way that I could understand. Electric charge is simple - it's just a real scalar quantity. Ignoring ...
16
votes
1answer
545 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\{ ...
15
votes
3answers
2k views

Why is the Symmetry Group for the Electroweak force $SU(2) \times U(1)$ and not $U(2)$?

Let me first say that I'm a layman who's trying to understand group theory and gauge theory, so excuse me if my question doesn't make sense. Before symmetry breaking, the Electroweak force has 4 ...
15
votes
2answers
1k views

What's the relationship between $SL(2,\mathbb{C})$, $SU(2)\times SU(2)$ and $SO(1,3)$?

I'm a beginner of QFT. Ref. 1 states that [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$. But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group ...
14
votes
2answers
1k 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 ...
13
votes
1answer
482 views

Covariant derivatives

I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \{X_i, Y_a\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \{X_i\}$ Lie algebra of $G$ has ...
13
votes
3answers
988 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...
12
votes
4answers
4k 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 ...
12
votes
3answers
5k views

What do the Pauli matrices mean?

All the introductions I've found to Pauli matrices so far simply state them and then start using them. Accompanying descriptions of their meaning seem frustratingly incomplete; I, at least, can't ...
12
votes
2answers
328 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 ...
12
votes
1answer
407 views

Lie group of Schrodinger Wave equation

In Ballentine's book on quantum mechanics (in 3rd chapter), he introduces the symmetry transformation of Galilean group associated with Schrodinger equation. Now the Galilean group as such has 10 ...
11
votes
1answer
337 views

Relation between cohomology and the BRST operator

Given a manifold $M$, we may define the $p$th de Rham cohomology group $H^p(M)$ as the quotient, $$C^p(M) \, / \, Z^p(M)$$ where $C^p$ and $Z^p$ are the groups of closed and exact $p$-forms ...
11
votes
3answers
282 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 ...
10
votes
2answers
4k 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 ...
10
votes
2answers
709 views

Lie algebra in simple terms [closed]

My question is regarding a vector space and Lie algebra. Why is it that whenever I read advanced physics texts I always hear about Lie algebra? What does it mean to "endow a vector space with a lie ...
10
votes
2answers
812 views

Geometric/Visual Interpretation of Virasoro Algebra

I've been trying to gain some intuition about Virasoro Algebras, but have failed so far. The Mathematical Definition seems to be clear (as found in http://en.wikipedia.org/wiki/Virasoro_algebra). I ...
10
votes
3answers
475 views

When do phase space functions' Poisson brackets inherit the Lie algebra structure of a symmetry?

I've seen several examples of phase space functions whose Poisson brackets (or Dirac brackets) have the same algebra as the Lie algebra of some symmetry. For example, for plain old particle motion in ...
9
votes
2answers
875 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 $$[J_{ab},J_{cd}] ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$ ...
9
votes
1answer
1k views

Generators of Poincare Groups

How can I determine the generators of the Poincare Group, $P(1,3)$ explicitly? Here $P(1,3)$ means a matrix Lie group.
9
votes
2answers
513 views

Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
9
votes
2answers
3k 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
2answers
798 views

Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning ...
9
votes
3answers
222 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 $OSp(8|4)$?
9
votes
1answer
597 views

Why does the eightfold way work?

Last year I attended an introductory particle physics course, in which the Eigthfold Way for classifying hadrons has been discussed. The main idea consists in grouping hadrons in multiplets (i.e ...
9
votes
2answers
665 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 ...
9
votes
1answer
203 views

Confusion about two definitions of anomalies

As I am currently studying for an exam about quantum field theory and string theory, I got confused about the notion of "anomalies" and how they are actually defined. Similar questions have already ...
9
votes
1answer
485 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 ...
8
votes
1answer
687 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
8
votes
2answers
2k views

Lie algebra and Lie group about quantum harmonic oscillator

We know that in the quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ are the annihilation and creation operators, and $H$ is the ...
8
votes
2answers
2k 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 ...
8
votes
2answers
918 views

How does non-Abelian gauge symmetry imply the quantization of the corresponding charges?

I read an unjustified treatment in a book, saying that in QED charge an not quantized by the gauge symmetry principle (which totally clear for me: Q the generator of $U(1)$ can be anything in ...
8
votes
2answers
900 views

Lie group Homomorphism $SU(2) \to SO(3)$

The Lie algebra of $ \mathfrak{so(3)} $ and $ \mathfrak{su(2)} $ are respectively $$ [L_i,L_j] = i\epsilon_{ij}^{\;\;k}L_k $$ $$ [\frac{\sigma_i}{2},\frac{\sigma_j}{2}] = ...
8
votes
4answers
429 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 ...
8
votes
2answers
186 views

Subgroup of Lorentz Group Generated by Boosts

It is common knowledge that a composition of boosts is not a boost, but involves a rotation. Further, in discussions of Thomas precession, it is often stated that boosting in $x$, then $y$, then back ...
7
votes
3answers
726 views

The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices

The Pauli spin matrices $$ \sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}), \qquad\qquad \sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
7
votes
2answers
2k views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
7
votes
2answers
153 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, ...
7
votes
1answer
1k 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 ...
7
votes
2answers
115 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 ...
7
votes
1answer
345 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 = ...
7
votes
1answer
134 views

Lie algebra of axial charges

Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices) $L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} ...
7
votes
4answers
393 views

How to prove that any rotation can be represented by 3 Euler angles

How can one prove that any rotation of a rigid object in 3-dimensional (3D) space can be represented by a sequence of three rotations around pre-fixed axes by 3 Euler angles? I see this statement in ...
7
votes
1answer
361 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group? [duplicate]

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...