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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Classical limit and generalized coherent states

In quantum optics coherent states introduced by Glauber have a localized probability distribution in classical phase-space with maximum following classical equations of motions. This is not a ...
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20 views

Casimir operators of the Poincare Group

Say, $P^2$ and $W^2$ are the Casimir operators of the Poincare Group. Should the commutator of these Casimir operators be zero, because then they would be independent of each other and form a unique ...
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28 views

Why are the azimuthal quantum number $\ell$ and magnetic quantum number $m$ integers? [duplicate]

Why are the azimuthal quantum number $\ell$ and magnetic quantum number $m$ integers?
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39 views

The Lie algebra of the Lorentz group is $su(2) \oplus su(2)$. Is there a similar relation for the algebra of the Poincare group?

It can be shown easily, by introducing new generators from the usual ones that we can think of the Lie algebra of the Lorentz group as being built up by two copies of the $SU(2)$ Lie algebra: $$ ...
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1answer
63 views

Where does in GUT symmetry breaking $U(1)$ come from?

In GUTs one starts with some larger group, like $SU(5)$, which is then broken into smaller groups, for example $$SU(5) ~\longrightarrow~ SU(3) \times SU(2) \times U(1)$$ This can be seen, for ...
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43 views

Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$? [migrated]

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
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1answer
27 views

Dimension and Basis properties of $SU(N)$

$SU(N)$ is the group of special unitary matrices of dimension $N$, i.e., the set of all unitary ($U^{\dagger}U=I$) $N\times N$ matrices with $\det(U)=1$. For $N=2$, these matrices are spanned by the ...
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27 views

How to write $su(3)$ Lie algebra as a sum of two subspaces? [migrated]

Let $K,F\subset\mathfrak{su}(3)$ be subspaces, such that $K \oplus F =\mathfrak{su}(3)$, and $K$ has a $\mathfrak{su}(2)$ structure. How can we show that $[K,K] = K$ (i.e., commutator of any two ...
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23 views

How to expand free energy of Heisenberg spin chain?

In Dasgupta & Ma's 1979 paper "Low-temperature properties of the random Heisenberg antiferromagnetic chain", they give the free energy of a few interacting Heisenberg spins on a chain. I can't ...
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0answers
43 views

About generator of $SU(2)$ flavor symmetry group

I am reading the textbook, "weak interactions" by Howard Georgi which can be founded in his homepage. Here i am trying to solve the problem 1b-2. The problem is given as follows. Consider the ...
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1answer
41 views

How to derive the form of the invariant spinor inner product?

So we have gamma matrices that satisfy the spacetime algebra relations, $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu}$. We know that if we set $\sigma^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ ...
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1answer
91 views

What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$?

I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. To me this implies ...
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1answer
64 views

Why is orbital angular momentum quantized according to $I=\frac h{2 \pi} \sqrt{l(l+1)}$?

I simply have no idea how this result is found $$I=\frac h{2 \pi} \sqrt{l(l+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that ...
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0answers
31 views

Name for the transformation into an accelerated frame?

A transformation into a frame that looks at an experiment from a rotated perspective is called a rotation. A transformation into a frame that moves with a different constant velocity is called a ...
2
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1answer
34 views

Coset construction of Tricritical Ising CFT

In http://iopscience.iop.org/1742-5468/2008/03/P03010 the authors state that the Tricritical Ising Model (TIM) CFT can be obtained from a Wess Zumino Witten construction based in the coset ...
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2answers
208 views

Why are the spin operators defined as they are?

$$\begin{align*}S_z &= \frac{\hbar}{2} \left(\left|+\right>\left<+\right| - \left|-\right>\left<-\right|\right)\\ S_y &= i\frac{\hbar}{2} \left(\left|-\right>\left<+\right| - ...
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1answer
83 views

About $SU(2)_L \times U(1)_L = U(2)_L $

In the many textbook of standard model, i encounter the relation \begin{align} SU(2)_L \times U(1)_L ~=~ U(2)_L. \end{align} Here $L$ means the left-handness. (It is a physical ...
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1answer
43 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 ...
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2answers
57 views

