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2 votes
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$SU(3)$ adjoint representation and irreducibility

The question fails to interpret its own definition correctly: It is correct that every vector $v$ in an irreducible representation is cyclic, which is the technical term for the span of the orbit ...
ACuriousMind's user avatar
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$SU(3)$ adjoint representation and irreducibility

So, the perhaps underwhelming answer is that the action of a group on an irreducible representation need not be transitive. As a trivial example, consider the action of $SU(2)$ on $v=|\uparrow\rangle|\...
TLDR's user avatar
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3 votes
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Unitary Representation of $\text{SO}(3)$ in Position Representation

The eigenvector $|x\rangle$ of the position operator $\hat X$ associated to the eigenvalue $x$ being defined by $$\hat X|x\rangle=x|x\rangle$$ the defining property of $U_R$ gives $$U_R^{-1}\hat XU_R|...
Christophe's user avatar
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5 votes
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$6j$-symbol example in Quantum Mechanics

It is a shame that most articles about $6j$-symbols (including Wikipedia) fail to mention for what they are actually used. They are used to transform between different coupling schemes in systems with ...
Thomas Fritsch's user avatar
1 vote

Character Table and Binary Basis

The final column is obtained by representing the abstract group as an isometry subgroup. Each set of quadratic polynomials (separated by a semicolon) span an irreducible representation whose type is ...
LPZ's user avatar
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0 votes

Wigner-Eckart theorem: Completeness relation

Your notation suggests that $j$ and $m$ index the rotation group representation vectors, which are of course complete. And for $\alpha$ you need to have just one value to apply the theorem, so whether ...
Jos Bergervoet's user avatar
2 votes
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Addition of angular momenta with relative coefficients

Well, assuming these represent rotation of a tensor product space, so operators of different subscripts commute, and they satisfy the rotation group Lie algebra, $$ [J^a_i,J^b_j]=i\epsilon^{abc} J^c_i ...
Cosmas Zachos's user avatar
4 votes
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Projector onto Adjoint and Singlet Representations for $SU(N)$

Indeed. Review for su(2), in your notation, where $\vec S$ are the normalized 3-vector generators, so $\vec \sigma /2$ for the fundamental and anti fundamental, so you have $$ \vec{S}_1\cdot \vec{S}_2=...
Cosmas Zachos's user avatar
0 votes

How do we know the creation and annihilation operators for angular momentum give rise to a complete basis?

The irreducible representations of the Lie algebra of $\rm SU(2)$ (defined by $[T_k,T_\ell]= i \varepsilon_{k \ell m}T_m$) can be classified by the "weights" $j=0,\, 1/2,\, 1,\, 3/2, \,2, \...
Hyperon's user avatar
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0 votes

Columns, rows, dotted, undotted, $SL(2, \mathbb{C})$ reps, and building Dirac spinors from Weyl spinors

First of all, I don't know the book to which you refer to. By the way in both block of formulas the third row does not follow correctly the index rules, since the index $\dot{B}$ appears twice up ...
Frederic Thomas's user avatar
2 votes

Degrees of freedom in $(A,B)$ representation of the Lorentz group

Now since this is a direct sum of a $2A+1$ dimensional representation with a $2B+1$ dimensional representation, the matrix representation of $(A,B)$ is $2A+2B+2$ dimensional, which is what I would ...
Cosmas Zachos's user avatar
0 votes

Real representation of smallest dimension of Clifford Algebra with $d$ generators

You need to put two things together here: First, the classification of the complex irreps of the Clifford algebra, and then the question when these have a real structure. The unique complex ...
ACuriousMind's user avatar
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1 vote
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Unitary representation of physics groups in Wu-Ki Tung book

Indeed, the author illustrates by explicit construction that, for noncompact groups (contrast to what you have learned in SO(3) !), their unitary representations (Hermitian irreps for the generators) ...
Cosmas Zachos's user avatar
0 votes

Algebraic definition of ground state

Perhaps a concrete example would help. Consider the $*$-algebra of complex $2\times 2$ matrices $\mathscr A:= \mathrm{Mat}_{2\times 2}(\mathbb C)$, with the star operation given by the conjugate ...
J. Murray's user avatar
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-2 votes

Algebraic definition of ground state

A bound state has negative eigenvalue with potential term, but a strict positive expectation in the free algebra of positive observables, so its not the ground state of the free Heisenberg algebra of ...
Roland F's user avatar
1 vote

In what sense are fields representations of the Poincare group?

Fields are reducible representations (by which I mean, fields are elements of the vector space on which a reducible representation of the Poincar'e group acts). On the other hand, particles are ...
Prahar's user avatar
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