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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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Physically, why don't we care about representations that differ only by a similarity transformation?

I was looking at how to derive the (1/2, 0) representation of the Lorentz group when acting on fields. Specifically, I'm interested in understanding the logic behind replacing the "symbols" $A,B$ with ...
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Why isn't $SO(n)/SO(n\!-\!1)$ a symmetric space?

It's my understanding that one way to define a symmetric space $G/H$ is by the commutation relations $$ [T^a, T^b] = f^{abc} T^c, \qquad [T^a, X^{\hat{b}}] = f^{a\hat{b}\hat{c}}X^{\hat{c}}, \qquad [X^{...
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Commutator of Lie Group Generators

This is from Maggiore's "A Modern Introduction to Field Theory", Page 15. I have a Lie group with matrix generators $$ T^{a}_{R}$$ Where $a$ takes values from 1 to the dimension of the Lie group. ...
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Half Witt algebra

I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule $$ \Big[ L_i ,L_j \Big]=\frac18 \frac{(2i+2j-1)(2j-2i)}{(2j+1)(2i+1)}L_{i+j-1}-\...
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Measuring the lorentz transform generators $J$, $K$, and providing evidence that photons have no internal continuous d.o.f

I am reading Weinberg's first QFT book. We looked for (and I suppose found) unitary representations of the Lorentz group: $$U(\Lambda) = 1 - i (\vec{\theta}\cdot\vec{J}-\vec{\eta}\cdot \vec{K})$$ ...
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Generator of 3D rotations in $\mathbb{C}^2 \otimes \mathbb{C}^2$

Let us consider a system of two spinors. The 3D rotation operator around the $\vec{n}$ axis in $\mathbb{C}^2$ is clearly $R(\theta) = \exp(i \frac{\theta}{2}\vec{n}\cdot\vec{\sigma})$. If I wish to ...
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Symmetry of the Batalin-Vilkovisky (BV) antibracket operation

Batalin and Vilkovisky define $^1$ an operation they call antibracket which is $$(F,H) = \Big(\frac{\partial_r F}{\partial \Phi^A}\Big) \Big(\frac{\partial_l H}{\partial \Phi^* _A} \Big) - \Big(\frac{...
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Prove that the Lorentz algebra $o(3,1)$ is simple / doesn't have invariant subalgebra [migrated]

I know how to prove that $o(4)$ is not a simple Lie algebra, which is finding explicitly two subalgebras that commute with each other. I can show for the Lorentz group that the analogous combination ...
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Relation Between Cross Product and Infinitesimal Rotations, Generators, Etc [duplicate]

Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group $SO(3)$. For example: $$\vec{\mathbf{...
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Books, papers, etc on Lorentz and Poincare groups/algebras/etc

I'm currently trying to learn more about the Lorentz- and Poincare Lie-algebras and the representation theory about them. But I'm really struggling with the material that we were given and I'm also ...
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Poincaré invariance of linearly polarised plane wave

I am reading a book that just quotes the Lie group generators and the discrete subgroups that leave a linearly polarised plane wave unchanged. And I have no idea how to derive them. Context The ...
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Products of Lie-Groups versus Lie-Group Extensions in Physics

The Standard Model of elementary particle physics is a gauge theory based on the Lie group $U(1) \times SU(2) \times SU(3)$. From the mathematical perspective I read that: Simple Lie groups have ...
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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 ...
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Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
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Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
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Is there an anticommutator relation for orbital angular momentum?

So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
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Bosonic representation of $SU(N)$: what values can $n_b$ take?

In Assa Auerbach's book on page 166, he describes the construction of a bosonic representation of $SU(N)$ where the generators $S^{mn} \rightarrow b^\dagger_m b_n$. I'm a bit confused about the ...
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About $(0,1/2)$ representations

While studying representations of Lorentz group, we get the generators to be $J_{i}$ - rotations and $K_{i}$ - boosts. We define $N_{i}^+$ and $N_{i}^-$ operators and these operators obey the same ...
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Which are the underlying Lie group and algebra related to the translation invariance in field theories?

I'm new to Physics SE. I've seen a lot of interesting questions and answers, and thought it will be very useful to participate a little. I'm currently stuck in a, probably, very simple matter, ...
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Role of the special linear Lie algebra in general relativity (GR)

The Lie derivative measures the difference between two paths in the timespace manifold, and hence the commutator bracket occurs naturally, as explained in the presentation What is a Tensor? Lesson 21: ...
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Isometry group on a coset manifold

In ''Einstein Gravity in a Nutshell'' Zee says ''On a coset manifold $G/H$, the isometry group is evidently just $G$'' when discussing the relation between the Killing vector fields and Lie ...
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Generators in Field Theory and Derivatives

Let's consider a representation of the multiplicative group $(0,\infty)$ on Minkowski space $\mathbb{R}^4$ by dilations. \begin{align} \rho:(0,\infty)&\rightarrow\text{GL}(\mathbb{R}^4)&\\ a ...
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Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-...
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Commutator generating transformations

Lately I am encountering the commutator of variations of the variables and I'm not quite sure about its physical meaning. Some examples. 1) "The composition of two supersymmetries generates a time ...
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Calculating adjoint representation of Lie group/algebra [duplicate]

How do I calculate adjoint representation of Lie group and Lie algebra? I would be thankful if anyone can give good example or general formula on calculating adjoint of any Lie group
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Decomposition of the complex conjugate of the fundamental representation of $SU(5)$ in $SU(3)\times SU(2)\times U(1)$

I know I can decompose the fundamental representation (denoted as $5$) of $SU(5)$ as: $$ (3,1)_{-2c/3} \oplus (1,2)_{c} $$ But how do I get the decomposition of the complex conjugate of this ...
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Is there a difference of sign conventions of Dirac Index between mathematics and physics?