Incorrect proof that all gauge theories are abelian

Consider a gauge field $W_\mu = W_\mu^{a} \tau_a$ where $\tau_a$ are the generators of the Lie algebra and $W_\mu^{a}$ just numbers. Then: $$ W^2 = W_\mu W^\mu = W_\mu^a\tau_a W^{\mu b} \tau_b = ...
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1answer
60 views

Question about angular momentum operator

To show that the eigenvalue to $L^2$ is proportional to $\hbar^2$ is shown from $L_z=xP_y-yP_x$ $p_y=-i\hbar\frac{\partial}{\partial y}$ $p_x=-i\hbar\frac{\partial}{\partial x}$ ...
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40 views

Uses of the accidental isomorphism $SO(5)\sim Sp(2)$?

Some of the accidental isomorphisms of low dimensional Lie algebras have very important applications in physics. The theory of angular momentum makes use of the fact that $SO(3)\sim SU(2)$. ...
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1answer
53 views

$SU(2)$ generators and creation annihilation operators

The algebraic method to find the irreducible representation of the $SU(2)$ group makes use of the operators: $$J_z\\J_+=\frac{1}{\sqrt{2}}(J_x+iJ_y)\\J_-=\frac{1}{\sqrt{2}}(J_x-iJ_y)$$ In the book ...
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1answer
37 views

What exactly are the ADE type of gauge theories?

What exactly are the ADE type of (susy) gauge theories? What exactly we mean, intuitively, the ADE singularities? What are their relation to brane constructions and do you have any references one ...
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1answer
108 views

Complete derivation of generator of rotations

I have been look all across the internet and every book I could find trying to get a full derivation of the generator of rotations and more specifically angular momentum as a generator of rotations. I ...
2
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2answers
96 views

Half-integer spin and infinitesimal rotations

On p. 692 of 'Quantum Mechanics' by Cohen-Tannoudji, he states that: Every finite rotation can be decomposed into an infinite number of infinitesimal rotations, since the angle of rotation can ...
3
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1answer
753 views

Proof of weaker Baker-Campbell-Hausdorff Formula [duplicate]

Prove the weaker form of the BCH Formula: $$e^A e^B = e^{A + B + \frac{1}{2}[A,B]} $$ with the assumption $[A, [B, A]] = 0; [B, [B,A]] = 0$ Start with $f(\lambda) = e^{\lambda A} e^{\lambda B} ...
2
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0answers
49 views

Finding Casimir operators for the Poincare group $ISO(1,2)$

I was asked to write the generators for translations and Lorentz-transforms in 1+2 dimensions and then to find the Casimir operators. For the generators I can take the same ones as in 1+3 case ...
3
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2answers
246 views

An identity of Pauli matrices

I am studying spin recently, and textbook gives some identities of Pauli matrices, one said that for any two unit vectors $\bf m$ and $\bf n$, $[\bf m \cdot \bf{\sigma},\bf {n \cdot \sigma}]= ...
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1answer
59 views

Is the fundamental representation of $SU(3)$ irreducible?

I want to check if the fundamental representation of $SU(3)$ is irreducible. The algebra is $$\mathbb{su}(3) = \{ m \in Mat(3,\mathbb{C} )\ |\ m = -m^+,\ Tr[m] = 0 \}$$ and I've found the generators. ...
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55 views

How would I relate $\Lambda=e^{-i\omega_{\mu\nu}J^{\mu\nu}/2}$ to the Lorentz boost matrix?

$\omega_{\mu\nu}$ contains infinitesimal parameters and $J^{\mu\nu}$ contains generators of boost and rotation. Any 4-vector transforms as $p^\mu=\Lambda^\mu_\nu p^\nu$. Starting from given ...
5
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1answer
191 views

Noether's Theorem: Lie algebra, Lie groups

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
2
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1answer
91 views

Schrodinger equation, commutative operators, and Symmetry

When solving Schrodinger's equation in 3D with a spherical laplacian you reach a point at which you introduce a separation constant and can see that the same eigenvalue satisfies the radial and ...
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1answer
78 views

What is a su(2) level k algebra

What is meant by su(2) level k algebra ? Is it a lie algebra of some lie group ? What is the relation with SU(2) group. I see it in the context of quantum hall edges. Googling and google-booking for a ...
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1answer
24 views

Why is it that every locally conformal transformation can be extended to a global conformal transformation for D>2?

