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
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20answers
30k 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 ...
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1answer
146 views

Quantum spin, tensor product: a long time relationship

Anyone who has studied quantum mechanics know the following relation: $ 2 \otimes 2 = 3 \oplus 1 $ But how did a man woke up and said "Hell yeah, I'll use tensor product of two spin $1/2$ to simulate ...
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0answers
27 views

There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
3
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1answer
49 views

Fierz identity for symplectic group

For the fundamental representation of $SU(N)$, there is a Fierz identity: $$ \sum_iT^i_{ab}T^i_{cd}=\frac{1}{2}\left(\delta_{ad}\delta_{bc}-\frac{1}{N}\delta_{ab}\delta_{cd}\right) $$ where $T^i$ is ...
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1answer
83 views

Parametrizing $SU(2)$ with Hermitian matrices

There is something that is not clear to me Here is what I know: Pauli matrices are $\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$, $\sigma_2 = \begin{pmatrix}0 & -i \\ i & 0\...
6
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1answer
99 views

If Poisson Bracket of Momentum and Position is non-zero, why no Uncertainty Principle?

In Hamiltonian classical mechanics, we have that the Poisson bracket of position and momentum satisfies $$\{q_i, p_j\} = \delta_{ij}$$ But this implies that momentum and position 'generate' changes ...
8
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1answer
461 views

Is $SU(2)\times U(1) = U(2)$?

In the many textbook of the Standard Model, I encounter the relation \begin{align} SU(2)_L \times U(1)_L = U(2)_L. \end{align} Here the subscript $L$ means the left-handness (i.e., the chirality of ...
0
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0answers
28 views

Generators of 2D global conformal group in terms of differential operators?

I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$....
7
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1answer
278 views

Topological/Geometrical justification for $\text{CFT}_2$ being special

It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n>2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are ...
5
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2answers
169 views

What is a Borel subalgebra?

Borel subalgebra appears here https://arxiv.org/abs/hep-th/9508170 in the context of quantum double of $SU(2)$. I request a layman explanation of what a Borel subalgebra is.
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4answers
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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 ...
1
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1answer
177 views

How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate ...
0
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1answer
152 views

Does invariance under infinite small transformation imply invariance to the finite one?

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it. The chiral transform is defined as: $$\psi(x) \rightarrow \psi'(x) =e^{i \...
0
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1answer
46 views

A simple calculation in Peskin's and Schroeder's QFT book on page 608 chapter 18

I am trying to calculate the term: $$(t^a)_{ij} (t^a)_{kl}$$ In the book it's written that it equals to $$A\delta_{il}\delta_{kj}+B\delta_{ij}\delta_{kl}$$ and from using equation (18.40) $$tr[t^a](...
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0answers
30 views

Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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0answers
48 views

Why use $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? [duplicate]

Why do we use the group $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? As far as I can tell, the $SL(3, \mathbb{R})$ is volume and orientation preserving, by the fact that it has unit ...
0
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0answers
42 views

Regarding notation used for infintesimal parameters of the Lorentz algebra and generators of the Lorentz group

I have a confusion regarding the notation that is used for infintesimal Lorentz transformations and the parameters that define the Lorentz transformation (used in various books such as Srednicki's and ...
2
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1answer
213 views

Levi-Civita tensor and the Lorentz group generators in the vector representation

In the vector representation of the Lorentz group its generators are given by - $$(J^{\mu\nu})_{\alpha\beta} = i(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha)$$ It can be ...
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3answers
89 views

Why the full conformal symmetry is $Vir\otimes \overline{Vir}$ instead of $Vir\oplus \overline{Vir}$

In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. ...
0
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0answers
19 views

Physics-y resource request on Killing-Cartan forms

Most books that treat nonabelian gauge theory do not contain detailed discussion on Killing-Cartan forms, they'll usually just say that in $\text{SU}(N)$ Yang-Mills theory, one can choose generators $...
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2answers
66 views

$3+3$ representation of $SO(4)$

In Zee's Group Theory in a Nutshell book, he says that the antisymmetric tensor $A^{ij}$ furnishes a 6 dimensional representation of $SO(4)$. He further argues that this 6 dimensional representation ...
9
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1answer
939 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 ...
2
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1answer
102 views

Why is $\rm{Conf}(\mathbb{R}^{1,1}) = \rm{Diff}(S^1) \times \rm{Diff}(S^1)$ and not $ \rm{Diff}(\mathbb{R}) \times \rm{Diff}(\mathbb{R})$?

