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Questions tagged [lie-algebra]

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 gauge group's Lie algebra correspond to Noether currents and conserved quantities.

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61
votes
20answers
31k views

Comprehensive book on group theory for physicists?

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
54
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3answers
5k 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 ...
52
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1answer
3k 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}(3)...
34
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2answers
2k views

Is there an elegant proof of the existence of Majorana spinors?

Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly: A priori, we are dealing with ...
31
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3answers
2k 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 ...
27
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2answers
5k 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 ...
27
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2answers
845 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 ...
24
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1answer
4k 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 ...
21
votes
4answers
17k 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 ...
21
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3answers
4k 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 ...
20
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2answers
3k 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 ...
20
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2answers
6k 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 ...
20
votes
1answer
329 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, ...
18
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2answers
9k 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 ...
18
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2answers
1k 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 ...
17
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1answer
988 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 ...
17
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3answers
2k 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}} = \...
16
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1answer
1k views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: $$[L_{\mu\nu},L_{\rho\sigma}]=i(\eta_{\mu\sigma}L_{\nu\rho}-\eta_{\mu\rho}L_{\nu\sigma}-\eta_{\nu\sigma}L_{\mu\rho}+\...
16
votes
1answer
703 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 ...
16
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2answers
1k 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. ...
16
votes
2answers
590 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 $...
16
votes
1answer
616 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
4answers
7k 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 ...
15
votes
2answers
2k 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 $\mathbb{...
15
votes
1answer
999 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 ...
14
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2answers
3k 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
4answers
2k views

Why Lie algebras if what we care about in physics are groups?

In physics, we want irreducible representations of the symmetry group of our system. However, one frequently sees representations of the corresponding Lie algebra being studied instead. Is it that in ...
13
votes
2answers
4k 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 ...
13
votes
1answer
538 views

What is “broken symmetry”?

For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this). This is my current ...
13
votes
3answers
1k 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 ...
12
votes
2answers
2k 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 $$[J_{ab},J_{cd}] ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$ ...
12
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2answers
3k views

What exactly is the connection between the Jacobi and Bianchi identities?

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0.$$ ...
11
votes
2answers
768 views

Invariance under boosts but not rotations?

I am aware that there are 6 independent infinitesimal Lorentz transformations that can be separated into 3 rotations and 3 boosts. Is it possible for a quantum field theory to be invariant under the ...
11
votes
3answers
1k views

Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$

I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example: Suppose I have a spin-1/2 particle, lets say an electron, ...
11
votes
2answers
3k 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}] = i\epsilon_{ij}^{\;\;k}\frac{...
11
votes
2answers
2k 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 ...
11
votes
1answer
2k views

The conjugate representation in $\mathfrak{su}(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $\mathfrak{su}(2)$ and that for the diagonal operator $T_3$, \begin{align} ...
11
votes
3answers
338 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
372 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 $$ [...
10
votes
2answers
2k 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 ...
10
votes
2answers
1k views

Why are the generators of rotation in the 4-dimensional Euclidean space correspond to rotations in a plane?

In three-dimensions, the rotation generators are represented by $J_1$, $J_2$ and $J_3$ where $1,2,3$ respectively stands for the generator of rotation about $x,y,z$ axes respectively. In general, in ...
10
votes
2answers
3k 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 ...
10
votes
1answer
3k 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.
10
votes
2answers
5k 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.
10
votes
2answers
7k 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 ...
10
votes
2answers
1k 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
2answers
491 views

Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...
10
votes
1answer
369 views

Reference recommendation for Projective representation, group cohomology, Schur's multiplier and central extension

Recently I read the chapter 2 of Weinberg's QFT vol1. I learned that in QM we need to study the projective representation of symmetry group instead of representation. It says that a Lie group can have ...
10
votes
4answers
2k 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 ...
9
votes
4answers
1k views

Matrix diagonalization in SU(2) and SO(3)

I'm currently using Nadri Jeevanjee's book on group theory for physicists to understand quantum mechanics. I came across these two pages that left me stuck: Example 4.19 $SU(2)$ and $SO(3)$ In ...