In D=2, we can have locally analytic transformations that cannot be globally well-defined. However, for CFTs in D>2, we have only the global group. Why is that? Also, is it a statement that depends ...
2
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1answer
59 views

Does $GL(N,\mathbb{R})$ own spinor representation? Which group is its covering group? (Kaku's QFT textbook)

In Kaku's QFT textbook page 54, there is a saying: $GL(N)$ does not have any finite-dimensional spinorial representation. This implicates that $GL(N)$ owns infinite-dimensional spinorial ...
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2answers
92 views

What's the degree of freedom of this kind of matrix?

We first have a unitary matrix $$\{a_{ij}\}\quad(n\times n)$$ I know how to calculate its degree of freedom, which is $n^2$ if we consider a real variable as one degree of freedom. Now we have a ...
3
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1answer
93 views

Subgroup of Lorentz Group Generated by Boosts

It is common knowledge that a composition of boosts is not a boost, but involves a rotation. Further, in discussions of Thomas precession, it is often stated that boosting in $x$, then $y$, then back ...
3
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1answer
68 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...
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1answer
47 views

What is the point of defining the lie algebra of the proper Lorentz group in a “covariant” way?

In Muller-Kirsten's book Introduction to Supersymmetry, the author first defines the proper Lorentz group's lie algebra basis in the standard manner - antisymmetric matrices consisting of $0$s and ...
4
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0answers
56 views

Decomposing a representation under a subgroup [closed]

I am trying to understand what is the method for decomposing representations of a group under one of its subgroups. I already had a look in Slansky, but I could not extract a concrete set of ...
5
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1answer
130 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
2
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1answer
71 views

How does the Lorentz group act on a 4-vector in the spinor-helicity formalism $p_{\alpha\dot{\alpha}}$?

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I ...
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0answers
24 views

Reference request: Relation between $Sp(N)$, $Spin(N) $, $SU(N)$ groups and physics [duplicate]

I want to understand the relationship of the so common $SU(N)$ and $SO(N)$ groups in physics with the symplectic group which I think is the double cover of the first and the Spin groups $Spin(N)$. Is ...
2
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1answer
65 views

Representations of subalgebra in the super virasoro algebra

In the Virasoro algebra, which is generated by $L_n$, one has the obvious subalgebra spanned by $L_{-1}$ ,$L_{1}$ and $L_{0}$ which is isomorphic to the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$. The ...
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1answer
119 views

Is there a connection between Lie Groups and observable quantities in physics?

Good evening everybody. I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...
3
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2answers
197 views

Lorentz algebra and its generators

I'm reading Maggiore's book A Modern Introduction to Quantum Field Theory and I'm getting a bit confused when he writes about Lorentz algebra: $$K^i = J^{i0},$$ ...
2
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1answer
74 views

Unitary representations of $SO(3)$ and $so(3)$

According to my skript: Quantum mechanic states $ψ ∈ \mathcal H$ changes under a rotation $R ∈ \text{SO(3)}, \vec{x} \rightarrow R\vec{x}$ according to $ψ \rightarrow U(R)ψ$, whereas $U(R)$ is a ...
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1answer
29 views

Why does non-abelianity implies a single coupling constant?

Why does a theory described by a non-abelian group has only a single coupling constant $g$? While on the other hand in an abelian theory, as Electromagnetism, each charged particle has its own charge ...
3
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3answers
408 views

In physics, what is the importance of distinguishing between a matrix and a group? [closed]

On the topic of Pauli matrices, I have noticed that some authors tend to use the term matrix and group interchangeably. I am asking because I do not see see any profound difference referring to the ...
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0answers
51 views

Proving a relation for representations of gauge groups [closed]

I have asked this question in Math Stackexchange as well. However, given that it is closely related to gauge theories studied by physicists who will probably be more familiar with the language and ...