The Minkowski metric for $\mathbb{R}^{1,1}$ is $$ ds^2 = dt^2 - dx^2 = du dv $$ for coordinates $$ u = t + x \hspace{1cm} v = t - x $$ If you do any coordinate transformation that acts independently ...
3
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2answers
124 views

Global conformal group in 2D Euclidean space

This is a rather naive question, but I was just wondering. I know that the local conformal algebra of 2d Euclidean space is the direct sum \begin{equation} \cal{L}_0\oplus\overline{\cal{L}_0}, \end{...
0
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0answers
93 views

Virasoro algebra commutation (part 2)

This was a sub-question in my previous post that I ask separately now. In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the ...
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2answers
36 views

Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
0
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1answer
78 views

Show that when angular momentum $L_x$ and $L_y$ commute with operator $G$, then $L_z$ also commutes with $G$

I want to prove that if Angular momentum $L_x$ and $L_y$ commute with an operator $G$, angular momentum $L_z$ also commutes with $G$. if $[L_x , G] = [L_y, G] = 0$ then $[L_z , G] = 0$ I know that $...
0
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0answers
27 views

Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is Calculate $[J_{(mn)}, J_{(pq)}]$. We are given that $[J_{(...
1
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1answer
96 views

Virasoro Algebra commutation

In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the central extension of the Witt algebra. They give the central extension $$\...
0
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2answers
26 views

Angular velocity is $\dot{g}$ carried to the identity element of the group

I was reading the example below from Arnolds book I can't really understand why the angular velocity is $\dot{g}$ carried to the identity element of the group. I would appreciate if someone who ...
0
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1answer
24 views

Anticommutativity of an anticommutator of supercharges

In this paper, equation 38 gives the ${\cal N}=2$ Super-Poincare (extended with the central extension $\mathcal{Z}$). The anticommutation relation of the two different supercharges is given as: $$\{Q^...
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1answer
65 views

Representation and Lie algebra of $SO(3)$

Studyng the book Group Theory in Physics of Wu-Ki Tung, I have read: "... every representation of the [$SO(3)$] group is automatically a representation of the corresponding Lie algebra, (...) a ...
0
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1answer
34 views

Lie Subgroups of $SL(2,\mathbb{R})$

I'm wondering about the Lie subgroups of $SL(2,\mathbb{R})$. It's Lie algebra is the algebra of real traceless matrices and has basis elements $$L_0 = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \...
0
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0answers
27 views

Legal values of spin-1 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

For the spin-1/ boson field $A_\mu$, we may choose it to be a vector which needs to be real $\mathbb{R}$ usually for photon field. The field strength $F= dA$ is also real. Same for the nonabelian ...
5
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1answer
104 views

Why is supersymmetry a continuous symmetry?

Supersymmetry feels like a discrete symmetry to me, since the fermions are turning into bosons, and vice versa. I understand there is an infinitesimal parameter involved in the transformations, but I ...
3
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2answers
1k views

Commutation relations of the generators of the Lorentz group

$$ J^{\mu\nu} = i(x^\mu\partial^\nu-x^\nu\partial^\mu). \tag{3.16}$$ We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group. To determine ...
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0answers
44 views

Commutation relation Lorentz Algebra

Related question, which I don't understand either. I think is easier to get the Lorentz group algebra as defined by Maggiore, $$ [J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} - \eta^{\...
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1answer
65 views

Representations of the rotation group

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can ...
0
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1answer
63 views

Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
0
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1answer
90 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
0
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2answers
55 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
2
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1answer
96 views

Factorization of exponential of broken generators in parametrization of scalar multiplet in non-abelian SSB

Describing abelian symmetry breaking in his book on gauge theories, after favouring a vacuum (whose expectation value is $v$) from the symmetric continuum, Quigg parametrize the complex scalar as $$ ...
3
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1answer
309 views

Hamiltonian symmetry Lie algebra

What is the connection between complete set of commuting observables and generators of the Lie group? I have a Hamiltonian written down in second quantized formalism and I also checked that it ...
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3answers
78 views

What does $X|n\rangle \propto |n+c\rangle$ mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right\rangle}$ I'm transcribing below (but see edit history for a scan) a calculation from pg 17 of this article on Lie groups and Lie algebras $$ [N,X]=cX....
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0answers
31 views

Is it always possible to move to the “Cartan Gauge”?

Forgive me for potentially coming up with a new name for what I am about to describe. Let's say we have a scalar field $\phi^a$ which transforms with respect to the adjoint representation of some Lie ...
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1answer
43 views

Gauge group of Electroweak theory

I am doing a question that asks me to identify the gauge groups of a Lagrangian with the field strength tensors $$\bf{F}_{\mu \nu} = \partial_{\mu}\bf{W}_{\nu} - \partial_{\nu} \bf{W}_{\mu} - g\bf{...
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 ...
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0answers
46 views

Error with generators of Lorentz group (basis of Lorentz Lie algebra) [closed]

Can someone help me figure out why my $J_y$ is incorrect? :/ It's supposed to be \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & -...
1
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2answers
83 views

Symplectic group $Sp(2N)$ in Srednicki's book

There is a question in Mark Srednicki's Book (Problem 24.4, p.160) about $Sp(2N)$, but I am not sure I understand the significance (application?) of this group. In that chapter, he talks about $SO(N)$ ...
6
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4answers
234 views

Why in QFT what really matters is $\exp(\mathfrak{so}(1,3))$ instead of $O(1,3)$?

In QFT fields are classified according to representations of the Lorentz group $O(1,3)$. Now, most books when getting into this say that in order to understand the representations of $O(1,3)$ we need ...