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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22
votes
3answers
634 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 ...
15
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 ...
4
votes
2answers
415 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 ...
3
votes
1answer
477 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, ...
8
votes
2answers
594 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
2answers
1k 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 ...
11
votes
2answers
529 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 ...
5
votes
2answers
445 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 ...
3
votes
1answer
478 views

Why is the value of spin +/- 1/2?

I understand how spin is defined in analogy with orbital angular momentum. But why must electron spin have magnetic quantum numbers $m_s=\pm \frac{1}{2}$ ? Sure, it has to have two values in ...
7
votes
2answers
658 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 ...
13
votes
2answers
417 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 ...
2
votes
2answers
198 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 ...
8
votes
3answers
583 views

Why is the Symmetry Group for the Electroweak force SU(2)xU(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 ...
5
votes
2answers
403 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 ...
3
votes
2answers
124 views

Unitrary groups and infinitesimal transformations - Schwingers way of deriving Lie groups

In Schwinger's source theory book, he suggests if $G_a$ are the hermitian generators of the Unitary group, then we have an infinitesimal transformation is given by : $$ G = \sum_{a=1}^n ...
3
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.
13
votes
2answers
671 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 ...
13
votes
2answers
711 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. ...
11
votes
1answer
178 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 ...
6
votes
1answer
422 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.
6
votes
2answers
277 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}] = ...
9
votes
1answer
350 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 ...
6
votes
2answers
161 views

Galilean, SE(3), Poincare groups - Central Extension

After having learnt that the Galilean (with its central extension) with an unitary operator $$ U = \sum_{i=1}^3\Big(\delta\theta_iL_i + \delta x_iP_i + \delta\lambda_iG_i +dtH\Big) + ...
4
votes
1answer
103 views

Conceptual question about field transformation

(c.f Conformal Field Theory by Di Francesco et al, p39) From another source, I understand the mathematical derivation that leads to eqn (2.126) in Di Francesco et al, however conceptually I do not ...
4
votes
1answer
95 views

Representations and transformations under an $SU(n)$ Lie groups?

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
3
votes
1answer
93 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 ...
1
vote
0answers
39 views

Spin matrix for various spacetime fields

Let $V^{\mu}$ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation $V'^{\mu} = \Lambda^{\mu}_{\,\,\,\nu}V^{\nu}$. We can write this like ...
6
votes
2answers
541 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 ...
3
votes
1answer
371 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 ...
2
votes
1answer
104 views

Real representation is physically real?

In Peskin & Schroder, Introduction to Quantum Field Theory equation (15.82) states that $$ t^a_{\bar{r}} = -(t^a_{r})^* = (t^a_{r})^T $$ Why is the representation which satisfies $$ ...
1
vote
2answers
66 views

Generator of the Special Conformal Transformation

In this thread Integrating the generator of the infinitesimal special conformal transformation, the generator of the 'flow' of the transformation is written as $$G_b = 2(b \cdot x)x - x^2 b,$$ where ...
2
votes
1answer
99 views

Charge of a field under the action of a group

What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?
2
votes
1answer
172 views

Similar masses and lifetimes of the $\Delta$ baryons

Why do the four spin 3/2 $\Delta$ baryons have nearly identical masses and lifetimes despite their very different $u$ and $d$ quark compositions?