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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4
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1answer
29 views

Group representations as vectors

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 ...
2
votes
2answers
219 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 ...
2
votes
1answer
50 views

Why does the raising and lowering operator not affect total angular momentum?

My notes define: $$ L_{\pm} = L_{x} \pm i L_{y} $$ and states: $$ [L_{z},L_{\pm}] = \pm \hbar L_{\pm} $$ I'm fine with this as it's easy to show the result with some ugly algebra. It then says: ...
0
votes
1answer
64 views

Are the Pauli matrices closed under commutation?

I tried to make a group multiplication table for the Pauli matrices, but I keep getting multiples in front of the elements. What am I doing wrong? I thought the Pauli matrices formed a group that was ...
4
votes
2answers
95 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 ...
4
votes
1answer
147 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 ...
3
votes
0answers
61 views

New Supersymmetry Algebra

We know that SUSY generators commute with translation $$ [P_\mu,Q_\alpha]=0 $$ I have some questions: What is this equation physical meaning? Is it possible to make "SUSY-like" generators that do ...
1
vote
0answers
16 views

Differenational in Lie groups [migrated]

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Here $\phi(t)$ is a one-parameter subgroup of the Lie group ...
1
vote
1answer
40 views

Correspondence between one-parameter subgroups of $G$ and $T_eG$

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
1
vote
0answers
47 views

The conjugate representation in $su(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $su(2)$ and that for the diagonal operator $T_3$, \begin{align} T_3\psi_1 ...
7
votes
0answers
51 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 ...
4
votes
1answer
107 views

The relationship between Lorentz Lie algebra and curvature

Here I transfered the question from the comment The relationship between spin and spinor curvature How $\mathcal{R}_{ab} = \frac{1}{4}R_{abst}\gamma^s \gamma^t$ is from $\Psi \mapsto \Psi + ...
1
vote
3answers
79 views

Generators of the Diffeomorphism Group

So what are the generators of a Diffeomorphism Group? For simplicity, let's consider $ Diff(R^2) $ (diffeomorphisms of the euclidean plane.) Diffeomorphisms are differentiable, invertible ...
8
votes
3answers
61 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 ...
2
votes
1answer
50 views

Symmetry transformation on Quantum Field

I stumbled upon this point several times, the latest beeing this question: Connection between conserved charge and the generator of a symmetry I want to understand, why Quantum fields transform under ...
5
votes
1answer
74 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 ...
2
votes
0answers
36 views

Left (right) invariant vector fields on superspace

I read Freed'd book on "Five lectures on supersymmetry". For any vector space $V$ with metric of signature $(1,n-1)$ he constructs super Lie algebra $$ L=V \oplus S^*, $$ where $S$ is space of ...
3
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0answers
36 views

Highest weight unitary representations of $psl(2|2)$

I'm having some trouble understanding how to extend representation theory from Lie algebras to super Lie algebras, in particular with $psl(2|2)$. Ultimately I'm interested in 2D quantum sigma models ...
2
votes
2answers
214 views

Lie algebra - basis for adjoint matrix products in SU(N)?

In $SU(N)$, the set of matrices in the fundamental representation plus the identity: $$ \left\{ \mathbf{1}, t^a \right\} $$ acts as a basis for generator products under matrix multiplication, such ...
7
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 ...
1
vote
0answers
32 views

Lie theory and particle physics [duplicate]

I have recently been reading Intro to Lie algebras and representation theory by Humphreys, and when I am finished I am interested in reading about Lie groups and Lie algebras and their applications to ...
2
votes
0answers
51 views

Possible Error in deriving conformal generator

My professor gave me the following derivation for the full generator of the Lorentz transformations. The starting point is to consider a subgroup of the conformal group that leaves the origin fixed ...
2
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0answers
46 views

The states of the adjoint representation correspond to the generators

I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\rangle$. What does ...
2
votes
2answers
89 views

Constructing SUSY algebra via index structure

Often in literature the SUSY algebra is simply given, but various books, for example Bailin and Love, goes through the trouble of showing how the SUSY commutation relations are the only possible ones ...
1
vote
1answer
84 views

What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
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vote
1answer
75 views

Verification of the Poincare Algebra

The generators of the Poincare group $P(1;3)$ are supposed to obey the following commutation relation to be verified: $$\left[ M^{\mu\nu}, P^{\rho} \right] = i \left(g^{\nu\rho} P^{\mu} - g^{\mu\rho} ...
4
votes
2answers
104 views

Why are the “coupling constants” constant?

The coupling constants (in the gauge theory) fix an inner product on the lie algebra of the gauge group and we use it to define strength of the fields. we are using ad-invariant inner products which ...
2
votes
2answers
148 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
2answers
672 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 ...
0
votes
1answer
96 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 ...
4
votes
1answer
114 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 ...
14
votes
2answers
468 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 ...
4
votes
2answers
136 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 $K_{\mu}$ ...
1
vote
1answer
78 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
56 views

Formalism and representation in Quantum Mechanics

I am just curious about the formalism of basic Quantum Mechanics. Lets take for instance the system of a spin-$\frac{1}{2}$ particle. The state of the particle is described by a vector in an abstract ...
5
votes
1answer
93 views

Kac-Moody algebras in 5 dimensional Kaluza-Klein theory

I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory. Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein ...
4
votes
2answers
122 views

Algebra, commutators and test functions

I am trying to make sense out of the algebra of the generators of the conformal group and I am running into some issues regarding how to calculate commutators. For instance, for translations of a ...
5
votes
1answer
341 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
2
votes
3answers
208 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 ...
1
vote
0answers
43 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 ...
2
votes
0answers
35 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 ...
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 ...
4
votes
0answers
84 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
108 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 $$ ...
8
votes
1answer
244 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
76 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 ...
3
votes
1answer
347 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 $(2,0)$?
12
votes
0answers
347 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\{ ...
16
votes
0answers
261 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 ...
6
votes
2answers
326 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}] = ...