The representation-theory tag has no wiki summary.
10
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2answers
130 views
When are there enough Casimirs?
I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators ...
1
vote
2answers
111 views
Doubts concerning Wigner's classification
Wigner classified particles in function of the eigenvalues of $P_\mu P^\mu$ and $W_\mu W^\mu$. Then, it can be proved that for massless particles spin values can be only $\pm s_{max}$. But for a ...
3
votes
2answers
208 views
Irreducible Representations Of Lorentz Group
In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64.
He define states as ...
3
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1answer
91 views
Different representations of the Lorentz algebra
I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
6
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3answers
563 views
What is the difference between a spinor and a vector or a tensor?
Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
1
vote
1answer
85 views
Second baryon octet
Let's temporarily ignore spin. If 3 denotes the standard representation of SU(3), 1 the trivial rep, 8 the adjoint rep and 10 the symmetric cube then it's well-known that
3 x 3 x 3 = 1 + 8 + 8 + 10
...
8
votes
1answer
187 views
What really are superselection sectors and what are they used for?
When reading the term superselection sector, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-)
But now I have read in this answer, that for ...
8
votes
2answers
954 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
4
votes
1answer
137 views
The use of Hall algebras in physics
I asked the same question in mo. I think maybe here there are more physics guys to help me.
I once read a statement (not memorized precisely) that a certain physics quantity between two states of ...
20
votes
3answers
783 views
Lie theory, Representations and particle physics
This is a question that has been posted at many different forums, I thought maybe someone here would have a better or more conceptual answer than I have seen before:
Why do physicists care about ...
2
votes
1answer
289 views
What's the mathematical background to the representation for Gaussian beams?
Background
A general optical system (not necessarily having an axis of
rotational symmetry) can be represented, for small deviations from a base ray,
by the matrix transfer equation,
$$
...
3
votes
1answer
90 views
Mathematically, how do we deduce that angular momentum is bounded?
So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$?
I.e. that m is bounded by j.
We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its ...
9
votes
1answer
297 views
Dimension of Dirac $\gamma$ matrices
While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices:
...
5
votes
2answers
102 views
Why aren't the spin-3/2 fields in the (3/2,0)+(0,3/2) representation?
Why is it that spin-$\frac 32$ fields are usually described to be in the $(\frac 12, \frac 12)\otimes[(\frac 12,0)\oplus(0,\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\frac ...
4
votes
1answer
184 views
Gauge invariance and the form of the Rarita-Schwinger action
in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field ...
2
votes
2answers
107 views
Representations of Lie algebras in physics
Why is an invariant vector subspace sometimes called a representation? For example in Lie algebras, say su(3), the subspace characterized by the highest weight (1,0) is an irreducible representation ...
20
votes
5answers
84 views
Which symmetric pure qudit states can be reached within local operations?
There are two pure symmetric states $|\psi\rangle$ and $|\phi\rangle$ of $n$ qudits. Is there any known set of invariants $\{I_i:i\in\{1,\ldots,k\}\}$ which is equal for both states iff ...
1
vote
0answers
71 views
Deriving term symbols from electron configuration using Young tableaux
Can somebody explain me how to derive all term symbols using Young tableaux? Our lecturer showed us but I couldn't quite understand it without any background on group theory. I have some vague ...
4
votes
1answer
148 views
Labelling representations using isospin and hypercharge
Can someone explain how isospin and hypercharge, can be used to label representations? What is the meaning of the term singlet, doublet etc in this context? In particular how can I use it to label ...
3
votes
0answers
78 views
Isospin and Hypercharge of the SU(2) bps monopole embedding
I am reading the paper Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups - Weinberg, Erick J .
In appendix C of this paper the author states, that the solution ...
2
votes
2answers
317 views
Why do many people say vector fields describe spin-1 particle but omit the spin-0 part?
We know a vector field is a $(\frac{1}{2},\frac{1}{2})$ representation of Lorentz group, which should describe both spin-1 and spin-0 particles. However many of the articles(mostly lecture notes) I've ...
4
votes
1answer
89 views
Massive excitations in Conformal Quantum Field Theory
Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of ...
8
votes
1answer
281 views
Representations of Lorentz Group
I'd be grateful if someone could check that my exposition here is correct, and then venture an answer to the question at the end!
$SO(3)$ has a fundamental representation (spin-1), and tensor product ...
11
votes
1answer
113 views
Majorana-like representation for mixed symmetric states?
Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)?
By Majorana representation I mean the decomposition of ...
9
votes
2answers
322 views
Can any rank tensor be decomposed into symmetric and anti-symmetric parts?
I know that rank 2 tensors can be decomposed as such. But I would like to know if this is possible for any rank tensors?
4
votes
1answer
194 views
Holstein-Primakoff and Dyson-Maleev representation
In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
1
vote
2answers
180 views
What does “the N of a group” mean?
In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the N" of a group, for example "a 24 of SU(5)" or "the 1" (the integer is usually typeset in ...
3
votes
1answer
187 views
Question on Sakurai's treatment of the Harmonic Oscillator:
In Section 2.3 of the second edition of Modern Quantum Mechanics (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\left|n\right> = (n-1)a\left|n\right>,$$ and states ...
2
votes
0answers
109 views
Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius
Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
2
votes
2answers
148 views
Multiplicity of eigenvalues of angular momentum
Reading Dirac's Principles of Quantum Mechanics, I encounter in ยง 36 (Properties of angular momentum) this fragment:
This is for a dynamical system with two angular momenta $\mathbf{m}_1$ and ...
4
votes
1answer
207 views
About 2+1 dimensional superconformal algebra
I would like to get some help in interpreting the main equation of the superconformal algebra (in $2+1$ dimenions) as stated in equation 3.27 on page 18 of this paper. I am familiar with supersymmetry ...
3
votes
1answer
164 views
Symmetrical Spinors and Symmetrical Tensors
In Quantum Electrodynamics by Landau and Lifshiz there is the following:
The correspondence between the spinor $\zeta^{\alpha \dot{\beta}}$ and
the 4-vector is a particular case of a general ...
3
votes
0answers
146 views
Some more questions on conformal spinors of $SO(n,2)$
This is somewhat of a continuation of my previous question.
I had stated there that a conformal spinor ($V$) of $SO(n,2)$ can be created by taking a direct sum of two $SO(n-1,1)$ spinors $Q$ and $S$ ...
5
votes
3answers
397 views
Why does spin have a discrete spectrum?
Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) continuous values, spin has a (probabilistic) discrete spectrum?
4
votes
1answer
164 views
Grassmann Variables Representation ?
It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a ...
4
votes
1answer
122 views
Eigenvalue of $L_z$
In section 4.3 of Griffths' "Introduction to Quantum Mechanics", just below Figure 4.6, the sentence begins
Let $\hbar \ell$ be the eigenvalue of $L_z$ at this top rung...
Why is this valid? ...
