# 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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### How to find all the roots of a Lie Algebra starting from Cartan marix, following Georgi's book?

I am reading chapter 8 of Georgi's book, in particular sections 8.8-8.11, where he shows a diagrammatic procedure to find all the roots starting from simple roots. I understand that at each step, we ...
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### What do quantum spin hamiltonians describe?

I've learned all particles are either fermions or bosons, obeying their respective operator algebras, and then I've seen Hamiltonians describing models carrying one of these two types of particles. So ...
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### How to show that two operators in the forms of vectors commute?

This is an exercise from Peskin&Schroeder's book. The exercise requires to show that $\textbf{J+}$ and $\textbf{J-}$ commute with each other. What is the exact meaning of commutation between ...
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### 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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### 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 ...
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### How to prove that given operators form an algebra? [closed]

I am new to Group theory and representations and I'm having trouble with this problem in an exercise: Given the two oscillator algebra $$[a, a^†] = 1$$ $$[b, b^†] = 1$$ $$[a, b] = 0$$ show that ...
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### Difficulty understanding Srednicki's derivation of the Lie Algebra Commutators of the Lorentz Group

I have just started reading through Srednicki's QFT in preparation for a few courses I am about to take. I have taken courses that have covered the basics of special relativity including ...
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### Evaluating $n \otimes_A n^*$ in $SU(n)$

In "Quantum Field Theory in a Nutshell" pg424 the author (Zee) writes: $$(n\oplus n^*)\otimes_A(n \oplus n^*)\quad\cong\quad(n^2-1)\oplus 1 \oplus n(n-1)/2 \oplus ((n(n-1))/2)^*$$ From what I ...
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### Infinitesimal transformation

I came across this statement in the book "Quantum Field theory and the Standard Model" by Schwartz. "We would now like to find all the representations of the Lorentz group. The Lorentz group ...
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The quadratic conformal Casimir in $d$-dimensional Euclidean space is given by C = \frac{1}{2}L_{\mu \nu}L^{\mu \nu} - D^2 -\frac{1}{2}\left(P^\mu K_\mu + K^\mu P_\mu \right) \end{...
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### On the homomorphism of the Lorentz algebra representation (1/2, 0)

I was reading this answer and I don't quite understand how the $\rho$ homomorphism works. The generators of the two copies of $\mathfrak{su}(2)$ in $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ are given ...
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### Show that the fundamental representation is a representation

I want to see that the fundamental representation is a representation. Suppose the structure constants $f^{abc}$ are given. We can assume there is at least one non-zero structure constant, otherwise ...
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### What are the relations between a ladder algebra and an $SU(2)$ algebra?

I am studying elementary models of the Quantum Hall Effect. I don't have a strong background in Lie algebras so I was hoping someone could elaborate on the following observation: For a free particle ...
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### Representation of the Lorentz group from representation of its Lie algebra [duplicate]

In many books on particle physics, it is first shown that the Lie algebra of the Lorentz group is isomorphic to $$su(2) \oplus su(2) ,$$ then by this fact, it is implicitly assumed that for each ...
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### Representation Theory in a Nutshell for Physicists? [duplicate]

Are there good references and introduction of Representation Theory as "Representation Theory in a Nutshell for Physicists"? For example, we hope that the book/ref contains the "introduction to ...
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### How do WZW coset models contain perturbations?

I've been studying the coset construction. As far as I understand it, the Sugawara energy momentum tensor is a way of embedding the virasoro algebra inside the Lie algebra of your original WZW model. ...
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### Symmetry of the Pauli group and its representations

The three traceless Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ are arbitrary in the sense that any three operators with the appropriate commutation relations can be represented with those matrices. ...