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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2
votes
0answers
38 views

What is 'heterotic string compactification'?

I've read that some exceptional groups arises in the context of 'heterotic string compactification'. Could someone explain (to a person studying physics but who doesn't know string theory) what ...
4
votes
2answers
298 views

Commutation relations of the generators of the conformal group

My question is from P.98 of the book by Di Francesco on Conformal Field theory. He gives the six non-vanishing commutation relations between the elements $P_{\mu}, D, L_{\mu \nu}$ and $K_{\mu}$ ...
4
votes
0answers
99 views

Wilson's Renormalization Group and Lie's Third Theorem

If you think of a one-parameter group of transformations along a curve in the plane as a (Lie) group, and the tangent vector to the curve as a generator of the curve we can intuitively understand ...
3
votes
1answer
121 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
1answer
132 views

Translation and Dilation transformations within the conformal group

I am using Di Francesco's book P.39. The equation that the generators of the transformations satisfy is given by: $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta w_a} \partial_{\mu} \Phi - \frac{\delta ...
6
votes
1answer
230 views

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

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 ...
5
votes
1answer
168 views

What is the relationship between Cyclic Coordinates and Killing Vector Fields?

My question is related to this question. There are three or four other questions on Killing Vector Fields here, however none of them that I have seen address my question. $\\$ I've been studying ...
4
votes
2answers
360 views

Why the generators of boosts transform like a vector under rotation?

$$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$ $$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$ $$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ ...
3
votes
0answers
106 views

Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group [duplicate]

So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can ...
19
votes
0answers
422 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 ...
4
votes
1answer
112 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
139 views

$SU(2)$ gauge symmetry

Take the Lagrangian with one fermion: $$ \mathcal{L} = -\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi$$ where the gauge covariant derivative $D_\mu = ...
3
votes
1answer
170 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 ...
5
votes
1answer
94 views

R-symmetry commutator

I've seen the claim made several placed; Terning's "Modern Supersymmetry" p. 5 on N=1 SUSY algebra states it as well as anyone: The SUSY algebra is invariant under a multiplication of $Q_\alpha$ ...
14
votes
2answers
749 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 ...
7
votes
2answers
393 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 ...
1
vote
1answer
76 views

Where do $L_+$ and $L_-$ live, if not in $\mathfrak{so(3)}$?

This question is continuation to the previous post. The lie algebra of $ \mathfrak{so(3)} $ is real Lie-algebra and hence, $ L_{\pm} = L_1 \pm i L_2 $ don't belong to $ \mathfrak{so(3)} $. However, ...
6
votes
2answers
565 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}] = ...
0
votes
3answers
102 views

Commutator summation notation

I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?
13
votes
3answers
1k 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 ...
2
votes
1answer
48 views

Mixing generators of different dimensionality

Reading a paper about compactified manifolds used in Kaluza Klein theories the author discusses in which ways you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of ...
2
votes
0answers
79 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[ ...
6
votes
2answers
247 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) + ...
6
votes
1answer
182 views

Question about the Noether charge algebra

I'm reading these notes - page 8 and 9 - and I'm a bit confused. If we consider a field $\phi$ (which can be either bosonic or fermionic) transforming as: \begin{equation} \phi(x) \rightarrow \phi(x) ...
1
vote
1answer
250 views

A roadmap for learning standard model of particle physics [duplicate]

Assuming that a person has understanding of theory of Lie groups, Lie algebras and basic quantum mechanics, what is the simplest route to gain a basic understanding of the SM of particle physics? Are ...
11
votes
1answer
255 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 ...
2
votes
1answer
214 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field ...
3
votes
1answer
213 views

Lie Algebra Generators

We know that rotations are performed via real and orthogonal matrices, $O^{T}=O^{-1}$. We can write $O$ as, (The proper rotations have unit determinant) $$O = \exp(A),$$ where $A^{T}=-A$. In three ...
5
votes
1answer
156 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 ...
2
votes
0answers
70 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 ...
1
vote
0answers
79 views

How are symmetries defined mathematically? [duplicate]

I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding ...
2
votes
2answers
226 views

Why do we require the generators of $\mathrm{SU(N)}$ gauge theories to be $N \times N$ matrices?

I have often read that the generators for $\mathrm{SU(N)}$ gauge theories must be $N \times N$ matrices; see for instance these notes at the top of page 3: ...
7
votes
1answer
113 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} ...
6
votes
2answers
464 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 ...
5
votes
1answer
116 views

Proving Lemma 4 in Georgi's Lie Algebra in Particle Physics 2nd p 251

The lemma 4 is given in the above picture. My question is, how to verify linear dependence (20.15) for diagram (a)? I tried to extend the matrix for the simple root in wikipedia $$ \left ...
2
votes
0answers
87 views

Group of translations in two dimensions - A weird treatment

Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$ G = p_1\delta x_1 +p_2 \delta x_2 $$ Now, we know clearly that this group is an Abelian ...
3
votes
2answers
156 views

Unitary 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 ...
6
votes
2answers
324 views

Number of the Generators of Poincare Group

It is said that the Poincare group, $P(3,1)$ has $10$ generators. $6$ of them are the generators of the Lorentz group, $O(3,1)$ and the other $4$ generators are the generators of $4D$ translational ...
2
votes
3answers
275 views

Number of Parameters of Lorentz Group

We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the ...
5
votes
2answers
311 views

Rotation Group and Lorentz Group

It is often stated that rotations in the 3 spatial dimensions are examples of Lorentz transformations. But Lorentz transformations form a group named the Lorentz Group, $O(1,3)$ which is a group a ...
4
votes
1answer
403 views

Scalar field transformation and generators

When we do a transformation (norm preserving one) for a given quantity, from what I have understood it seems like there is a representation of the group element for each quantity depending how they ...
8
votes
1answer
788 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.
2
votes
1answer
123 views

Does a Super Noether Theorem exist?

I am wondering if an extension of Noether theorem to supergroups exists. In particular the analogy with the usual case should be that supersymmmetries are in 1 to 1 correspondence to certain ...
2
votes
0answers
64 views

Quantum Field Theory and Lie Theory [duplicate]

I am reading Vol.1 of "The Quantum Theory Of Fields" by S. Weinberg. However I have come to a halt when connected Lie groups were introduced. I have solid knowledge in elementary group theory and ...
2
votes
1answer
75 views

1-dimensional Ring geometry - Group of Translations

I considered a Ring-like one dimensional geometry. In this, if we fix an origin (at some point on the circumference), we can think of set of all displacements along the circumference to form a vector ...
24
votes
3answers
924 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 ...
2
votes
2answers
118 views

Different definition of SL(2,R) algebra?

I'm looking into $SL(2,\mathbb{R})$ group and it's algebra. I found on line that the $sl(2,\mathbb{R})$ algebra is given by the two by two real matrices of trace zero. This Lie algebra has dimension ...
0
votes
1answer
136 views

Triangle inequality Clebsch-Gordan coeffcients

The Clebsch-Gordan coefficients can only be non-zero if the triangle inequality holds: $$\vert j_1-j_2 \vert \le j \le j_1+j_2$$ In my syllabus they give the following proof: $$-j \le m \le j$$ $$-j_1 ...
2
votes
1answer
363 views

Lorentz group representation and transformation of “vectors”

Let $P$ be the parity operator of the Lorentz group, $$P=\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}$$ the commutation relations ...
1
vote
0answers
82 views

Action of conformal generators on fields

I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am ...