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3answers
51 views

Conceptualization and modelling of spin

I'm trying to get a decent understanding of the bell inequality, and so am trying to understand spin both conceptually and mathematically. When I picture spin, I imagine a sphere rotating about its ...
8
votes
1answer
196 views

Why are non-Abelian gauge theories Lorentz invariant quantum mechanically?

I seem to be missing something regarding why Yang-Mills theories are Lorentz invariant quantum mechanically. Start by considering QED. If we just study the physics of a massless $U(1)$ gauge field ...
5
votes
1answer
179 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
5
votes
1answer
257 views

Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

Happy holidays, everyone! The following is part question, part visual gallery, and part classical mechanics problem. Inspired by snow over the weekend I began simulating the vibrations of the ...
3
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1answer
85 views

What does Weinberg–Witten theorem want to express?

Weinberg-Witten theorem states that massless particles (either composite or elementary) with spin $j > 1/2$ cannot carry a Lorentz-covariant current, while massless particles with spin $j > 1$ ...
2
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1answer
73 views

Vector spaces for the irreducible representations of the Lorentz Group

The definition of a representation is a map to the space of linear operators over a vector space. My question is: What are the corresponding vector spaces for the $(0,0)$ Representation ...
1
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1answer
50 views

Irreducible Representations from Cartan Matrix

I need some help understanding how you can construct the irreducible representations of an algebra from knowing its roots or its Cartan matrix, which I am being told you most certainly can. ...
-1
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1answer
40 views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
5
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0answers
32 views

Motivating Irreducibility of Hilbert Space for Quantization Axioms

In the context of geometric quantization, we usually look for a map from the Poisson algebra of classical observables to the algebra of quantum observables (or rather, a sub-algebra of the classical ...
4
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0answers
279 views

Deducing Young Tableaux from symmetries

I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to ...
4
votes
0answers
120 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
3
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0answers
45 views

Highest weight unitary representations of $psl(2|2)$

I'm having some trouble understanding how to extend representation theory from Lie algebras to super Lie algebras, in particular with $psl(2|2)$. Ultimately I'm interested in 2D quantum sigma models ...
3
votes
0answers
78 views

The states of the adjoint representation correspond to the generators

I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\rangle$. What does ...
3
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0answers
56 views

Subgroups of the Clifford Group

We recall the definition of a Clifford group (over $n$ qubits) is the set of unitary transformations: $$\{U: UPU^\dagger\in\mathcal{P}\}$$ where $\mathcal{P}$ denotes the corresponding Pauli group ...
3
votes
0answers
98 views

Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group

So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can ...
3
votes
0answers
33 views

$\mathcal{N}=2$ spin $1/2$ supermultiplet

In Freedman and Van Proeyen's Supergravity, in the footnote on pg. 128, they say There is a subtle hermiticity requirement for $\mathcal{N}=2$, which requires the multiplet $(-1/2,0,0,1/2)$ must ...
3
votes
0answers
54 views

$\mathcal{N}=4$ SUSY in $d=3$ versus $\mathcal{N}=2$ in $d=4$

Which is the field content of the hypermultiplet and the vector multiplet in $\mathcal{N}=4 \ d=3$ Supersymmmetry? Is it correct to state that $\mathcal{N}=4$ in $d=3$ has $8$ supercharges, (since ...
3
votes
0answers
113 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 ...
3
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0answers
126 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 ...
3
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0answers
167 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 $S$ ...
2
votes
0answers
58 views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
2
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0answers
48 views

Young Tableau Projectors: Does the order of symmetric and anti-symmetric projectors matter?

Given a Young Tableau we find the irreducible basis of an arbitrary tensor by projecting, The projectors are usually defined as first symmetrise over the row entries and then anti-symmetrise over the ...
2
votes
0answers
100 views

Two spinor tensors and Maxwell's equations

Let's have two symmetric (by the indices) spinor tensors $F_{ab}, F_{\dot {a}\dot {b}}$ and conditions $$ F_{ab}, \partial^{\dot {a} a}F_{ab} = 0, \quad F_{\dot {a}\dot {b}}, \partial^{\dot ...
2
votes
0answers
65 views

State space of strings: Spin-1 particles in the conformal gauge?

I obviously have a problem with basics of group theory. consider an open string in flat spacetime. there are usually two common gauge to solve the classical problem and quantize the strings: ...
1
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0answers
23 views

What is a Chiral Algebra for a group?

What do we mean by the Chiral Algebra for a group G (SO(3) etc )? Do you know a reference suitable for physicists? Thank you
1
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0answers
33 views

How does independence of the basic bras affect the choice of numbers used to represent a ket?

On page 54 of Dirac's book, The Principles of Quantum Mechanics, he states: Take an orthogonal representation with basic bras $\langle\lambda_1\lambda_2...\lambda_u|$, labelled by parameters ...
1
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0answers
45 views

In SUSY, why do fermions and gauge bosons in the same multiplet both transform in the adjoint representation of the gauge group?

I'm trying to understand a certain point about supersymmetry. We are dealing with a N=1 (i.e, one supersymmetric flavour), massless, four dimensional theory. Then the vector multiplet consists of a ...
1
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0answers
53 views

The conjugate representation in $su(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $su(2)$ and that for the diagonal operator $T_3$, \begin{align} T_3\psi_1 ...
1
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0answers
29 views

Asymptotics of the Wigner 6j Symbol

So, in doing some numerical computations in QFT, I've run into the following Wigner 6j-Symbol: $ \left\{ \begin{array}{ccc} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2} ...
1
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0answers
32 views

(A,B)-Representation of Lorentz Group: Coefficient functions of fields

I have a question regarding the construction of general causal fields in Weinberg's book on quantum field theory. In his conventions a field that transforms according to the irreducible (A,B) ...
1
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0answers
114 views

I want to decompose a tensor product using Littlewood-Richardson rule, How do I find the component of this in each irreducible space?

Let me set up the notation I am using. $(abc,de)$ denotes the standard Young tableau where the first row is $abc$ and the second row is $de$. Each young tableau corresponds to the young symmetriser, ...
1
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0answers
83 views

Dirac representation between matter and anti-matter

If $|\psi \rangle$ represents a wavefunction through a column matrix and $\langle \psi |$ represents the dual vector in a row matrix and $|\psi \rangle \langle \psi | = \rho $ is the probability ...
1
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0answers
141 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 ...
0
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0answers
24 views

What experiments don't confirm Poincare group-based representation of particle behavior

What experiments don't confirm Poincare group-based representation of particle behavior? Sources: http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory ...
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0answers
31 views

Transformation from waves to matter

Let $|\psi \rangle$ represents a wavefunction through and $\langle \psi |$ represent the dual vector. Now there are things such as matter waves, could $|\psi \rangle$ represents a matter wavefunction, ...