In section 12.6.2 of Nakahara, on a four dimensional manifold, the index of a twisted Dirac operator is given by $$\mathrm{Ind}(D\!\!\!\!/_{A})=\frac{-1}{8\pi^{2}}\int_{M}\mathrm{Tr}(F\wedge F)+\frac{...
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Questions on how Wilson loops relate to field & charge conservation, and lattice QFT

The path-ordered exponential from which the Wilson loop is traced is, crudely, $$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$ which returns a matrix $\...
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Another way to write the Einstein-Hilbert action?

Let's take a look at the equation for the Riemann tensor in terms of an arbitrary 1-form: $$\nabla_{\mu}\nabla_{\nu}A_{\alpha}-\nabla_{\nu}\nabla_{\mu}A_{\alpha}=R_{\mu\nu\alpha}^{\quad\:\delta}A_{\...
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$SU\left(N\right)$ Dynkin labels, how to compute

Let $V$ be somecomplex irreducible representation of $SU\left(N\right)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of ...
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Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
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Weights of $SU\left(5\right)$ representation

Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights? Are these the correct Dynkin labels ...
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Why do we use infinitesimal forms of operators?

In many undergraduate texts on quantum mechanics (I'm using Modern Quantum Mechanics 2nd Edition by Sakurai as reference here), the treatment of angular momentum goes something along the lines of: ...
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How do the properties of a Lie group (represented as a manifold) manifest in the metric tensor of that manifold?

I know this is a math question; however, physicists are more likely to be familiar with what I'm asking (also, I'm directly trying to utilize it in the context of general relativity). I may have ...
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1answer
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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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Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
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Commutator of Gauge Transformations for Yang-Mills Theory

Following the conventions of "Quantum Field Theory and the Standard Model" by Schwartz, we have that for Yang-Mills Theory, an infinitesimal gauge transformation acts like $$\delta_{\alpha} A = d\...
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Physical meaning of the Casimir operators of Poincarè algebra

If one considers the algebra $su(2)$, it is well known that the Casimir Operator is $$ C=L_1^2+L_2^2+L_3^2. $$ It corresponds to the total angular momentum and correctly is a conserved quantity. ...
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Symmetric tensor product decomposition of $su(2)$

Taking the tensor product of two spin 1 representations of $su(2)$ yields $$1 \otimes 1 = 0 \oplus 1 \oplus 2.$$ What changes if instead we take the symmetric tensor product $1 \odot 1$ of these ...
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Books about Group Theory [duplicate]

I was looking for a book to complement the lecture notes of the course for a more intuitive approach to the subject and full of examples (mathematical), because the handouts seem only a bunch of ...
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Racah's Derivation of CG coefficients

I have what is hopefully a quick question. In Racah's 1942 paper Theory of Complex Spectra II, the author utilizes the action of $J_+$ on the Clebsch Gordan expression: $$ \begin{align*} &|j m\...
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1answer
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Why is the $\mathrm{SU}(2)$ algebra taken over the complex field?

The lie algebra of $\mathrm{SU}(n)$ is composed by the $n \times n$ antihermitian matrix with null trace over the real field, but physicists prefer to use hermitian matrix. Does this mean taking the ...
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Why we need $SU(2)$ symmetry? When we use it? [closed]

I am trying to learn Quantum mechanics and I am familiar with Pauli matrice but not with group theory. I want to understand SU2 symmetry in common language. When we talk about Pauli matrix x we simply ...
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Notation question for exponential form of Lorentz transformations

So my question is, when we write the Lorentz transformation in the following form \begin{equation} \Lambda = e^{- \frac{\mathrm{i}}{2} \omega ^{\rho \sigma} M_{\rho \sigma}} \end{equation} Since the ...
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Relation between Lie algebra of conformal Killing vector fields and conformal algebra

I'm new to conformal transformations and I have a question. Following the book of Barrett O'Neill "Semi-Riemannian geometry with applications to relativity", there is a Lie anti-isomorphism between ...
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$\mathfrak{su}(3)$ structure constants

The $\mathfrak{su}(3)$ structure constants $f^{abc}$ are defined by $$[T^a,T^b] = i f^{abc} T^c,$$ with $T^a$ being the generators of the group $\mathrm{SU}(3)$. They are usually written out in a very ...
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How to interpret the extra indices of the generators of the Lorentz algebra in Peskin & Schroeder?

In Peskin & Schroeder p.39 they introduce the 4x4-matrices $$\left(\mathcal{J}^{\mu\nu}\right)_{\alpha\beta} = i \left(\delta^{\mu}_{\;\alpha} \delta^{\nu}_{\;\beta} - \delta^{\mu}_{\;\beta}\...
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How to prove commutation relation between charge and current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Proceeding equation 5.54, there is a statement which says Then from Lorentz covariance, we can include the ...
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Gauge transformations and Covariant derivatives commute

I would like to understand the statement "Gauge transformations and Covariant derivatives commute on fields on which the algebra is closed off-shell" which was taken from section 11.2.1 (page 223)...
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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{...